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        <title>Stefan Wullems</title>
        <link>https://stefanwullems.com</link>
        <description>Writing on SEO, content analysis, and building things on the web.</description>
        <lastBuildDate>Tue, 18 Aug 2026 12:54:33 GMT</lastBuildDate>
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            <title>Stefan Wullems</title>
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            <link>https://stefanwullems.com</link>
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        <copyright>All rights reserved 2026</copyright>
        <item>
            <title><![CDATA[Content Inventory Analysis - Interactive SOP]]></title>
            <link>https://stefanwullems.com/blog/content-inventory</link>
            <guid isPermaLink="false">https://stefanwullems.com/blog/content-inventory</guid>
            <pubDate>Mon, 29 Sep 2025 00:00:00 GMT</pubDate>
            <content:encoded><![CDATA[<p>Create a content inventory interactively using Ahrefs data to classify, rank, and evaluate your website content.</p>
<p><strong>Goal</strong>
Provide a clear process for conducting a content inventory analysis using Ahrefs data to classify, rank, and evaluate website content.</p>
<p><strong>Ideal Outcome</strong>
You generate a complete content inventory with quartile rankings, composite scores, and efficiency metrics that highlight star performers, optimization candidates, and underperformers.</p>
<p><strong>Why this is Important</strong>
This analysis creates a roadmap for content optimization. It helps you decide which pages to amplify, which to improve, and which to remove, ensuring resources are used effectively.</p>
<p><strong>Where this is done</strong>
Inside <strong>Ahrefs Site Explorer</strong> and the Content Inventory Analysis tool.</p>
<p><strong>When this is done</strong>
Whenever you need to assess your or your competitor’s website’s content performance, typically once per quarter or after publishing a significant batch of new content.</p>
<p><strong>Who does this</strong>
Content strategists, SEO specialists, or marketing team members responsible for content performance.</p>
<p><strong>Prerequisites or Requirements</strong></p>
<ul>
<li>Website domain to analyze (own site or competitor)</li>
<li>Access to <strong>Ahrefs Site Explorer</strong></li>
<li><strong>CSV export</strong> from Ahrefs with correct settings</li>
</ul>
<h2>Step 1. Export Data from Ahrefs</h2>
<ol>
<li>Open <strong>Ahrefs Site Explorer</strong>.</li>
<li>Navigate to <strong>Top Pages</strong>.</li>
<li>Set date range to <strong>“Don’t compare”</strong> to prevent errors in the export.</li>
<li>Enable <strong>“SERP titles”</strong> for proper page identification.</li>
<li>Export as <strong>CSV with UTF-8 encoding</strong>.</li>
<li>Select <strong>all available rows</strong> for the most reliable dataset.</li>
</ol>
<h2>Step 2. Upload Data</h2>
<div class="mb-8"><div class="border-b border-zinc-200 pb-4 dark:border-zinc-700"><div class="flex items-center space-x-3"><span class="flex h-5 w-5 items-center justify-center rounded-full text-sm font-medium bg-teal-500 text-white dark:bg-teal-600 dark:text-white">1</span><div><h3 class="text-xl font-semibold text-zinc-900 dark:text-zinc-100">Data Upload</h3><p class="mt-1 text-sm text-zinc-600 dark:text-zinc-400">Upload your Ahrefs CSV export</p></div></div></div><div class="mt-6"><div class="space-y-6"><div class="space-y-6"><div><p class="mb-4 text-zinc-700 dark:text-zinc-300">Select your Ahrefs CSV file to upload.</p></div><div class="rounded-lg border-2 border-dashed p-6 text-center transition-colors border-zinc-300 hover:border-zinc-400 dark:border-zinc-600 dark:hover:border-zinc-500"><input type="file" accept=".csv" class="hidden"><div class="space-y-2"><svg class="mx-auto h-8 w-8 text-zinc-400 dark:text-zinc-500" fill="none" viewBox="0 0 24 24" stroke="currentColor"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M7 16a4 4 0 01-.88-7.903A5 5 0 1115.9 6L16 6a5 5 0 011 9.9M15 13l-3-3m0 0l-3 3m3-3v12"></path></svg><p class="text-sm text-zinc-600 dark:text-zinc-400">Drop your CSV file here, or<!-- --> <button class="text-teal-600 underline hover:text-teal-700 dark:text-teal-400 dark:hover:text-teal-300">browse to upload</button></p><p class="text-xs text-zinc-500 dark:text-zinc-400">CSV files up to 10MB</p></div></div></div></div></div></div>
<h2>Step 3. Run Content Inventory Analysis</h2>
<div class="mb-8"><div class="border-b border-zinc-200 pb-4 dark:border-zinc-700"><div class="flex items-center space-x-3"><span class="flex h-5 w-5 items-center justify-center rounded-full text-sm font-medium bg-zinc-300 text-zinc-500 dark:bg-zinc-600 dark:text-zinc-400">2</span><div><h3 class="text-xl font-semibold text-zinc-900 dark:text-zinc-100">Content Inventory Analysis</h3><p class="mt-1 text-sm text-zinc-600 dark:text-zinc-400">Generate quartile rankings and composite performance scores for your content</p></div></div></div><div class="mt-6"><div class="space-y-6"><button disabled="" class="not:disabled:hover:bg-zinc-50 not:disabled:dark:hover:bg-zinc-700 w-full rounded-md border border-zinc-300 bg-white px-4 py-2 text-sm font-medium text-zinc-700 focus:ring-2 focus:ring-teal-500 focus:outline-none disabled:cursor-not-allowed disabled:opacity-50 dark:border-zinc-600 dark:bg-zinc-800 dark:text-zinc-300">Run Analysis</button></div></div></div>
<blockquote>
<p>💡 <strong>Tip</strong>: Quartiles divide your content into four buckets (Q1–Q4). Instead of obsessing over raw numbers, quartiles show relative performance. If a page lands in Q4, it outperforms 75% of your content. If it lands in Q1, it underperforms compared to 75% of your portfolio. This way you focus on patterns, not isolated datapoints.</p>
</blockquote>
<h2>Step 4. Interpret</h2>
<ol>
<li>
<p>Check <strong>2–8 point composite scores</strong>:</p>
<ul>
<li><strong>6–8</strong>: High-performing content; amplify and maintain.</li>
<li><strong>4–5</strong>: Mid-level performers; prioritize optimization.</li>
<li><strong>2–3</strong>: Underperformers; consider overhaul, consolidation, or removal.</li>
</ul>
</li>
<li>
<p>Identify <strong>efficiency champions</strong> with high <em>referring domains</em> and low <em>traffic</em>.</p>
<blockquote>
<p>Example: A page with 300+ referring domains but very little traffic signals “earned links” (people linked because of quality or originality, not just rankings).</p>
</blockquote>
</li>
<li>
<p>Spot <strong>low-efficiency pages</strong> where traffic doesn’t translate into authority. These may need targeted outreach or better internal linking.</p>
</li>
</ol>
<blockquote>
<p>💡 <strong>Context</strong>: Referring domains per visit helps separate <em>passive link-building</em> (pages rank high, so they naturally get links) from <em>active link-worthiness</em> (pages get links because they truly resonate). Especially useful when analyzing competitor sites — it highlights which posts genuinely gained traction, not just rankings.</p>
</blockquote>]]></content:encoded>
        </item>
        <item>
            <title><![CDATA[Content Time Analysis - Interactive SOP]]></title>
            <link>https://stefanwullems.com/blog/content-time-analysis</link>
            <guid isPermaLink="false">https://stefanwullems.com/blog/content-time-analysis</guid>
            <pubDate>Mon, 29 Sep 2025 00:00:00 GMT</pubDate>
            <content:encoded><![CDATA[<p>This Standard Operating Procedure (SOP) explains how to run a time-based performance analysis with Ahrefs data. It reveals how your content performs relative to its publication date and helps you identify improving or declining content over time.</p>
<p><strong>Goal</strong>
Measure content performance relative to publication dates instead of static site-wide averages.</p>
<p><strong>Ideal Outcome</strong>
You generate CSV outputs with performance indices, publication dates, and traffic metrics. Each page is benchmarked against content published before it, allowing you to clearly spot overperforming and underperforming content.</p>
<p><strong>Why this is Important</strong>
Static numbers (like “50 referring domains”) don’t mean the same thing at every stage of a site’s life. A brand-new site earning 50 links might be extraordinary, while a mature site earning the same could be average. Time-based analysis shows how impressive a result was <em>for that moment in the site’s history</em>.</p>
<p>💡 <strong>Competitive advantage</strong>: Few analysts account for temporal context. Looking at <em>relative performance over time</em> means you see opportunities others miss — for example, identifying older posts that were unusually strong for their period even if their absolute numbers look modest.</p>
<p><strong>Where this is done</strong>
<strong>Ahrefs Site Explorer</strong> for data exports, and the <strong>Content Time Analysis tool</strong> for processing and visualization.</p>
<p><strong>When this is done</strong>
Quarterly, or after major content launches, to monitor trends and build a temporal performance database.</p>
<p><strong>Who does this</strong>
Content Strategist or SEO Specialist.</p>
<h2>Prerequisites or Requirements</h2>
<ul>
<li><strong>Access to Ahrefs Site Explorer</strong></li>
<li><strong>Domain to analyze</strong> (yours or a competitor’s)</li>
<li><strong>CSV export with correct settings</strong> (see Step 1)</li>
<li><strong>Time to fetch published dates</strong> (5–15 minutes depending on dataset size)</li>
</ul>
<h2>Step 1: Upload Ahrefs Data</h2>
<ol>
<li>In <strong>Ahrefs Site Explorer</strong>, go to <strong>Top Pages</strong>.</li>
<li>Set date range to <strong>“Don’t compare”</strong>.</li>
<li>Enable <strong>“SERP titles”</strong>.</li>
<li>Export as <strong>CSV with UTF-8 encoding</strong>.</li>
<li>Include <strong>all available rows</strong>.</li>
</ol>
<div class="mb-8"><div class="border-b border-zinc-200 pb-4 dark:border-zinc-700"><div class="flex items-center space-x-3"><span class="flex h-5 w-5 items-center justify-center rounded-full text-sm font-medium bg-teal-500 text-white dark:bg-teal-600 dark:text-white">1</span><div><h3 class="text-xl font-semibold text-zinc-900 dark:text-zinc-100">Data Upload</h3><p class="mt-1 text-sm text-zinc-600 dark:text-zinc-400">Upload your Ahrefs CSV export</p></div></div></div><div class="mt-6"><div class="space-y-6"><div class="space-y-6"><div><p class="mb-4 text-zinc-700 dark:text-zinc-300">Select your Ahrefs CSV file to upload.</p></div><div class="rounded-lg border-2 border-dashed p-6 text-center transition-colors border-zinc-300 hover:border-zinc-400 dark:border-zinc-600 dark:hover:border-zinc-500"><input type="file" accept=".csv" class="hidden"><div class="space-y-2"><svg class="mx-auto h-8 w-8 text-zinc-400 dark:text-zinc-500" fill="none" viewBox="0 0 24 24" stroke="currentColor"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M7 16a4 4 0 01-.88-7.903A5 5 0 1115.9 6L16 6a5 5 0 011 9.9M15 13l-3-3m0 0l-3 3m3-3v12"></path></svg><p class="text-sm text-zinc-600 dark:text-zinc-400">Drop your CSV file here, or<!-- --> <button class="text-teal-600 underline hover:text-teal-700 dark:text-teal-400 dark:hover:text-teal-300">browse to upload</button></p><p class="text-xs text-zinc-500 dark:text-zinc-400">CSV files up to 10MB</p></div></div></div></div></div></div>
<h2>Step 2: Configure Published Date Fetching</h2>
<ol>
<li>Start the date-fetching process in the tool.</li>
<li>Set <strong>delay to 1000–2000ms</strong> for balanced speed and politeness.</li>
<li>Expect the process to take <strong>5–15 minutes</strong> depending on dataset size.</li>
<li>Monitor progress in real-time.</li>
</ol>
<p>💡 Publication dates let you calculate performance indices that normalize “how impressive” a result was at the moment it was published — not years later when expectations had changed.</p>
<div class="mb-8"><div class="border-b border-zinc-200 pb-4 dark:border-zinc-700"><div class="flex items-center space-x-3"><span class="flex h-5 w-5 items-center justify-center rounded-full text-sm font-medium bg-zinc-300 text-zinc-500 dark:bg-zinc-600 dark:text-zinc-400">2</span><div><h3 class="text-xl font-semibold text-zinc-900 dark:text-zinc-100">Published Date Configuration</h3><p class="mt-1 text-sm text-zinc-600 dark:text-zinc-400">Configure and fetch published dates for temporal analysis</p></div></div></div><div class="mt-6"><div class="space-y-6"><div class="space-y-6"><div><p class="mb-4 text-zinc-700 dark:text-zinc-300">Configure published date fetching for your<!-- --> <strong>0<!-- --> pages</strong>.</p></div><div class="space-y-4"><div class="ml-7 space-y-4"><div><label for="publishedDateDelay" class="mb-2 block text-sm font-medium text-zinc-700 dark:text-zinc-300">Delay between requests (milliseconds)</label><select id="publishedDateDelay" class="w-full rounded-md border border-zinc-300 px-3 py-2 shadow-sm focus:border-teal-500 focus:ring-2 focus:ring-teal-500 focus:outline-none dark:border-zinc-600 dark:bg-zinc-800 dark:text-zinc-100"><option value="200">200ms - Fast (may be blocked by some sites)</option><option value="500" selected="">500ms - Balanced (recommended)</option><option value="1000">1000ms - Conservative (slower but safer)</option></select><p class="mt-1 text-xs text-zinc-500 dark:text-zinc-400">Higher delays are safer but take longer. 500ms works well for most websites.</p></div><div class="rounded-md border border-teal-200 bg-teal-50 p-4 dark:border-teal-800 dark:bg-teal-900/20"><h4 class="mb-2 text-sm font-medium text-teal-800 dark:text-teal-200">Estimated Time</h4><p class="text-sm text-teal-700 dark:text-teal-300">With <!-- -->500<!-- -->ms delay: approximately<!-- --> <strong>0<!-- --> minutes</strong> <!-- -->for <!-- -->0<!-- --> pages</p></div></div></div><button class="rounded-md px-4 py-2 text-sm font-medium focus:ring-2 focus:ring-offset-2 focus:outline-none bg-teal-600 text-white hover:bg-teal-700 focus:ring-teal-500 dark:bg-teal-600 dark:hover:bg-teal-700">Fetch Published Dates</button></div></div></div></div>
<h2>Step 3: Run Time-Based Analysis</h2>
<p>💡 <strong>Why median matters</strong>: Using the <em>median</em> prevents one viral hit from distorting the benchmark. For example, a post earning 200 links on a site that normally earns 1–2 would artificially inflate the average, making other posts look weaker than they really were. The median resists these distortions.</p>
<div class="mb-8"><div class="border-b border-zinc-200 pb-4 dark:border-zinc-700"><div class="flex items-center space-x-3"><span class="flex h-5 w-5 items-center justify-center rounded-full text-sm font-medium bg-zinc-300 text-zinc-500 dark:bg-zinc-600 dark:text-zinc-400">3</span><div><h3 class="text-xl font-semibold text-zinc-900 dark:text-zinc-100">Time-Based Performance Analysis</h3><p class="mt-1 text-sm text-zinc-600 dark:text-zinc-400">Configure analysis strategy and generate temporal performance insights</p></div></div></div><div class="mt-6"><div class="space-y-6"><div class="space-y-6"><div class="space-y-4"><div class="space-y-2"><label for="strategy" class="block text-sm font-medium text-zinc-700 dark:text-zinc-300">Analysis Strategy</label><select id="strategy" class="w-full rounded-md border border-zinc-300 px-3 py-2 shadow-sm focus:border-teal-500 focus:ring-2 focus:ring-teal-500 focus:outline-none dark:border-zinc-600 dark:bg-zinc-800 dark:text-zinc-100"><option value="average" selected="">Average (Recommended) - Traditional mathematical mean</option><option value="median">Median - More stable with outliers, use only with larger websites</option></select></div><button disabled="" class="w-full rounded-md px-4 py-2 text-sm font-medium focus:ring-2 focus:ring-offset-2 focus:outline-none cursor-not-allowed bg-zinc-300 text-zinc-500 dark:bg-zinc-600 dark:text-zinc-400">Run Time-Based Analysis</button></div></div></div></div></div>
<h2>Step 4: Interpreting Results</h2>
<ul>
<li><strong>Performance Indices</strong>:<!-- -->
<ul>
<li><strong>&gt; 1.5</strong> = exceptional performers</li>
<li><strong>1.0–1.5</strong> = solid performers</li>
<li><strong>0.5–1.0</strong> = underperformers</li>
<li><strong>&lt; 0.5</strong> = poor performers</li>
</ul>
</li>
<li><strong>Historical context:</strong> A post with 40 referring domains in 2015 might be more impressive than one with 57 in 2019, because expectations rise as a site grows.</li>
<li><strong>Relative performance:</strong> Indices show how surprising or impressive a result was <em>for that period</em>.</li>
<li><strong>Sorting advantage:</strong> Sorting by index, not absolute values, surfaces hidden gems — posts that would be overlooked if you only scanned the top-10 by links.</li>
</ul>
<h3>Optimization Strategies</h3>
<ul>
<li><strong>High performers (Index &gt; 1.2):</strong> Replicate and refresh these as models for future content.</li>
<li><strong>Underperformers (Index &lt; 0.8):</strong> Revisit for updates or consolidation.</li>
</ul>
<p>Time-based analysis shows not just <em>what</em> content performed, but <em>how impressive it was at the time</em>. That perspective reveals hidden winners, uncovers overlooked opportunities, and gives you a research edge competitors rarely use.</p>]]></content:encoded>
        </item>
        <item>
            <title><![CDATA[Linear/Quadratic Approximation Cheatsheet]]></title>
            <link>https://stefanwullems.com/blog/linear-quadratic-approximation-cheatsheet</link>
            <guid isPermaLink="false">https://stefanwullems.com/blog/linear-quadratic-approximation-cheatsheet</guid>
            <pubDate>Tue, 18 Aug 2026 00:00:00 GMT</pubDate>
            <content:encoded><![CDATA[<nav aria-label="Table of contents" class="not-prose mt-8 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h2 class="text-xs font-medium tracking-wider text-zinc-500 uppercase dark:text-zinc-400">Contents</h2><div class="mt-4"><ul class="space-y-2"><li><a href="#definitions" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1</span><span class="font-medium">Definitions</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#linear-approximation" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.1</span><span class="text-sm">Linear approximation</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#slope-intuition" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.1.1</span><span class="text-sm">Slope intuition</span></a></li><li><a href="#reconstructing-a-and-b" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.1.2</span><span class="text-sm">Reconstructing a and b</span></a></li></ul></div></li><li><a href="#quadratic-approximation" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.2</span><span class="text-sm">Quadratic approximation</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#parabola-intuition" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.2.1</span><span class="text-sm">Parabola intuition</span></a></li><li><a href="#reconstructing-a-b-and-c" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">1.2.2</span><span class="text-sm">Reconstructing a, b and c</span></a></li></ul></div></li></ul></div></li><li><a href="#cheatsheet" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2</span><span class="font-medium">Cheatsheet</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#near-0" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1</span><span class="text-sm">Near 0</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#trigonometric" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1</span><span class="text-sm">Trigonometric</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#fn-sin" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.1</span><span class="text-sm">sin x</span></a></li><li><a href="#fn-cos" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.2</span><span class="text-sm">cos x</span></a></li><li><a href="#fn-tan" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.3</span><span class="text-sm">tan x</span></a></li><li><a href="#fn-sec" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.4</span><span class="text-sm">sec x</span></a></li><li><a href="#fn-cot" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.5</span><span class="text-sm">cot x</span></a></li><li><a href="#fn-csc" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.6</span><span class="text-sm">csc x</span></a></li></ul></div></li><li><a href="#exponents-and-logarithms" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2</span><span class="text-sm">Exponents and logarithms</span></a><div class="ml-4 border-l border-zinc-200 pl-3 dark:border-zinc-800"><ul class="mt-1 space-y-1"><li><a href="#fn-exp" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.1</span><span class="text-sm">eˣ</span></a></li><li><a href="#fn-ln" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.2</span><span class="text-sm">ln(1 + x)</span></a></li><li><a href="#fn-pow" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.3</span><span class="text-sm">(1 + x)ʳ</span></a></li><li><a href="#fn-apow" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.4</span><span class="text-sm">aˣ</span></a></li></ul></div></li></ul></div></li></ul></div></li><li><a href="#combining-approximations" class="group flex gap-3 text-zinc-600 transition hover:text-teal-600 dark:text-zinc-400 dark:hover:text-teal-400"><span class="w-14 shrink-0 font-mono text-sm text-zinc-400 dark:text-zinc-500">3</span><span class="font-medium">Combining approximations</span></a></li></ul></div></nav>
<h2 id="definitions" class="scroll-mt-24">1. Definitions</h2>
<h3 id="linear-approximation" class="scroll-mt-24">1.1 Linear approximation</h3>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mi>x</mi></mrow><annotation encoding="application/x-tex">f(x) \approx f(0) + f'(0)\,x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">x</span></span></span></span></span>
<p>Replace the curve with the straight line that matches it at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span> in <strong>height</strong> and <strong>slope</strong>.</p>
<h4 id="slope-intuition" class="scroll-mt-24">1.1.1 Slope intuition</h4>
<p>A line through <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0, f(0))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">))</span></span></span></span> needs one more number: its slope. Pick the slope wrong and the line peels away from the curve immediately. Pick <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> and it stays glued for as long as possible.</p>
<div class="not-prose mt-6 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><p class="text-sm text-zinc-600 dark:text-zinc-400">Fixing the height at <span class="font-mono">f(0) = 1</span>, slide the slope <span class="font-mono">b</span> and watch the gap. Only one value hugs the curve.</p><div class="mt-5 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of e^x with its approximations near zero"><line x1="10" x2="630" y1="305.9743252901553" y2="305.9743252901553" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><line x1="320" x2="320" y1="10" y2="330" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><path d="M10.00 293.08 L11.29 293.00 L12.58 292.91 L13.88 292.83 L15.17 292.75 L16.46 292.67 L17.75 292.58 L19.04 292.50 L20.33 292.42 L21.62 292.33 L22.92 292.25 L24.21 292.16 L25.50 292.07 L26.79 291.99 L28.08 291.90 L29.38 291.81 L30.67 291.72 L31.96 291.63 L33.25 291.54 L34.54 291.45 L35.83 291.36 L37.13 291.27 L38.42 291.18 L39.71 291.08 L41.00 290.99 L42.29 290.90 L43.58 290.80 L44.87 290.71 L46.17 290.61 L47.46 290.51 L48.75 290.42 L50.04 290.32 L51.33 290.22 L52.63 290.12 L53.92 290.02 L55.21 289.92 L56.50 289.82 L57.79 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280.25 L34.54 280.11 L35.83 279.96 L37.13 279.82 L38.42 279.67 L39.71 279.53 L41.00 279.39 L42.29 279.24 L43.58 279.10 L44.87 278.95 L46.17 278.81 L47.46 278.66 L48.75 278.52 L50.04 278.37 L51.33 278.23 L52.63 278.08 L53.92 277.94 L55.21 277.80 L56.50 277.65 L57.79 277.51 L59.08 277.36 L60.37 277.22 L61.67 277.07 L62.96 276.93 L64.25 276.78 L65.54 276.64 L66.83 276.50 L68.13 276.35 L69.42 276.21 L70.71 276.06 L72.00 275.92 L73.29 275.77 L74.58 275.63 L75.88 275.48 L77.17 275.34 L78.46 275.19 L79.75 275.05 L81.04 274.91 L82.33 274.76 L83.62 274.62 L84.92 274.47 L86.21 274.33 L87.50 274.18 L88.79 274.04 L90.08 273.89 L91.38 273.75 L92.67 273.61 L93.96 273.46 L95.25 273.32 L96.54 273.17 L97.83 273.03 L99.12 272.88 L100.42 272.74 L101.71 272.59 L103.00 272.45 L104.29 272.30 L105.58 272.16 L106.88 272.02 L108.17 271.87 L109.46 271.73 L110.75 271.58 L112.04 271.44 L113.33 271.29 L114.62 271.15 L115.92 271.00 L117.21 270.86 L118.50 270.72 L119.79 270.57 L121.08 270.43 L122.38 270.28 L123.67 270.14 L124.96 269.99 L126.25 269.85 L127.54 269.70 L128.83 269.56 L130.13 269.41 L131.42 269.27 L132.71 269.13 L134.00 268.98 L135.29 268.84 L136.58 268.69 L137.88 268.55 L139.17 268.40 L140.46 268.26 L141.75 268.11 L143.04 267.97 L144.33 267.82 L145.63 267.68 L146.92 267.54 L148.21 267.39 L149.50 267.25 L150.79 267.10 L152.08 266.96 L153.38 266.81 L154.67 266.67 L155.96 266.52 L157.25 266.38 L158.54 266.24 L159.83 266.09 L161.13 265.95 L162.42 265.80 L163.71 265.66 L165.00 265.51 L166.29 265.37 L167.58 265.22 L168.88 265.08 L170.17 264.93 L171.46 264.79 L172.75 264.65 L174.04 264.50 L175.33 264.36 L176.63 264.21 L177.92 264.07 L179.21 263.92 L180.50 263.78 L181.79 263.63 L183.08 263.49 L184.38 263.35 L185.67 263.20 L186.96 263.06 L188.25 262.91 L189.54 262.77 L190.83 262.62 L192.13 262.48 L193.42 262.33 L194.71 262.19 L196.00 262.04 L197.29 261.90 L198.58 261.76 L199.87 261.61 L201.17 261.47 L202.46 261.32 L203.75 261.18 L205.04 261.03 L206.33 260.89 L207.63 260.74 L208.92 260.60 L210.21 260.46 L211.50 260.31 L212.79 260.17 L214.08 260.02 L215.38 259.88 L216.67 259.73 L217.96 259.59 L219.25 259.44 L220.54 259.30 L221.83 259.15 L223.13 259.01 L224.42 258.87 L225.71 258.72 L227.00 258.58 L228.29 258.43 L229.58 258.29 L230.88 258.14 L232.17 258.00 L233.46 257.85 L234.75 257.71 L236.04 257.57 L237.33 257.42 L238.62 257.28 L239.92 257.13 L241.21 256.99 L242.50 256.84 L243.79 256.70 L245.08 256.55 L246.38 256.41 L247.67 256.26 L248.96 256.12 L250.25 255.98 L251.54 255.83 L252.83 255.69 L254.13 255.54 L255.42 255.40 L256.71 255.25 L258.00 255.11 L259.29 254.96 L260.58 254.82 L261.88 254.67 L263.17 254.53 L264.46 254.39 L265.75 254.24 L267.04 254.10 L268.33 253.95 L269.63 253.81 L270.92 253.66 L272.21 253.52 L273.50 253.37 L274.79 253.23 L276.08 253.09 L277.38 252.94 L278.67 252.80 L279.96 252.65 L281.25 252.51 L282.54 252.36 L283.83 252.22 L285.13 252.07 L286.42 251.93 L287.71 251.78 L289.00 251.64 L290.29 251.50 L291.58 251.35 L292.88 251.21 L294.17 251.06 L295.46 250.92 L296.75 250.77 L298.04 250.63 L299.33 250.48 L300.63 250.34 L301.92 250.20 L303.21 250.05 L304.50 249.91 L305.79 249.76 L307.08 249.62 L308.38 249.47 L309.67 249.33 L310.96 249.18 L312.25 249.04 L313.54 248.89 L314.83 248.75 L316.13 248.61 L317.42 248.46 L318.71 248.32 L320.00 248.17 L321.29 248.03 L322.58 247.88 L323.88 247.74 L325.17 247.59 L326.46 247.45 L327.75 247.31 L329.04 247.16 L330.33 247.02 L331.62 246.87 L332.92 246.73 L334.21 246.58 L335.50 246.44 L336.79 246.29 L338.08 246.15 L339.38 246.00 L340.67 245.86 L341.96 245.72 L343.25 245.57 L344.54 245.43 L345.83 245.28 L347.13 245.14 L348.42 244.99 L349.71 244.85 L351.00 244.70 L352.29 244.56 L353.58 244.42 L354.88 244.27 L356.17 244.13 L357.46 243.98 L358.75 243.84 L360.04 243.69 L361.33 243.55 L362.63 243.40 L363.92 243.26 L365.21 243.11 L366.50 242.97 L367.79 242.83 L369.08 242.68 L370.37 242.54 L371.67 242.39 L372.96 242.25 L374.25 242.10 L375.54 241.96 L376.83 241.81 L378.13 241.67 L379.42 241.52 L380.71 241.38 L382.00 241.24 L383.29 241.09 L384.58 240.95 L385.88 240.80 L387.17 240.66 L388.46 240.51 L389.75 240.37 L391.04 240.22 L392.33 240.08 L393.63 239.94 L394.92 239.79 L396.21 239.65 L397.50 239.50 L398.79 239.36 L400.08 239.21 L401.38 239.07 L402.67 238.92 L403.96 238.78 L405.25 238.63 L406.54 238.49 L407.83 238.35 L409.12 238.20 L410.42 238.06 L411.71 237.91 L413.00 237.77 L414.29 237.62 L415.58 237.48 L416.88 237.33 L418.17 237.19 L419.46 237.05 L420.75 236.90 L422.04 236.76 L423.33 236.61 L424.63 236.47 L425.92 236.32 L427.21 236.18 L428.50 236.03 L429.79 235.89 L431.08 235.74 L432.38 235.60 L433.67 235.46 L434.96 235.31 L436.25 235.17 L437.54 235.02 L438.83 234.88 L440.13 234.73 L441.42 234.59 L442.71 234.44 L444.00 234.30 L445.29 234.16 L446.58 234.01 L447.87 233.87 L449.17 233.72 L450.46 233.58 L451.75 233.43 L453.04 233.29 L454.33 233.14 L455.63 233.00 L456.92 232.85 L458.21 232.71 L459.50 232.57 L460.79 232.42 L462.08 232.28 L463.38 232.13 L464.67 231.99 L465.96 231.84 L467.25 231.70 L468.54 231.55 L469.83 231.41 L471.13 231.27 L472.42 231.12 L473.71 230.98 L475.00 230.83 L476.29 230.69 L477.58 230.54 L478.88 230.40 L480.17 230.25 L481.46 230.11 L482.75 229.96 L484.04 229.82 L485.33 229.68 L486.62 229.53 L487.92 229.39 L489.21 229.24 L490.50 229.10 L491.79 228.95 L493.08 228.81 L494.38 228.66 L495.67 228.52 L496.96 228.37 L498.25 228.23 L499.54 228.09 L500.83 227.94 L502.13 227.80 L503.42 227.65 L504.71 227.51 L506.00 227.36 L507.29 227.22 L508.58 227.07 L509.88 226.93 L511.17 226.79 L512.46 226.64 L513.75 226.50 L515.04 226.35 L516.33 226.21 L517.63 226.06 L518.92 225.92 L520.21 225.77 L521.50 225.63 L522.79 225.48 L524.08 225.34 L525.38 225.20 L526.67 225.05 L527.96 224.91 L529.25 224.76 L530.54 224.62 L531.83 224.47 L533.13 224.33 L534.42 224.18 L535.71 224.04 L537.00 223.90 L538.29 223.75 L539.58 223.61 L540.88 223.46 L542.17 223.32 L543.46 223.17 L544.75 223.03 L546.04 222.88 L547.33 222.74 L548.63 222.59 L549.92 222.45 L551.21 222.31 L552.50 222.16 L553.79 222.02 L555.08 221.87 L556.38 221.73 L557.67 221.58 L558.96 221.44 L560.25 221.29 L561.54 221.15 L562.83 221.01 L564.13 220.86 L565.42 220.72 L566.71 220.57 L568.00 220.43 L569.29 220.28 L570.58 220.14 L571.88 219.99 L573.17 219.85 L574.46 219.70 L575.75 219.56 L577.04 219.42 L578.33 219.27 L579.63 219.13 L580.92 218.98 L582.21 218.84 L583.50 218.69 L584.79 218.55 L586.08 218.40 L587.38 218.26 L588.67 218.12 L589.96 217.97 L591.25 217.83 L592.54 217.68 L593.83 217.54 L595.13 217.39 L596.42 217.25 L597.71 217.10 L599.00 216.96 L600.29 216.81 L601.58 216.67 L602.88 216.53 L604.17 216.38 L605.46 216.24 L606.75 216.09 L608.04 215.95 L609.33 215.80 L610.63 215.66 L611.92 215.51 L613.21 215.37 L614.50 215.22 L615.79 215.08 L617.08 214.94 L618.38 214.79 L619.67 214.65 L620.96 214.50 L622.25 214.36 L623.54 214.21 L624.83 214.07 L626.13 213.92 L627.42 213.78 L628.71 213.64 L630.00 213.49" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><label class="mt-5 block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">1 + b·x</span><span class="font-mono text-zinc-800 dark:text-zinc-200">b<!-- --> = <!-- -->0.40</span></span><input type="range" min="-0.5" max="2.5" step="0.05" class="mt-2 w-full accent-teal-500" value="0.4"></label><div class="mt-4 flex flex-wrap items-baseline justify-between gap-3 border-t border-zinc-200 pt-4 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">average gap over [−1, 1]</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">0.3452</span></div><button type="button" class="mt-4 rounded-full bg-teal-500 px-4 py-1.5 text-sm text-white transition hover:bg-teal-400">Snap to f′(0)</button></div>
<h4 id="reconstructing-a-and-b" class="scroll-mt-24">1.1.2 Reconstructing a and b</h4>
<p>The other way to read the formula: linear approximation is a search for the best <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">a + bx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span></span></span></span>.</p>
<p>Write <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">P(x) = a + bx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span></span></span></span> and demand it agree with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span>:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>a</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mtext>&nbsp;</mtext><mi>a</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>b</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mtext>&nbsp;</mtext><mi>b</mi><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
P(0) &amp;= a &amp;&amp;\Rightarrow\ a = f(0) \\
P'(0) &amp;= b &amp;&amp;\Rightarrow\ b = f'(0)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7em;vertical-align:-1.1em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6em"><span style="top:-3.76em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-2.26em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6em"><span style="top:-3.76em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">a</span></span></span><span style="top:-2.26em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6em"><span style="top:-3.6em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span><span style="top:-2.1em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6em"><span style="top:-3.76em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">⇒</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-2.26em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">⇒</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1em"><span></span></span></span></span></span></span></span></span></span></span></span>
<p>There is no choice left — matching height and slope <em>forces</em> both coefficients. So if you hand the formula something that already is a line, it must hand the same line back:</p>
<div class="not-prose mt-6 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><div class="grid gap-4 sm:grid-cols-2"><label class="block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">a</span><span class="font-mono text-zinc-800 dark:text-zinc-200">2.00</span></span><input type="range" min="-3" max="3" step="0.25" class="mt-2 w-full accent-teal-500" value="2"></label><label class="block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">b</span><span class="font-mono text-zinc-800 dark:text-zinc-200">-1.00</span></span><input type="range" min="-3" max="3" step="0.25" class="mt-2 w-full accent-teal-500" value="-1"></label></div><div class="mt-6"><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">f(x)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2 − 1x</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">f(0)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">f′(0)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">-1</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">f(0) + f′(0)·x</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2 − 1x</span></div></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">The last row always matches the first. Approximating something that is already a line hands the line straight back.</p></div>
<h3 id="quadratic-approximation" class="scroll-mt-24">1.2 Quadratic approximation</h3>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mi>x</mi><mo>+</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x) \approx f(0) + f'(0)\,x + \frac{f''(0)}{2}x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:2.1149em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4289em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>
<p>Same idea, one more term: match the curve at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span> in <strong>height</strong>, <strong>slope</strong>, and <strong>curvature</strong>.</p>
<h4 id="parabola-intuition" class="scroll-mt-24">1.2.1 Parabola intuition</h4>
<p>A line can only match height and slope; it is always straight while the curve bends. Adding an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> term lets the approximation bend too, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f''(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> says how much.</p>
<div class="not-prose mt-6 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><p class="text-sm text-zinc-600 dark:text-zinc-400">Fixing height and slope, slide the curvature coefficient<!-- --> <span class="font-mono">c</span>. 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323.31 L52.63 322.95 L53.92 322.59 L55.21 322.23 L56.50 321.87 L57.79 321.51 L59.08 321.15 L60.37 320.79 L61.67 320.42 L62.96 320.06 L64.25 319.70 L65.54 319.34 L66.83 318.98 L68.13 318.62 L69.42 318.26 L70.71 317.90 L72.00 317.53 L73.29 317.17 L74.58 316.81 L75.88 316.45 L77.17 316.09 L78.46 315.73 L79.75 315.37 L81.04 315.01 L82.33 314.64 L83.62 314.28 L84.92 313.92 L86.21 313.56 L87.50 313.20 L88.79 312.84 L90.08 312.48 L91.38 312.12 L92.67 311.75 L93.96 311.39 L95.25 311.03 L96.54 310.67 L97.83 310.31 L99.12 309.95 L100.42 309.59 L101.71 309.23 L103.00 308.86 L104.29 308.50 L105.58 308.14 L106.88 307.78 L108.17 307.42 L109.46 307.06 L110.75 306.70 L112.04 306.34 L113.33 305.97 L114.62 305.61 L115.92 305.25 L117.21 304.89 L118.50 304.53 L119.79 304.17 L121.08 303.81 L122.38 303.45 L123.67 303.08 L124.96 302.72 L126.25 302.36 L127.54 302.00 L128.83 301.64 L130.13 301.28 L131.42 300.92 L132.71 300.56 L134.00 300.19 L135.29 299.83 L136.58 299.47 L137.88 299.11 L139.17 298.75 L140.46 298.39 L141.75 298.03 L143.04 297.67 L144.33 297.30 L145.63 296.94 L146.92 296.58 L148.21 296.22 L149.50 295.86 L150.79 295.50 L152.08 295.14 L153.38 294.78 L154.67 294.41 L155.96 294.05 L157.25 293.69 L158.54 293.33 L159.83 292.97 L161.13 292.61 L162.42 292.25 L163.71 291.89 L165.00 291.52 L166.29 291.16 L167.58 290.80 L168.88 290.44 L170.17 290.08 L171.46 289.72 L172.75 289.36 L174.04 288.99 L175.33 288.63 L176.63 288.27 L177.92 287.91 L179.21 287.55 L180.50 287.19 L181.79 286.83 L183.08 286.47 L184.38 286.10 L185.67 285.74 L186.96 285.38 L188.25 285.02 L189.54 284.66 L190.83 284.30 L192.13 283.94 L193.42 283.58 L194.71 283.21 L196.00 282.85 L197.29 282.49 L198.58 282.13 L199.87 281.77 L201.17 281.41 L202.46 281.05 L203.75 280.69 L205.04 280.32 L206.33 279.96 L207.63 279.60 L208.92 279.24 L210.21 278.88 L211.50 278.52 L212.79 278.16 L214.08 277.80 L215.38 277.43 L216.67 277.07 L217.96 276.71 L219.25 276.35 L220.54 275.99 L221.83 275.63 L223.13 275.27 L224.42 274.91 L225.71 274.54 L227.00 274.18 L228.29 273.82 L229.58 273.46 L230.88 273.10 L232.17 272.74 L233.46 272.38 L234.75 272.02 L236.04 271.65 L237.33 271.29 L238.62 270.93 L239.92 270.57 L241.21 270.21 L242.50 269.85 L243.79 269.49 L245.08 269.13 L246.38 268.76 L247.67 268.40 L248.96 268.04 L250.25 267.68 L251.54 267.32 L252.83 266.96 L254.13 266.60 L255.42 266.24 L256.71 265.87 L258.00 265.51 L259.29 265.15 L260.58 264.79 L261.88 264.43 L263.17 264.07 L264.46 263.71 L265.75 263.35 L267.04 262.98 L268.33 262.62 L269.63 262.26 L270.92 261.90 L272.21 261.54 L273.50 261.18 L274.79 260.82 L276.08 260.46 L277.38 260.09 L278.67 259.73 L279.96 259.37 L281.25 259.01 L282.54 258.65 L283.83 258.29 L285.13 257.93 L286.42 257.57 L287.71 257.20 L289.00 256.84 L290.29 256.48 L291.58 256.12 L292.88 255.76 L294.17 255.40 L295.46 255.04 L296.75 254.67 L298.04 254.31 L299.33 253.95 L300.63 253.59 L301.92 253.23 L303.21 252.87 L304.50 252.51 L305.79 252.15 L307.08 251.78 L308.38 251.42 L309.67 251.06 L310.96 250.70 L312.25 250.34 L313.54 249.98 L314.83 249.62 L316.13 249.26 L317.42 248.89 L318.71 248.53 L320.00 248.17 L321.29 247.81 L322.58 247.45 L323.88 247.09 L325.17 246.73 L326.46 246.37 L327.75 246.00 L329.04 245.64 L330.33 245.28 L331.62 244.92 L332.92 244.56 L334.21 244.20 L335.50 243.84 L336.79 243.48 L338.08 243.11 L339.38 242.75 L340.67 242.39 L341.96 242.03 L343.25 241.67 L344.54 241.31 L345.83 240.95 L347.13 240.59 L348.42 240.22 L349.71 239.86 L351.00 239.50 L352.29 239.14 L353.58 238.78 L354.88 238.42 L356.17 238.06 L357.46 237.70 L358.75 237.33 L360.04 236.97 L361.33 236.61 L362.63 236.25 L363.92 235.89 L365.21 235.53 L366.50 235.17 L367.79 234.81 L369.08 234.44 L370.37 234.08 L371.67 233.72 L372.96 233.36 L374.25 233.00 L375.54 232.64 L376.83 232.28 L378.13 231.92 L379.42 231.55 L380.71 231.19 L382.00 230.83 L383.29 230.47 L384.58 230.11 L385.88 229.75 L387.17 229.39 L388.46 229.03 L389.75 228.66 L391.04 228.30 L392.33 227.94 L393.63 227.58 L394.92 227.22 L396.21 226.86 L397.50 226.50 L398.79 226.14 L400.08 225.77 L401.38 225.41 L402.67 225.05 L403.96 224.69 L405.25 224.33 L406.54 223.97 L407.83 223.61 L409.12 223.25 L410.42 222.88 L411.71 222.52 L413.00 222.16 L414.29 221.80 L415.58 221.44 L416.88 221.08 L418.17 220.72 L419.46 220.35 L420.75 219.99 L422.04 219.63 L423.33 219.27 L424.63 218.91 L425.92 218.55 L427.21 218.19 L428.50 217.83 L429.79 217.46 L431.08 217.10 L432.38 216.74 L433.67 216.38 L434.96 216.02 L436.25 215.66 L437.54 215.30 L438.83 214.94 L440.13 214.57 L441.42 214.21 L442.71 213.85 L444.00 213.49 L445.29 213.13 L446.58 212.77 L447.87 212.41 L449.17 212.05 L450.46 211.68 L451.75 211.32 L453.04 210.96 L454.33 210.60 L455.63 210.24 L456.92 209.88 L458.21 209.52 L459.50 209.16 L460.79 208.79 L462.08 208.43 L463.38 208.07 L464.67 207.71 L465.96 207.35 L467.25 206.99 L468.54 206.63 L469.83 206.27 L471.13 205.90 L472.42 205.54 L473.71 205.18 L475.00 204.82 L476.29 204.46 L477.58 204.10 L478.88 203.74 L480.17 203.38 L481.46 203.01 L482.75 202.65 L484.04 202.29 L485.33 201.93 L486.62 201.57 L487.92 201.21 L489.21 200.85 L490.50 200.49 L491.79 200.12 L493.08 199.76 L494.38 199.40 L495.67 199.04 L496.96 198.68 L498.25 198.32 L499.54 197.96 L500.83 197.60 L502.13 197.23 L503.42 196.87 L504.71 196.51 L506.00 196.15 L507.29 195.79 L508.58 195.43 L509.88 195.07 L511.17 194.71 L512.46 194.34 L513.75 193.98 L515.04 193.62 L516.33 193.26 L517.63 192.90 L518.92 192.54 L520.21 192.18 L521.50 191.82 L522.79 191.45 L524.08 191.09 L525.38 190.73 L526.67 190.37 L527.96 190.01 L529.25 189.65 L530.54 189.29 L531.83 188.92 L533.13 188.56 L534.42 188.20 L535.71 187.84 L537.00 187.48 L538.29 187.12 L539.58 186.76 L540.88 186.40 L542.17 186.03 L543.46 185.67 L544.75 185.31 L546.04 184.95 L547.33 184.59 L548.63 184.23 L549.92 183.87 L551.21 183.51 L552.50 183.14 L553.79 182.78 L555.08 182.42 L556.38 182.06 L557.67 181.70 L558.96 181.34 L560.25 180.98 L561.54 180.62 L562.83 180.25 L564.13 179.89 L565.42 179.53 L566.71 179.17 L568.00 178.81 L569.29 178.45 L570.58 178.09 L571.88 177.73 L573.17 177.36 L574.46 177.00 L575.75 176.64 L577.04 176.28 L578.33 175.92 L579.63 175.56 L580.92 175.20 L582.21 174.84 L583.50 174.47 L584.79 174.11 L586.08 173.75 L587.38 173.39 L588.67 173.03 L589.96 172.67 L591.25 172.31 L592.54 171.95 L593.83 171.58 L595.13 171.22 L596.42 170.86 L597.71 170.50 L599.00 170.14 L600.29 169.78 L601.58 169.42 L602.88 169.06 L604.17 168.69 L605.46 168.33 L606.75 167.97 L608.04 167.61 L609.33 167.25 L610.63 166.89 L611.92 166.53 L613.21 166.17 L614.50 165.80 L615.79 165.44 L617.08 165.08 L618.38 164.72 L619.67 164.36 L620.96 164.00 L622.25 163.64 L623.54 163.28 L624.83 162.91 L626.13 162.55 L627.42 162.19 L628.71 161.83 L630.00 161.47" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-sky-500"></path></svg></div><label class="mt-5 block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">1 + x + c·x²</span><span class="font-mono text-zinc-800 dark:text-zinc-200">c<!-- --> = <!-- -->0.00</span></span><input type="range" min="-1" max="1.5" step="0.05" class="mt-2 w-full accent-teal-500" value="0"></label><div class="mt-4 flex flex-wrap items-baseline justify-between gap-3 border-t border-zinc-200 pt-4 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">average gap over [−1, 1]</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">0.1770</span></div><button type="button" class="mt-4 rounded-full bg-teal-500 px-4 py-1.5 text-sm text-white transition hover:bg-teal-400">Snap to f″(0)/2</button></div>
<h4 id="reconstructing-a-b-and-c" class="scroll-mt-24">1.2.2 Reconstructing a, b and c</h4>
<p>Now the search is for the best <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">a + bx + cx^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. Write <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">P(x) = a + bx + cx^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>a</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mtext>&nbsp;</mtext><mi>a</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>b</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mtext>&nbsp;</mtext><mi>b</mi><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>P</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mi>c</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>⇒</mo><mtext>&nbsp;</mtext><mi>c</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
P(0) &amp;= a &amp;&amp;\Rightarrow\ a = f(0) \\
P'(0) &amp;= b &amp;&amp;\Rightarrow\ b = f'(0) \\
P''(0) &amp;= 2c &amp;&amp;\Rightarrow\ c = \tfrac{1}{2}f''(0)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.2051em;vertical-align:-1.8526em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3526em"><span style="top:-4.5126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.0126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-1.5074em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8526em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3526em"><span style="top:-4.5126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">a</span></span></span><span style="top:-3.0126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">b</span></span></span><span style="top:-1.5074em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord">2</span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8526em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3526em"><span style="top:-4.3577em"><span class="pstrut" style="height:2.8451em"></span><span class="mord"></span></span><span style="top:-2.8577em"><span class="pstrut" style="height:2.8451em"></span><span class="mord"></span></span><span style="top:-1.3526em"><span class="pstrut" style="height:2.8451em"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8526em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3526em"><span style="top:-4.5126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">⇒</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.0126em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">⇒</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-1.5074em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">⇒</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8526em"><span></span></span></span></span></span></span></span></span></span></span></span>
<p>That <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> is not a fudge factor — it is there because <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P''(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> comes out as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>c</mi></mrow><annotation encoding="application/x-tex">2c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span><span class="mord mathnormal">c</span></span></span></span> rather than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span></span></span></span>.</p>
<div class="not-prose mt-6 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><div class="grid gap-4 sm:grid-cols-3"><label class="block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">a</span><span class="font-mono text-zinc-800 dark:text-zinc-200">2.00</span></span><input type="range" min="-3" max="3" step="0.25" class="mt-2 w-full accent-teal-500" value="2"></label><label class="block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">b</span><span class="font-mono text-zinc-800 dark:text-zinc-200">-1.00</span></span><input type="range" min="-3" max="3" step="0.25" class="mt-2 w-full accent-teal-500" value="-1"></label><label class="block"><span class="flex items-baseline justify-between text-sm text-zinc-600 dark:text-zinc-400"><span class="font-mono">c</span><span class="font-mono text-zinc-800 dark:text-zinc-200">0.50</span></span><input type="range" min="-3" max="3" step="0.25" class="mt-2 w-full accent-teal-500" value="0.5"></label></div><div class="mt-6"><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">g(x)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2 − 1x + 0.5x²</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">g(0)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">g′(0)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">-1</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">g″(0)</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">1</span></div><div class="flex items-baseline justify-between gap-4 border-t border-zinc-200 py-2 dark:border-zinc-800"><span class="text-sm text-zinc-500 dark:text-zinc-400">g(0) + g′(0)·x + g″(0)·x²/2</span><span class="font-mono text-sm text-zinc-800 dark:text-zinc-200">2 − 1x + 0.5x²</span></div></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">The last row always matches the first. Note<!-- --> <span class="font-mono">g″(0) = 2c</span> — the ÷2 in the formula is exactly what undoes that and recovers<!-- --> <span class="font-mono">c</span>.</p></div>
<p>Set <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span> and the quadratic formula collapses into the linear one. Linear approximation is the quadratic with curvature switched off, which is why a function with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f''(0)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span> has the same linear and quadratic approximation.</p>
<h2 id="cheatsheet" class="scroll-mt-24">2. Cheatsheet</h2>
<h3 id="near-0" class="scroll-mt-24">2.1 Near 0</h3>
<p>Each entry lists <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f''(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span>, then both approximations, then the values against a direct calculation, then the graphs.</p>
<h4 id="trigonometric" class="scroll-mt-24">2.1.1 Trigonometric</h4>
<section id="fn-sin" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.1</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7512em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6679em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.010000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.010000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.010000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">1.67e-7</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.099833</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.100000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.100000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000167</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.479426</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.500000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.020574</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.841471</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.158529</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" 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L520.21 50.47 L521.50 49.70 L522.79 48.93 L524.08 48.16 L525.38 47.39 L526.67 46.61 L527.96 45.84 L529.25 45.07 L530.54 44.30 L531.83 43.53 L533.13 42.76 L534.42 41.99 L535.71 41.22 L537.00 40.44 L538.29 39.67 L539.58 38.90 L540.88 38.13 L542.17 37.36 L543.46 36.59 L544.75 35.82 L546.04 35.05 L547.33 34.28 L548.63 33.50 L549.92 32.73 L551.21 31.96 L552.50 31.19 L553.79 30.42 L555.08 29.65 L556.38 28.88 L557.67 28.11 L558.96 27.33 L560.25 26.56 L561.54 25.79 L562.83 25.02 L564.13 24.25 L565.42 23.48 L566.71 22.71 L568.00 21.94 L569.29 21.17 L570.58 20.39 L571.88 19.62 L573.17 18.85 L574.46 18.08 L575.75 17.31 L577.04 16.54 L578.33 15.77 L579.63 15.00 L580.92 14.23 L582.21 13.45 L583.50 12.68 L584.79 11.91 L586.08 11.14 L587.38 10.37" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>sin x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Odd function, so f″(0) = 0 and the quadratic term vanishes — the quadratic approximation is the linear one.</p></section>
<section id="fn-cos" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.2</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7512em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4831em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4831em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.999950</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.999950</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">4.17e-10</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.995004</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.995000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">4.17e-6</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.877583</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.875000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.002583</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.540302</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.040302</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of cos x with its approximations near zero"><line x1="10" x2="630" y1="243.21280011403098" y2="243.21280011403098" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><line x1="320" x2="320" y1="10" y2="330" 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gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>cos x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Flat at 0, so the linear approximation is just the constant 1. All the information is in the quadratic term.</p></section>
<section id="fn-tan" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.3</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mop"><span class="mop">sec</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sec</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.010000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.010000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.010000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">3.33e-7</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.100335</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.100000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.100000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000335</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.546302</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.500000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.046302</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.557408</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.557408</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of tan x with its approximations near zero"><line x1="10" 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stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>tan x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Also odd, so again the quadratic term vanishes.</p></section>
<section id="fn-sec" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.4</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.9024em;vertical-align:-0.0833em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">tan</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8191em;"><span style="top:-3.068em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8141em;"></span><span class="mop"><span class="mop">sec</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4831em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4831em;"></span><span class="mop">sec</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000050</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000050</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">2.08e-9</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.005021</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.005000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">2.09e-5</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.139494</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.125000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.014494</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.850816</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.350816</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of sec x with its approximations near zero"><line x1="10" x2="630" y1="311.81460878640615" y2="311.81460878640615" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><line x1="320" x2="320" y1="10" y2="330" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><path d="M10.00 46.92 L11.29 68.37 L12.58 86.60 L13.88 102.28 L15.17 115.92 L16.46 127.88 L17.75 138.45 L19.04 147.87 L20.33 156.32 L21.62 163.93 L22.92 170.83 L24.21 177.10 L25.50 182.84 L26.79 188.10 L28.08 192.95 L29.38 197.42 L30.67 201.57 L31.96 205.42 L33.25 209.01 L34.54 212.36 L35.83 215.50 L37.13 218.44 L38.42 221.20 L39.71 223.80 L41.00 226.26 L42.29 228.57 L43.58 230.76 L44.87 232.84 L46.17 234.81 L47.46 236.68 L48.75 238.46 L50.04 240.15 L51.33 241.77 L52.63 243.31 L53.92 244.78 L55.21 246.19 L56.50 247.53 L57.79 248.83 L59.08 250.06 L60.37 251.25 L61.67 252.39 L62.96 253.49 L64.25 254.54 L65.54 255.56 L66.83 256.53 L68.13 257.47 L69.42 258.38 L70.71 259.26 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293.08 L369.08 293.08 L370.37 293.08 L371.67 293.08 L372.96 293.08 L374.25 293.08 L375.54 293.08 L376.83 293.08 L378.13 293.08 L379.42 293.08 L380.71 293.08 L382.00 293.08 L383.29 293.08 L384.58 293.08 L385.88 293.08 L387.17 293.08 L388.46 293.08 L389.75 293.08 L391.04 293.08 L392.33 293.08 L393.63 293.08 L394.92 293.08 L396.21 293.08 L397.50 293.08 L398.79 293.08 L400.08 293.08 L401.38 293.08 L402.67 293.08 L403.96 293.08 L405.25 293.08 L406.54 293.08 L407.83 293.08 L409.12 293.08 L410.42 293.08 L411.71 293.08 L413.00 293.08 L414.29 293.08 L415.58 293.08 L416.88 293.08 L418.17 293.08 L419.46 293.08 L420.75 293.08 L422.04 293.08 L423.33 293.08 L424.63 293.08 L425.92 293.08 L427.21 293.08 L428.50 293.08 L429.79 293.08 L431.08 293.08 L432.38 293.08 L433.67 293.08 L434.96 293.08 L436.25 293.08 L437.54 293.08 L438.83 293.08 L440.13 293.08 L441.42 293.08 L442.71 293.08 L444.00 293.08 L445.29 293.08 L446.58 293.08 L447.87 293.08 L449.17 293.08 L450.46 293.08 L451.75 293.08 L453.04 293.08 L454.33 293.08 L455.63 293.08 L456.92 293.08 L458.21 293.08 L459.50 293.08 L460.79 293.08 L462.08 293.08 L463.38 293.08 L464.67 293.08 L465.96 293.08 L467.25 293.08 L468.54 293.08 L469.83 293.08 L471.13 293.08 L472.42 293.08 L473.71 293.08 L475.00 293.08 L476.29 293.08 L477.58 293.08 L478.88 293.08 L480.17 293.08 L481.46 293.08 L482.75 293.08 L484.04 293.08 L485.33 293.08 L486.62 293.08 L487.92 293.08 L489.21 293.08 L490.50 293.08 L491.79 293.08 L493.08 293.08 L494.38 293.08 L495.67 293.08 L496.96 293.08 L498.25 293.08 L499.54 293.08 L500.83 293.08 L502.13 293.08 L503.42 293.08 L504.71 293.08 L506.00 293.08 L507.29 293.08 L508.58 293.08 L509.88 293.08 L511.17 293.08 L512.46 293.08 L513.75 293.08 L515.04 293.08 L516.33 293.08 L517.63 293.08 L518.92 293.08 L520.21 293.08 L521.50 293.08 L522.79 293.08 L524.08 293.08 L525.38 293.08 L526.67 293.08 L527.96 293.08 L529.25 293.08 L530.54 293.08 L531.83 293.08 L533.13 293.08 L534.42 293.08 L535.71 293.08 L537.00 293.08 L538.29 293.08 L539.58 293.08 L540.88 293.08 L542.17 293.08 L543.46 293.08 L544.75 293.08 L546.04 293.08 L547.33 293.08 L548.63 293.08 L549.92 293.08 L551.21 293.08 L552.50 293.08 L553.79 293.08 L555.08 293.08 L556.38 293.08 L557.67 293.08 L558.96 293.08 L560.25 293.08 L561.54 293.08 L562.83 293.08 L564.13 293.08 L565.42 293.08 L566.71 293.08 L568.00 293.08 L569.29 293.08 L570.58 293.08 L571.88 293.08 L573.17 293.08 L574.46 293.08 L575.75 293.08 L577.04 293.08 L578.33 293.08 L579.63 293.08 L580.92 293.08 L582.21 293.08 L583.50 293.08 L584.79 293.08 L586.08 293.08 L587.38 293.08 L588.67 293.08 L589.96 293.08 L591.25 293.08 L592.54 293.08 L593.83 293.08 L595.13 293.08 L596.42 293.08 L597.71 293.08 L599.00 293.08 L600.29 293.08 L601.58 293.08 L602.88 293.08 L604.17 293.08 L605.46 293.08 L606.75 293.08 L608.04 293.08 L609.33 293.08 L610.63 293.08 L611.92 293.08 L613.21 293.08 L614.50 293.08 L615.79 293.08 L617.08 293.08 L618.38 293.08 L619.67 293.08 L620.96 293.08 L622.25 293.08 L623.54 293.08 L624.83 293.08 L626.13 293.08 L627.42 293.08 L628.71 293.08 L630.00 293.08" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>sec x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">The mirror of cos x — same flat linear part, opposite sign on the quadratic term.</p></section>
<section id="fn-cot" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.5</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">cot</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-4"><div class="rounded-lg bg-white px-4 py-3 dark:bg-zinc-950/50"><span><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6151em;"></span><span class="mop">cot</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></span></div><p class="mt-4 text-sm text-zinc-600 dark:text-zinc-400">No approximation at 0. cot x blows up there, so there is no finite value, slope, or curvature to match.</p></div></section>
<section id="fn-csc" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.1.6</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">csc</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span></span></h5><div class="mt-4"><div class="rounded-lg bg-white px-4 py-3 dark:bg-zinc-950/50"><span><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mop">csc</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></span></div><p class="mt-4 text-sm text-zinc-600 dark:text-zinc-400">Same problem as cot x — undefined at 0, so no expansion exists there.</p></div></section>
<h4 id="exponents-and-logarithms" class="scroll-mt-24">2.1.2 Exponents and logarithms</h4>
<section id="fn-exp" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.1</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.010050</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.010000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.010050</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">1.67e-7</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.105171</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.100000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.105000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000171</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.648721</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.500000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.625000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.023721</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">2.718282</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">2.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">2.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.218282</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of e^x with its approximations near zero"><line x1="10" x2="630" y1="305.9743252901553" y2="305.9743252901553" class="stroke-zinc-300 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dark:bg-zinc-300"></span>e^x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Every derivative is e^x, so every coefficient at 0 is 1. This is the cleanest one to remember.</p></section>
<section id="fn-ln" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.2</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:2.2574em;vertical-align:-0.936em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.009950</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.010000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.009950</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">3.31e-7</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.095310</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.100000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.095000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000310</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.405465</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">0.500000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.375000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.030465</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">0.693147</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">0.500000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.193147</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of ln(1 + x) with its approximations near zero"><line x1="10" x2="630" y1="87.77249905993995" y2="87.77249905993995" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><line x1="320" x2="320" y1="10" y2="330" class="stroke-zinc-300 dark:stroke-zinc-700" stroke-width="1"></line><path d="M114.62 314.03 L115.92 283.13 L117.21 265.05 L118.50 252.23 L119.79 242.28 L121.08 234.15 L122.38 227.28 L123.67 221.33 L124.96 216.08 L126.25 211.38 L127.54 207.13 L128.83 203.25 L130.13 199.68 L131.42 196.38 L132.71 193.30 L134.00 190.42 L135.29 187.72 L136.58 185.17 L137.88 182.76 L139.17 180.48 L140.46 178.30 L141.75 176.23 L143.04 174.25 L144.33 172.35 L145.63 170.53 L146.92 168.78 L148.21 167.10 L149.50 165.48 L150.79 163.91 L152.08 162.40 L153.38 160.94 L154.67 159.52 L155.96 158.15 L157.25 156.82 L158.54 155.53 L159.83 154.27 L161.13 153.05 L162.42 151.86 L163.71 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L429.79 64.09 L431.08 63.81 L432.38 63.53 L433.67 63.25 L434.96 62.97 L436.25 62.70 L437.54 62.42 L438.83 62.14 L440.13 61.86 L441.42 61.58 L442.71 61.30 L444.00 61.02 L445.29 60.75 L446.58 60.47 L447.87 60.19 L449.17 59.91 L450.46 59.63 L451.75 59.35 L453.04 59.07 L454.33 58.79 L455.63 58.52 L456.92 58.24 L458.21 57.96 L459.50 57.68 L460.79 57.40 L462.08 57.12 L463.38 56.84 L464.67 56.57 L465.96 56.29 L467.25 56.01 L468.54 55.73 L469.83 55.45 L471.13 55.17 L472.42 54.89 L473.71 54.62 L475.00 54.34 L476.29 54.06 L477.58 53.78 L478.88 53.50 L480.17 53.22 L481.46 52.94 L482.75 52.66 L484.04 52.39 L485.33 52.11 L486.62 51.83 L487.92 51.55 L489.21 51.27 L490.50 50.99 L491.79 50.71 L493.08 50.44 L494.38 50.16 L495.67 49.88 L496.96 49.60 L498.25 49.32 L499.54 49.04 L500.83 48.76 L502.13 48.49 L503.42 48.21 L504.71 47.93 L506.00 47.65 L507.29 47.37 L508.58 47.09 L509.88 46.81 L511.17 46.53 L512.46 46.26 L513.75 45.98 L515.04 45.70 L516.33 45.42 L517.63 45.14 L518.92 44.86 L520.21 44.58 L521.50 44.31 L522.79 44.03 L524.08 43.75 L525.38 43.47 L526.67 43.19 L527.96 42.91 L529.25 42.63 L530.54 42.36 L531.83 42.08 L533.13 41.80 L534.42 41.52 L535.71 41.24 L537.00 40.96 L538.29 40.68 L539.58 40.40 L540.88 40.13 L542.17 39.85 L543.46 39.57 L544.75 39.29 L546.04 39.01 L547.33 38.73 L548.63 38.45 L549.92 38.18 L551.21 37.90 L552.50 37.62 L553.79 37.34 L555.08 37.06 L556.38 36.78 L557.67 36.50 L558.96 36.23 L560.25 35.95 L561.54 35.67 L562.83 35.39 L564.13 35.11 L565.42 34.83 L566.71 34.55 L568.00 34.27 L569.29 34.00 L570.58 33.72 L571.88 33.44 L573.17 33.16 L574.46 32.88 L575.75 32.60 L577.04 32.32 L578.33 32.05 L579.63 31.77 L580.92 31.49 L582.21 31.21 L583.50 30.93 L584.79 30.65 L586.08 30.37 L587.38 30.10 L588.67 29.82 L589.96 29.54 L591.25 29.26 L592.54 28.98 L593.83 28.70 L595.13 28.42 L596.42 28.15 L597.71 27.87 L599.00 27.59 L600.29 27.31 L601.58 27.03 L602.88 26.75 L604.17 26.47 L605.46 26.19 L606.75 25.92 L608.04 25.64 L609.33 25.36 L610.63 25.08 L611.92 24.80 L613.21 24.52 L614.50 24.24 L615.79 23.97 L617.08 23.69 L618.38 23.41 L619.67 23.13 L620.96 22.85 L622.25 22.57 L623.54 22.29 L624.83 22.02 L626.13 21.74 L627.42 21.46 L628.71 21.18 L630.00 20.90" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>ln(1 + x)</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Written as ln(1 + x) rather than ln x precisely so that the expansion point sits at 0 — ln x has no expansion there.</p></section>
<section id="fn-pow" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.3</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span></span></span></span></span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">x</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.355em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.0278em;">r</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="mb-5 flex flex-wrap items-center gap-2"><span class="mr-1 font-mono text-sm text-zinc-500 dark:text-zinc-400">r<!-- --> =</span><button type="button" aria-pressed="true" class="rounded-full px-3 py-1 font-mono text-sm transition bg-teal-500 text-white">1/2</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">−1/2</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">−1</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">2</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">3</button></div><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.004988</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.005000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.004987</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">6.21e-8</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.048809</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.050000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.048750</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">5.88e-5</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.224745</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.250000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.218750</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.005995</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.414214</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.500000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.375000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.039214</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg 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L570.58 65.29 L571.88 64.86 L573.17 64.44 L574.46 64.01 L575.75 63.58 L577.04 63.15 L578.33 62.73 L579.63 62.30 L580.92 61.87 L582.21 61.45 L583.50 61.02 L584.79 60.59 L586.08 60.16 L587.38 59.74 L588.67 59.31 L589.96 58.88 L591.25 58.45 L592.54 58.03 L593.83 57.60 L595.13 57.17 L596.42 56.74 L597.71 56.32 L599.00 55.89 L600.29 55.46 L601.58 55.04 L602.88 54.61 L604.17 54.18 L605.46 53.75 L606.75 53.33 L608.04 52.90 L609.33 52.47 L610.63 52.04 L611.92 51.62 L613.21 51.19 L614.50 50.76 L615.79 50.33 L617.08 49.91 L618.38 49.48 L619.67 49.05 L620.96 48.62 L622.25 48.20 L623.54 47.77 L624.83 47.34 L626.13 46.92 L627.42 46.49 L628.71 46.06 L630.00 45.63" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>(1 + x)^r</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">The workhorse. r = 1/2 gives √(1+x), r = −1 gives 1/(1+x). At r = 2 the quadratic approximation is exact — (1+x)² already is a quadratic, so there is nothing to discard.</p></section>
<section id="fn-apow" class="not-prose mt-8 scroll-mt-24 rounded-2xl border border-zinc-200 bg-zinc-50 p-5 sm:p-6 dark:border-zinc-800 dark:bg-zinc-900/60"><h5 class="flex items-baseline gap-3"><span class="font-mono text-sm text-zinc-400 dark:text-zinc-500">2.1.2.4</span><span class="text-xl text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></h5><div class="mt-5 overflow-x-auto"><table class="w-full min-w-[24rem] text-left text-sm"><tbody><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span></span></td></tr><tr class="border-t border-zinc-200 first:border-t-0 dark:border-zinc-800"><td class="py-2 pr-6"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8984em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">ln</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span></span></td><td class="py-2 text-right"><span class="text-zinc-800 dark:text-zinc-200"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.8984em;"></span><span class="mop"><span class="mop">ln</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span></span></td></tr></tbody></table></div><dl class="mt-5 grid gap-3 sm:grid-cols-2"><div class="rounded-lg border border-teal-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-teal-600 uppercase dark:text-teal-400">linear</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.6944em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span></span></dd></div><div class="rounded-lg border border-sky-500/30 bg-white px-4 py-3 dark:bg-zinc-950/50"><dt class="text-xs font-medium tracking-wide text-sky-600 uppercase dark:text-sky-400">quadratic</dt><dd class="mt-1"><span class="text-zinc-900 dark:text-zinc-100"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.3629em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0179em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">n</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="katex-sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></dd></div></dl><div class="mt-5"><div class="mb-5 flex flex-wrap items-center gap-2"><span class="mr-1 font-mono text-sm text-zinc-500 dark:text-zinc-400">a<!-- --> =</span><button type="button" aria-pressed="true" class="rounded-full px-3 py-1 font-mono text-sm transition bg-teal-500 text-white">2</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">3</button><button type="button" aria-pressed="false" class="rounded-full px-3 py-1 font-mono text-sm transition bg-zinc-200 text-zinc-600 hover:bg-zinc-300 dark:bg-zinc-800 dark:text-zinc-400 dark:hover:bg-zinc-700">10</button></div><div class="overflow-x-auto"><table class="w-full min-w-[30rem] text-left text-sm"><thead><tr class="text-zinc-500 dark:text-zinc-400"><th class="py-2 pr-4 font-medium">x</th><th class="py-2 pr-4 text-right font-medium">direct</th><th class="py-2 pr-4 text-right font-medium">linear</th><th class="py-2 pr-4 text-right font-medium">quadratic</th><th class="py-2 text-right font-medium">quad. error</th></tr></thead><tbody><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.000000</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.000000</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.000000</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.01</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.006956</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.006931</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.006955</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">5.56e-8</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.071773</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.069315</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.071717</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">5.65e-5</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">0.5</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">1.414214</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.346574</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.406630</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.007583</td></tr><tr class="border-t border-zinc-200 dark:border-zinc-800"><td class="py-2 pr-4 font-mono text-zinc-700 dark:text-zinc-300">1</td><td class="py-2 pr-4 text-right font-mono text-zinc-800 dark:text-zinc-200">2.000000</td><td class="py-2 pr-4 text-right font-mono text-teal-600 dark:text-teal-400">1.693147</td><td class="py-2 pr-4 text-right font-mono text-sky-600 dark:text-sky-400">1.933374</td><td class="py-2 text-right font-mono text-zinc-500 dark:text-zinc-400">0.066626</td></tr></tbody></table></div><div class="mt-6 rounded-xl bg-white p-3 dark:bg-zinc-950/50"><svg viewBox="0 0 640 340" class="h-auto w-full" role="img" aria-label="Graph of a^x with its approximations near zero"><line x1="10" x2="630" y1="328.24175824175825" y2="328.24175824175825" 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L480.17 175.35 L481.46 174.92 L482.75 174.49 L484.04 174.06 L485.33 173.63 L486.62 173.20 L487.92 172.77 L489.21 172.33 L490.50 171.90 L491.79 171.47 L493.08 171.04 L494.38 170.61 L495.67 170.18 L496.96 169.75 L498.25 169.32 L499.54 168.89 L500.83 168.46 L502.13 168.03 L503.42 167.60 L504.71 167.16 L506.00 166.73 L507.29 166.30 L508.58 165.87 L509.88 165.44 L511.17 165.01 L512.46 164.58 L513.75 164.15 L515.04 163.72 L516.33 163.29 L517.63 162.86 L518.92 162.42 L520.21 161.99 L521.50 161.56 L522.79 161.13 L524.08 160.70 L525.38 160.27 L526.67 159.84 L527.96 159.41 L529.25 158.98 L530.54 158.55 L531.83 158.12 L533.13 157.68 L534.42 157.25 L535.71 156.82 L537.00 156.39 L538.29 155.96 L539.58 155.53 L540.88 155.10 L542.17 154.67 L543.46 154.24 L544.75 153.81 L546.04 153.38 L547.33 152.95 L548.63 152.51 L549.92 152.08 L551.21 151.65 L552.50 151.22 L553.79 150.79 L555.08 150.36 L556.38 149.93 L557.67 149.50 L558.96 149.07 L560.25 148.64 L561.54 148.21 L562.83 147.77 L564.13 147.34 L565.42 146.91 L566.71 146.48 L568.00 146.05 L569.29 145.62 L570.58 145.19 L571.88 144.76 L573.17 144.33 L574.46 143.90 L575.75 143.47 L577.04 143.03 L578.33 142.60 L579.63 142.17 L580.92 141.74 L582.21 141.31 L583.50 140.88 L584.79 140.45 L586.08 140.02 L587.38 139.59 L588.67 139.16 L589.96 138.73 L591.25 138.30 L592.54 137.86 L593.83 137.43 L595.13 137.00 L596.42 136.57 L597.71 136.14 L599.00 135.71 L600.29 135.28 L601.58 134.85 L602.88 134.42 L604.17 133.99 L605.46 133.56 L606.75 133.12 L608.04 132.69 L609.33 132.26 L610.63 131.83 L611.92 131.40 L613.21 130.97 L614.50 130.54 L615.79 130.11 L617.08 129.68 L618.38 129.25 L619.67 128.82 L620.96 128.38 L622.25 127.95 L623.54 127.52 L624.83 127.09 L626.13 126.66 L627.42 126.23 L628.71 125.80 L630.00 125.37" fill="none" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="stroke-teal-500"></path></svg></div><div class="mt-4 flex flex-wrap items-center gap-x-6 gap-y-3"><span class="flex items-center gap-2 text-sm text-zinc-600 dark:text-zinc-400"><span class="h-0.5 w-4 rounded-full bg-zinc-400 dark:bg-zinc-300"></span>a^x</span><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-teal-500"></span>linear</label><label class="flex cursor-pointer items-center gap-2 text-sm text-zinc-600 select-none dark:text-zinc-400"><input type="checkbox" class="h-4 w-4 rounded border-zinc-300 accent-teal-500 dark:border-zinc-600" checked=""><span class="h-0.5 w-4 rounded-full bg-sky-500"></span>quadratic</label></div><label class="mt-4 block"><span class="flex items-baseline justify-between text-sm text-zinc-500 dark:text-zinc-400"><span>zoom (window ±)</span><span class="font-mono">1.50</span></span><input type="range" min="0.1" max="3" step="0.05" class="mt-2 w-full accent-teal-500" value="1.5"></label></div><p class="mt-5 border-t border-zinc-200 pt-4 text-sm text-zinc-600 dark:border-zinc-800 dark:text-zinc-400">Generalises e^x — set a = e and ln a = 1, recovering 1 + x + x²/2.</p></section>
<h2 id="combining-approximations" class="scroll-mt-24">3. Combining approximations</h2>
<p>Rather than differentiating a complicated function twice, build its approximation out of the entries above. The one move that covers most cases: <strong>substitute, then truncate.</strong></p>
<table><thead><tr><th>operation</th><th>rule</th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f + g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span></span></span></span></td><td>add the approximations</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>⋅</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">k \cdot f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.0315em">k</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span></span></span></span></td><td>scale the approximation by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.0315em">k</span></span></span></span></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \cdot g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span></span></span></span></td><td>multiply the approximations, then truncate</td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \circ g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(0) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span></td><td>substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span></span></span></span>'s approximation into <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span></span></span></span>'s, then truncate</td></tr></tbody></table>
<p><strong>Truncate</strong> means: discard every power above the order you are working to — <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> and up for linear, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> and up for quadratic.</p>
<p>For composition there is also the direct form, which is just the chain rule:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≈</mo><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mtext> </mtext><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext><mi>x</mi></mrow><annotation encoding="application/x-tex">f(g(x)) \approx f(g(0)) + f'(g(0))\,g'(0)\,x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">x</span></span></span></span></span>
<p>Two conditions worth keeping in mind:</p>
<ul>
<li><strong>The inside function must land near 0.</strong> Every entry above is an expansion around <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span>, so substitution is only legal when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(0) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">0</span></span></span></span>.</li>
<li><strong>Truncating is not optional.</strong> The product of two linear approximations contains an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> term that is not trustworthy — the inputs were only accurate to first order, so the second-order information in the product is noise.</li>
</ul>
<p>That second point is worth seeing. Multiplying two linear approximations gives</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>x</mi><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>x</mi><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">[</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">]</mo><mi>x</mi><mo>+</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\big(f(0) + f'(0)x\big)\big(g(0) + g'(0)x\big) = f(0)g(0) + \big[f'(0)g(0) + f(0)g'(0)\big]x + f'(0)g'(0)x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em"></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord mathnormal">x</span><span class="mord"><span class="delimsizing size1">)</span></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord mathnormal">x</span><span class="mord"><span class="delimsizing size1">)</span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord"><span class="delimsizing size1">]</span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span>
<p>Drop the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> and the coefficient of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(0)g(0) + f(0)g'(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> — exactly <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(fg)'(0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.1076em">f</span><span class="mord mathnormal" style="margin-right:0.0359em">g</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span></span></span></span> from the product rule. The shortcut agrees with the definition, which is why it is safe.</p>]]></content:encoded>
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