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<div toc=""><ul toc=""><li entry="h1"><a href="#chapter-3%20%7C%20applications%20of%20differentiation">Chapter 3 | Applications of Differentiation </a></li><li entry="h2"><a href="#3.1-%7C%20maximum%20and%20minimum%20values">3.1 | Maximum and Minimum Values</a></li><li entry="h3"><a href="#the-extreme%20value%20theorem">The extreme Value Theorem</a></li><li entry="h3"><a href="#the-extreme%20value%20theorem">The extreme Value Theorem</a></li><li entry="h3"><a href="#fermat%E2%80%99s-theorem">Fermat’s Theorem</a></li><li entry="h3"><a href="#fermat%E2%80%99s-theorem">Fermat’s Theorem</a></li><li entry="h3"><a href="#fermat%E2%80%99s-theorem%20(in%20terms%20of%20critical%20numbers)">Fermat’s Theorem (in terms of critical numbers)</a></li><li entry="h3"><a href="#critical-number">Critical Number</a></li><li entry="h3"><a href="#the-closed%20interval%20method">The Closed Interval Method</a></li><li entry="h3"><a href="#the-closed%20interval%20method">The Closed Interval Method</a></li><li entry="h2"><a href="#3.2-%7C%20the%20mean%20value%20theorem">3.2 | The Mean Value Theorem</a></li><li entry="h3"><a href="#rolle%E2%80%99s-theorem">Rolle’s Theorem</a></li><li entry="h3"><a href="#the-mean%20value%20theorem">The Mean Value Theorem</a></li><li entry="h3"><a href="#miscellaneous">Miscellaneous</a></li><li entry="h3"><a href="#theorem-(%C2%A73.2.5)">Theorem (§3.2.5)</a></li><li entry="h3"><a href="#corollary-(%C2%A73.2.7)">Corollary (§3.2.7)</a></li><li entry="h2"><a href="#3.3-%7C%20how%20derivatives%20affect%20the%20shape%20of%20a%20graph">3.3 | How Derivatives Affect the Shape of a Graph</a></li><li entry="h3"><a href="#what-does%20stem:%5Bf%5E%5Cprime%5D%20say%20about-f-?">What Does stem:[f^\prime] Say about <span inline="" math-inline="">\(f\)</span>?</a></li><li entry="h3"><a href="#increasing/decreasing-(i/d)%20test">Increasing/Decreasing (I/D) Test</a></li><li entry="h3"><a href="#the-first%20derivative%20test">The First Derivative Test</a></li><li entry="h3"><a href="#what-does-f%5E%7B%5Cprime%5Cprime%7D-say%20about-f-?">What Does <span inline="" math-inline="">\(f^{\prime\prime}\)</span> Say about <span inline="" math-inline="">\(f\)</span>?</a></li><li entry="h3"><a href="#concavity-test">Concavity Test</a></li><li entry="h4"><a href="#concavity-test%20(alternative)">Concavity Test (Alternative)</a></li><li entry="h3"><a href="#regarding-concavity%20and%20extrema">Regarding Concavity and Extrema</a></li><li entry="h4"><a href="#the-following%20three%20scenarios%20are%20possible:">The following three scenarios are possible:</a></li><li entry="h5"><a href="#the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20changing%20from%20negative%20to%20positive">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is changing from negative to positive</a></li><li entry="h5"><a href="#the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20changing%20from%20positive%20to%20negative.">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is changing from positive to negative.</a></li><li entry="h5"><a href="#the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20not%20changing%20at-x%20=%20c">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is not changing at <span inline="" math-inline="">\(x = c\)</span></a></li><li entry="h4"><a href="#the-inflection%20points%20test">The Inflection Points Test</a></li><li entry="h4"><a href="#what-f%5E%7B%5Cprime%5Cprime%7D-says-about-f%5E%7B%5Cprime%7D-(alternative)">What <span inline="" math-inline="">\(f^{\prime\prime}\)</span> says about <span inline="" math-inline="">\(f^{\prime}\)</span> <small>(Alternative)</small></a></li><li entry="h3"><a href="#inflection-points">Inflection Points</a></li><li entry="h3"><a href="#inflection-points%20(alternative)">Inflection Points (Alternative)</a></li><li entry="h3"><a href="#miscellaneous">Miscellaneous</a></li><li entry="h3"><a href="#the-second%20derivative%20test">The Second Derivative Test</a></li><li entry="h2"><a href="#horizontal-asymptotes-section">3.4 | Limits at Infinity; Horizontal Asymptotes</a></li><li entry="h3"><a href="#miscellaneous">Miscellaneous</a></li><li entry="h4"><a href="#%5Clim_%7Bx-%5Cto%20%5Cinfty%7D%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)"><span inline="" math-inline="">\(\lim_{x \to \infty} \sin\left(\frac{1}{x}\right)\)</span></a></li><li entry="h3"><a href="#theorem-(3.4)%20#4">Theorem (3.4) #4</a></li><li entry="h2"><a href="#3.5-%7C%20summary%20curve%20sketching">3.5 | Summary Curve Sketching</a></li><li entry="h3"><a href="#curve-sketching%20steps:">Curve Sketching Steps:</a></li><li entry="h4"><a href="#domain">Domain</a></li><li entry="h4"><a href="#intercepts">Intercepts</a></li><li entry="h4"><a href="#symmetry">Symmetry</a></li><li entry="h4"><a href="#asymptotes">Asymptotes</a></li><li entry="h5"><a href="#vertical-asymptotes">Vertical Asymptotes</a></li><li entry="h5"><a href="#horizontal-asymptotes">Horizontal Asymptotes</a></li><li entry="h6"><a href="#to-find%20horizontal%20asymptotes">To find horizontal asymptotes</a></li><li entry="h5"><a href="#slant-asymptotes">Slant Asymptotes</a></li><li entry="h4"><a href="#the-first%20derivative">The First Derivative</a></li><li entry="h5"><a href="#intervals-of%20increase%20or%20decrease">Intervals of Increase or Decrease</a></li><li entry="h5"><a href="#local-maximum%20and%20minimum%20values:">Local Maximum and Minimum Values:</a></li><li entry="h4"><a href="#the-second%20derivative">The Second Derivative</a></li><li entry="h5"><a href="#concavity-and%20points%20of%20inflection">Concavity and Points of Inflection</a></li><li entry="h4"><a href="#sketch-the%20curve">Sketch the Curve</a></li><li entry="h3"><a href="#miscellaneous">Miscellaneous</a></li><li entry="h4"><a href="#first-and%20second%20derivative%20for%20curve%20shape">First and second derivative for curve shape</a></li><li entry="h2"><a href="#3.6-%7C%20calculus%20with%20graphing%20calculators%20(skipped)">3.6 | Calculus with Graphing Calculators (Skipped)</a></li><li entry="h2"><a href="#3.7-%7C%20optimization%20problems">3.7 | Optimization Problems</a></li><li entry="h3"><a href="#first-derivative%20test%20for%20absolute%20extreme%20values">First Derivative Test for Absolute Extreme Values</a></li><li entry="h2"><a href="#3.8-%7C%20newtons%20method">3.8 | Newtons Method</a></li><li entry="h2"><a href="#3.9-%7C%20anti-derivatives">3.9 | Anti-Derivatives</a></li></ul></div>
</header>
<h1 id="chapter-3%20%7C%20applications%20of%20differentiation">Chapter 3 | Applications of Differentiation </h1>
<h2 id="3.1-%7C%20maximum%20and%20minimum%20values">3.1 | Maximum and Minimum Values</h2>
<section class="note-block block"><p>An **extremum** (or extreme value) of a function is a point at which a **maximum** or **minimum** value of the function is obtained in some interval.</p></section>
<p>Consider Figure 3.1.1:</p>
<div img-row="">
<img src="https://colbyn.github.io/school-notes/3.1.1a.995a7351.png">
<img src="https://colbyn.github.io/school-notes/3.1.1b.f0454ea7.png">
<img src="https://colbyn.github.io/school-notes/3.1.1c.06a430ed.png">
</div>
<ul>
<li>Figure (a) has a maximum, but no minimum, as the interval over which the function is defined is open.</li>
<li>Figure (b), the function has a minimum, but no maximum; there is a discontinuity in the "natural'' place for the maximum to occur.</li>
<li>Figure (c) has both a maximum and a minimum; note that the function is continuous and the interval on which it is defined is closed.</li>
</ul>
<h3 id="the-extreme%20value%20theorem">The extreme Value Theorem</h3>
<section class="note-block block"><h3 id="the-extreme%20value%20theorem">The extreme Value Theorem</h3><p>
If <span inline="" math-inline="">\(f\)</span> is continuous on a closed interval <span inline="" math-inline="">\([a, b]\)</span>, then <span inline="" math-inline="">\(f\)</span> attains an absolute maximum value stem:[f(c)] and an absolute minimum value <span inline="" math-inline="">\(f(d)\)</span> at some numbers <span inline="" math-inline="">\(c\)</span> and <span inline="" math-inline="">\(d\)</span> in <span inline="" math-inline="">\([a, b]\)</span>.
</p><section class="note-block block"><p>
Functions continuous on a closed interval always attain extreme values.
</p></section></section>
<h3 id="fermat%E2%80%99s-theorem">Fermat’s Theorem</h3>
<section class="note-block block"><h3 id="fermat%E2%80%99s-theorem">Fermat’s Theorem</h3><p>
If <span inline="" math-inline="">\(f\)</span> has a local maximum or minimum at <span inline="" math-inline="">\(c\)</span>, and if <span inline="" math-inline="">\(f^\prime(c)\)</span> exists, then <span inline="" math-inline="">\(f^\prime(c) = 0\)</span>.
</p></section>
<p><b>In terms of critical numbers, Fermat’s Theorem can be rephrased as follows:</b></p>
<section class="note-block block"><h3 id="fermat%E2%80%99s-theorem%20(in%20terms%20of%20critical%20numbers)">Fermat’s Theorem (in terms of critical numbers)</h3><p>
If f has a local maximum or minimum at c, then c is a critical number of f.
</p></section>
<h3 id="critical-number">Critical Number</h3>
<p>A critical number of a function <span inline="" math-inline="">\(f\)</span> is a number <span inline="" math-inline="">\(c\)</span> in the domain of <span inline="" math-inline="">\(f\)</span> such that either <texf^\prime(c) 0="" ==""> or <span inline="" math-inline="">\(f^\prime(c)\)</span> does not exist.</texf^\prime(c)></p>
<section class="note-block block"><p>
* *Critical numbers* of <span inline="" math-inline="">\(f\)</span> occur when <span inline="" math-inline="">\(f^\prime(c) = 0\)</span>, or when <span inline="" math-inline="">\(f^\prime(c) = \mathrm{undefined}\)</span>.
* Not every critical number gives rise to a maximum or a minimum.
</p></section>
<p>To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number by (7)] or it occurs at an endpoint of the interval, as we see from the examples in Figure 8. Thus the following three-step procedure always works. **See The Closed Interval Method.**</p>
<h3 id="the-closed%20interval%20method">The Closed Interval Method</h3>
<section class="note-block block"><h3 id="the-closed%20interval%20method">The Closed Interval Method</h3><p>
To find the absolute maximum and minimum values of a continuous function <span inline="" math-inline="">\(f\)</span> on a closed interval <span inline="" math-inline="">\([a, b]\)</span>:
</p><ul>
<li>Find the values of <span inline="" math-inline="">\(f\)</span> at the critical numbers of <span inline="" math-inline="">\(f\)</span> in <span inline="" math-inline="">\((a, b)\)</span>. I.e. *critical numbers* of <span inline="" math-inline="">\(f\)</span> occur when <span inline="" math-inline="">\(f^\prime(c) = 0\)</span>, or when stem:[f^\prime(c) = \mathrm{undefined}]</li>
<li>Find the values of <span inline="" math-inline="">\(f\)</span> at the endpoints of the interval. I.e. <span inline="" math-inline="">\(f(a)\)</span> and <span inline="" math-inline="">\(f(b)\)</span>.</li>
<li>The **largest** of the values from **Steps 1 and 2** is the absolute maximum value; the **smallest** of these values is the absolute minimum value.</li>
</ul></section>
<p>I.e. the maximum/minimum of <span inline="" math-inline="">\(f^\prime(c) = 0\)</span>, <span inline="" math-inline="">\(f(a)\)</span>, and <span inline="" math-inline="">\(f(b)\)</span>.</p>
<p>Using sudo code, the 'The Closed Interval Method' can be defined as the result of:</p>
<div block="" math-block="">$$\begin{equation}
\begin{split}
\mathrm{maximum} &= \mathrm{max}\lbrack f(a), f^\prime(c) = 0, f(b)\rbrack \\
\mathrm{minimum} &= \mathrm{min}\lbrack f(a), f^\prime(c) = 0, f(b)\rbrack
\end{split}
\end{equation}$$</div>
<h2 id="3.2-%7C%20the%20mean%20value%20theorem">3.2 | The Mean Value Theorem</h2>
<ul>
<li><a href="https://www.dpmms.cam.ac.uk/~wtg10/meanvalue.html">What is the point of the mean value theorem?</a></li>
<li>The mean value theorem says that the average speed of the car (the slope of the secant line) is equal to the instantaneous speed (slope of the tangent line) at some point(s) in the interval.</li>
</ul>
<blockquote>
In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.
</blockquote>
<p>For any function that is continuous <span inline="" math-inline="">\([a, b]\)</span> and differentiable on <span inline="" math-inline="">\((a,b)\)</span>, there exists some <span inline="" math-inline="">\(c\)</span> in the interval <span inline="" math-inline="">\((a,b)\)</span> such that the secant joining the endpoints of the <span inline="" math-inline="">\([a, b]\)</span> is parallel to the tangent at <span inline="" math-inline="">\(c\)</span>:</p>
<img block="" style="max-width: 300px;" src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Mvt2.svg/520px-Mvt2.svg.png">
<section class="note-block block"><p>
The function f attains the slope of the secant between a and b as the derivative at the point <span inline="" math-inline="">\({\displaystyle \xi \in (a,b)}\)</span>:
</p><img block="" style="max-width: 300px;" src="https://upload.wikimedia.org/wikipedia/commons/9/94/Mittelwertsatz3.svg"><p>
It is also possible that there are multiple tangents parallel to the secant:
</p><img block="" style="max-width: 300px;" src="https://upload.wikimedia.org/wikipedia/commons/f/f9/Mittelwertsatz6.svg"></section>
<p>We will see that many of the results of this chapter depend on one central fact, which is called the Mean Value Theorem. But to arrive at the Mean Value Theorem we first need the following result.</p>
<section class="note-block block"><h3 id="rolle%E2%80%99s-theorem">Rolle’s Theorem</h3><p>
Let f be a function that satisfies the following three hypotheses:
</p><ol>
<li><span inline="" math-inline="">\(f\)</span> is continuous on the closed interval <span inline="" math-inline="">\([a, b]\)</span>.</li>
<li><span inline="" math-inline="">\(f\)</span> is differentiable on the open interval <span inline="" math-inline="">\((a, b)\)</span>.</li>
<li><span inline="" math-inline="">\(f(a) = f(b)\)</span><br>I.e. the x_1 and x_2 map to the same y value.</li>
</ol><p>
Then there is a number <span inline="" math-inline="">\(c\)</span> in <span inline="" math-inline="">\((a, b)\)</span>, such that </p><texf^\prime(c) 0="" =="">
</texf^\prime(c)></section>
<section class="note-block block"><h3 id="the-mean%20value%20theorem">The Mean Value Theorem</h3><p>
Let <span inline="" math-inline="">\(f\)</span> be a function that satisfies the following hypotheses:
</p><ol>
<li><span inline="" math-inline="">\(f\)</span> is continuous on the closed interval <span inline="" math-inline="">\([a, b]\)</span>.</li>
<li><span inline="" math-inline="">\(f\)</span> is differentiable on the open interval <span inline="" math-inline="">\((a, b)\)</span>.</li>
</ol><p>
Then there is a number <span inline="" math-inline="">\(c\)</span> in <span inline="" math-inline="">\((a, b)\)</span>, such that:
<span inline="" math-inline="">\(\begin{equation}
\begin{split}
f^\prime(c) &= \frac{f(b) - f(a)}{b - a}
\end{split}
\end{equation}\)</span>
Or, equivalently:
<span inline="" math-inline="">\(\begin{equation}
\begin{split}
f(b) - f(a) &= f^\prime(c)(b - a)
\end{split}
\end{equation}\)</span>
Where the tangent at <span inline="" math-inline="">\(c\)</span> is parallel to the secant line through the endpoints <span inline="" math-inline="">\((a, f(a))\)</span> and stem:[(b, f(b))].
</p></section>
<section class="note-block block"><p>
The Mean Value Theorem is an example of what is called an existence theorem. Like the Intermediate Value Theorem, the Extreme Value Theorem, and Rolle’s Theorem, it guarantees that there exists a number with a certain property, but it doesn’t tell us how to find the number.
</p></section>
<h3 id="miscellaneous">Miscellaneous</h3>
<section class="note-block block"><h3 id="theorem-(%C2%A73.2.5)">Theorem (§3.2.5)</h3><p>
If stem:[f^\prime(x) = 0] for all x in an interval <span inline="" math-inline="">\((a, b)\)</span>, then f is constant on <span inline="" math-inline="">\((a, b)\)</span>.
</p></section>
<section class="note-block block"><h3 id="corollary-(%C2%A73.2.7)">Corollary (§3.2.7)</h3><p>
if stem:[f^\prime(x) = g^\prime(x)] for all stem:[x] in an interval <span inline="" math-inline="">\((a, b)\)</span>, then stem:[f - g] is constant on <span inline="" math-inline="">\((a, b)\)</span>; that is, stem:[f(x) = g(x) + c] where <span inline="" math-inline="">\(c\)</span> is a constant.
</p><section class="note-block block"><p>
Corollary 7 says that if two functions have the same derivatives on an interval, then their graphs must be vertical translations of each other there. In other words, the graphs have the same shape, but could be shifted up or down.
</p></section></section>
<h2 id="3.3-%7C%20how%20derivatives%20affect%20the%20shape%20of%20a%20graph">3.3 | How Derivatives Affect the Shape of a Graph</h2>
<p>Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives.
</p>
<p>Because stem:[f^\prime(x)] represents the slope of the curve stem:[y = f(x)] at the point stem:[(x, f(x))], it tells us the direction in which the curve proceeds at each point. So it is reasonable to expect that information about stem:[f^\prime(x)] will provide us with information about stem:[f(x)].</p>
<h3 id="what-does%20stem:%5Bf%5E%5Cprime%5D%20say%20about-f-?">What Does stem:[f^\prime] Say about <span inline="" math-inline="">\(f\)</span>?</h3>
<section class="note-block block"><h3 id="increasing/decreasing-(i/d)%20test">Increasing/Decreasing (I/D) Test</h3><ul>
<li>
<p>If <span inline="" math-inline="">\(f^\prime(x) > 0\)</span> on an interval, then <span inline="" math-inline="">\(f(x)\)</span> is increasing on that interval.</p>
<small>I.e. the tangent lines (in the <span inline="" math-inline="">\(f(x)\)</span> interval) have positive slope.<small>
</small></small></li>
<li>
<p>If <span inline="" math-inline="">\(f^\prime(x) < 0\)</span> on an interval, then <span inline="" math-inline="">\(f(x)\)</span> is decreasing on that interval.</p>
<small>I.e. the tangent lines (in the <span inline="" math-inline="">\(f(x)\)</span> interval) have negative slope.</small>
</li>
</ul></section>
<section class="note-block block"><h3 id="the-first%20derivative%20test">The First Derivative Test</h3><p>
Suppose that c is a critical number of a continuous function f:
</p><ul>
<li>If <span inline="" math-inline="">\(f^\prime\)</span> changes from <b>positive to negative</b> at <span inline="" math-inline="">\(c\)</span><sup>†</sup>, then <span inline="" math-inline="">\(f\)</span> has a local <b>maximum</b> at <span inline="" math-inline="">\(c\)</span>.</li>
<li>If <span inline="" math-inline="">\(f^\prime\)</span> changes from <b>negative to positive</b> at <span inline="" math-inline="">\(c\)</span>, then <span inline="" math-inline="">\(f\)</span> has a local <b>minimum</b> at <span inline="" math-inline="">\(c\)</span>.</li>
<li>If <span inline="" math-inline="">\(f^\prime\)</span> is <b>positive to the left and right</b> of <span inline="" math-inline="">\(c\)</span>, or <b>negative to the left and right</b> of <span inline="" math-inline="">\(c\)</span>, then <span inline="" math-inline="">\(f\)</span> has <b>no local maximum or minimum</b> at <span inline="" math-inline="">\(c\)</span>.</li>
</ul><section class="note-block block"><p>[†]: 'at <span inline="" math-inline="">\(c\)</span>' means from the left and right of the given point.</p></section></section>
<h3 id="what-does-f%5E%7B%5Cprime%5Cprime%7D-say%20about-f-?">What Does <span inline="" math-inline="">\(f^{\prime\prime}\)</span> Say about <span inline="" math-inline="">\(f\)</span>?</h3>
<section class="note-block block"><p>
If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.
</p></section>
<section class="note-block block"><h3 id="concavity-test">Concavity Test</h3><ul>
<li>If <span inline="" math-inline="">\(f^{\prime\prime}(x) > 0\)</span> for all x in I, then the graph of <span inline="" math-inline="">\(f\)</span> is concave upward on I.</li>
<li>If <span inline="" math-inline="">\(f^{\prime\prime}(x) < 0\)</span> for all x in I, then the graph of <span inline="" math-inline="">\(f\)</span> is concave downward on I.</li>
</ul><div style="display: flex; justify-content: space-around;">
<img block="" style="max-width: 400px;" src="https://qph.fs.quoracdn.net/main-qimg-4e6db5cc9fb301df3daae9f6c99ccb22">
</div><h4 id="concavity-test%20(alternative)">Concavity Test (Alternative)</h4><p>
For a function <span inline="" math-inline="">\(f(x)\)</span> with derivatives <span inline="" math-inline="">\(f^\prime\)</span> and <span inline="" math-inline="">\(f^{\prime\prime}\)</span> on an interval the following holds:
</p><ul>
<li><span inline="" math-inline="">\(f\)</span> is concave up ⇔ <span inline="" math-inline="">\(f^\prime\)</span> is increasing ⇔ <span inline="" math-inline="">\(f^{\prime\prime}(x)\)</span> is positive</li>
<li><span inline="" math-inline="">\(f\)</span> is concave down ⇔ <span inline="" math-inline="">\(f^\prime\)</span> is decreasing ⇔ <span inline="" math-inline="">\(f^{\prime\prime}(x)\)</span> is negative</li>
</ul></section>
<section class="note-block block"><h3 id="regarding-concavity%20and%20extrema">Regarding Concavity and Extrema</h3><h4 id="the-following%20three%20scenarios%20are%20possible:">The following three scenarios are possible:</h4><section class="note-block block"><h5 id="the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20changing%20from%20negative%20to%20positive">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is changing from negative to positive</h5><p>This means that the function is concave down before <span inline="" math-inline="">\(c\)</span>, concave up after <span inline="" math-inline="">\(c\)</span> and has an inflection point at <span inline="" math-inline="">\(x = c\)</span>.</p><p>For Example:</p><div style="display: flex; justify-content: space-around;">
<img block="" width="300px" src="https://colbyn.github.io/school-notes/image--014.6406b545.png" style="min-width: 0; max-width: 300px;">
</div></section><section class="note-block block"><h5 id="the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20changing%20from%20positive%20to%20negative.">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is changing from positive to negative.</h5><p>This means that the function is concave up before <span inline="" math-inline="">\(c\)</span>, concave down after <span inline="" math-inline="">\(c\)</span>, and has an inflection point at <span inline="" math-inline="">\(x = c\)</span>.</p><p>For Example:</p><div style="display: flex; justify-content: space-around;">
<img block="" width="300px" src="https://colbyn.github.io/school-notes/image--016.d1051dcb.png" style="min-width: 0; max-width: 300px;">
</div></section><section class="note-block block"><h5 id="the-sign%20of-f%5E%7B%5Cprime%5Cprime%7D-is%20not%20changing%20at-x%20=%20c">The sign of <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is not changing at <span inline="" math-inline="">\(x = c\)</span></h5><p>(I.e. it is either positive both before or after <span inline="" math-inline="">\(c\)</span> or negative both before or after <span inline="" math-inline="">\(c\)</span>)</p><p>In this case, f does not have an inflection point at <span inline="" math-inline="">\(x = c\)</span>.</p><p>For Example:</p><div style="display: flex; justify-content: space-around;">
<img block="" width="300px" src="https://colbyn.github.io/school-notes/image--018.99c84033.png" style="min-width: 0; max-width: 300px;">
<img block="" width="300px" src="https://colbyn.github.io/school-notes/image--020.f0572c05.png" style="min-width: 0; max-width: 300px;">
</div><section class="note-block block"><p>
The existence of the third case demonstrates that a function does not necessarily have an inflection point at a critical point of <span inline="" math-inline="">\(f^\prime\)</span>.
</p></section></section><p>Using the number line test for <span inline="" math-inline="">\(f^{\prime\prime}\)</span> one can both determine the intervals on which <span inline="" math-inline="">\(f\)</span> is concave up/down as well as classify the critical point of <span inline="" math-inline="">\(f^{\prime}\)</span> into three categories matching the three cases above and determine the inflection points.</p><section class="note-block block"><h4 id="the-inflection%20points%20test">The Inflection Points Test</h4><p>
To determine the inflection points a differentiable function <span inline="" math-inline="">\(f(x)\)</span></p><ol>
<li>Find <span inline="" math-inline="">\(f^{\prime\prime}\)</span>.</li>
<li>Set it to zero and find all the critical points of <span inline="" math-inline="">\(f^{\prime}(x)\)</span></li>
<li>
<span>Use the number line to classify the critical points of <span inline="" math-inline="">\(f^{\prime}\)</span> into the three cases</span>
<ul>
<li>if <span inline="" math-inline="">\(f^{\prime\prime}\)</span> changes sign at <span inline="" math-inline="">\(c\)</span>, <span inline="" math-inline="">\(f\)</span> has an inflection point at c</li>
<li>if <span inline="" math-inline="">\(f^{\prime\prime}\)</span> does not change the sign at <span inline="" math-inline="">\(c\)</span>, <span inline="" math-inline="">\(f\)</span> does not have an inflection point at <span inline="" math-inline="">\(c\)</span></li>
</ul>
</li>
</ol></section><section class="note-block block"><h4 id="what-f%5E%7B%5Cprime%5Cprime%7D-says-about-f%5E%7B%5Cprime%7D-(alternative)">What <span inline="" math-inline="">\(f^{\prime\prime}\)</span> says about <span inline="" math-inline="">\(f^{\prime}\)</span> <small>(Alternative)</small></h4><p>Concavity of the function can also be used to determine if there is an <b>extreme value</b> or not at a <b>critical point</b> of <span inline="" math-inline="">\(f\)</span>.</p><ul>
<li>a function with a relative <b>minimum</b> is <b>concave up</b> on an interval around it</li>
<li>a function with a relative <b>maximum</b> is <b>concave down</b> on an interval around it</li>
</ul><p>
Remember:
</p><div style="display: flex; justify-content: space-around;">
<img block="" style="max-width: 400px;" src="https://qph.fs.quoracdn.net/main-qimg-4e6db5cc9fb301df3daae9f6c99ccb22">
</div><section class="note-block block"><p>Thus, if <span inline="" math-inline="">\(c\)</span> is a critical point and the second derivative at <span inline="" math-inline="">\(c\)</span> is <b>positive</b>, that means that the function is <b>concave up</b> around <span inline="" math-inline="">\(c\)</span>. Thus, there is a relative <b>minimum</b> at <span inline="" math-inline="">\(c\)</span>.</p><p>Conversely, if <span inline="" math-inline="">\(c\)</span> is a critical point and the second derivative at <span inline="" math-inline="">\(c\)</span> is <b>negative</b>, that means that the function is <b>concave down</b> around <span inline="" math-inline="">\(c\)</span>. Thus, there is a relative <b>maximum</b> at <span inline="" math-inline="">\(c\)</span>.</p><section class="note-block block"><p>If <span inline="" math-inline="">\(f^{\prime\prime} = 0\)</span> this test is inconclusive (i.e. probably an inflection point).</p></section><div style="display: flex; justify-content: space-around;">
<img block="" style="max-width: 400px;" src="https://colbyn.github.io/school-notes/image--030.2a04d47c.png">
</div><p><small>This procedure of determining the extreme values is known as the Second Derivative Test.</small></p></section></section></section>
<section class="note-block block"><h3 id="inflection-points">Inflection Points</h3><section class="note-block block"><p>When the second derivative is zero, it corresponds to a possible inflection point.</p></section><p>
A point stem:[P] on a curve stem:[y = f(x)] is called an inflection point if <span inline="" math-inline="">\(f\)</span> is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at stem:[P].
</p><h3 id="inflection-points%20(alternative)">Inflection Points (Alternative)</h3><section class="note-block block"><p>
If <span inline="" math-inline="">\(f(x)\)</span> has an inflection point at <span inline="" math-inline="">\(x = c\)</span>, then <span inline="" math-inline="">\(f^{\prime\prime}(c) = 0\)</span> or <span inline="" math-inline="">\(f^{\prime\prime}(c)\)</span> does not exist.
</p></section><section class="note-block block"><p>
Example: Determine the intervals on which the function with the graph on the right defined on interval (a, ∞) is concave up/down.
</p><div style="display: flex; justify-content: space-around;">
<img block="" width="300px" src="https://colbyn.github.io/school-notes/image--012.8dd6b9cc.png" style="min-width: 0; max-width: 300px;">
</div><section class="note-block block"><p>Solution: The function is concave up on the interval (a,b) and concave down on the interval (b, ∞).</p></section></section><h3 id="miscellaneous">Miscellaneous</h3><section class="note-block block"><p>Quadratic functions have no points of inflection:</p><div style="display: flex; justify-content: space-around;">
<img style="width: 200px;" src="https://colbyn.github.io/school-notes/x-square.35cca5f4.png">
</div></section></section>
<section class="note-block block"><h3 id="the-second%20derivative%20test">The Second Derivative Test</h3><p>
Suppose <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is continuous near c:
</p><ul>
<li>If <texf^\prime(c) 0="" ==""> and stem:[f^{\prime\prime}(c) > 0], then <span inline="" math-inline="">\(f\)</span> has a local minimum at <span inline="" math-inline="">\(c\)</span>.</texf^\prime(c)></li>
<li>If <texf^\prime(c) 0="" ==""> and stem:[f^{\prime\prime}(c) < 0], then <span inline="" math-inline="">\(f\)</span> has a local maximum at <span inline="" math-inline="">\(c\)</span>.</texf^\prime(c)></li>
</ul><section class="note-block block"><p>
The Second Derivative Test is inconclusive when stem:[f^{\prime\prime}(c) = 0]. In other words, at such a point there might be a maximum, there might be a minimum, or there might be neither (as in Example 6). This test also fails when stem:[f^{\prime\prime}(c)] does not exist. In such cases the First Derivative Test must be used. In fact, even when both tests apply, the First Derivative Test is often the easier one to use.
</p></section><section class="note-block block"><p>
The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which <span inline="" math-inline="">\(f^\prime(x) = 0\)</span> and the second derivative is positive at this point, then <span inline="" math-inline="">\(f\)</span> has a local minimum here.
</p></section></section>
<h2 id="horizontal-asymptotes-section">3.4 | Limits at Infinity; Horizontal Asymptotes</h2>
<section class="note-block block"><div block="" math-block="">$$\begin{equation}
\begin{split}
\lim_{x \to \infty} f(x) &= \lim_{x \to \infty^{+}} f(x) = A \\
\lim_{x \to -\infty} f(x) &= B
\end{split}
\end{equation}$$</div><p>
Which makes sense, because <span inline="" math-inline="">\(-\infty \not\equiv \infty\)</span>. Since, in the case of horizontal asymptotes:
</p><ul>
<li><span inline="" math-inline="">\(\infty\)</span> implies going to the <em>right</em></li>
<li><span inline="" math-inline="">\(-\infty\)</span> implies going to the <em>left</em></li>
</ul><p>
Or rather is unbounded in the given direction.
</p></section>
<section class="note-block block"><p>
**The professor said:** you can't just apply the limit laws,
such as for:
</p><div block="" math-block="">$$\lim_{x \to \infty} \frac{x^2 + 3x + 5}{x^3 + 3x + 5}$$</div><p>
Because we don't know if it is continuous.
</p></section>
<section class="note-block block"><h3 id="miscellaneous">Miscellaneous</h3><section class="note-block block"><ul>
<div block="" math-block="">$$\infty \cdot -\infty = -\infty$$</div>
<div block="" math-block="">$$\lim_{x \to 0} \left( \frac{\sin(x)}{x} \right) = 1$$</div>
<div block="" math-block="">$$\lim_{x \to \infty} \left( \frac{\sin(x)}{x} \right) = 0$$</div>
<div block="" math-block="">$$\lim_{x \to \infty} \sin(x) = 0$$</div>
</ul></section><h4 id="%5Clim_%7Bx-%5Cto%20%5Cinfty%7D%20%5Csin%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)"><span inline="" math-inline="">\(\lim_{x \to \infty} \sin\left(\frac{1}{x}\right)\)</span></h4><div block="" math-block="">$$\begin{equation}
\begin{split}
-1 \leq \sin \left( \frac{1}{x} \right) \leq 1 \\
-\frac{1}{\frac{1}{|x|}} \leq \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} \leq \frac{1}{\frac{1}{|x|}} \\
-|x| \leq \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} \leq |x|
\end{split}
\end{equation}$$</div><p><span inline="" math-inline="">\(\)</span></p></section>
<section class="note-block block"><h3 id="theorem-(3.4)%20#4">Theorem (3.4) #4</h3><div block="" math-block="">$$ \begin{equation}
\begin{split}
\lim_{x \to \infty} \left( \frac{1}{x^{\mathbb{R}}} \right) &= 0 &\;\; \text{if} &\; \mathbb{R} > 0 \\
\lim_{x \to -\infty} \left( \frac{1}{x^{\mathbb{R}}} \right) &= 0 &\;\; \text{if} &\; \mathbb{R} > 0
\end{split}
\end{equation} $$</div></section>
<h2 id="3.5-%7C%20summary%20curve%20sketching">3.5 | Summary Curve Sketching</h2>
<section class="note-block block"><p>For e.g. horizontal asymptotes, don't forgot to add <span inline="" math-inline="">\(y = x\)</span>, don't just say <span inline="" math-inline="">\(x\)</span></p></section>
<section class="note-block block"><h3 id="curve-sketching%20steps:">Curve Sketching Steps:</h3><ol>
<li>Domain</li>
<li>Intercepts</li>
<li>Symmetry</li>
<li>Asymptotes</li>
<li>The First Derivative</li>
<li>The Second Derivative</li>
<li>Sketch the Curve</li>
</ol><section no-title="" class="note-block block"><h4 id="domain">Domain</h4><section class="note-block block"><p>
Check that there is no division with 0, or even roots of negative numbers.
</p></section></section><section no-title="" class="note-block block"><h4 id="intercepts">Intercepts</h4><ol>
<li>Find the y-intercepts</li>
<li>Find the x-intercepts</li>
</ol></section><section no-title="" class="note-block block"><h4 id="symmetry">Symmetry</h4><p><img src="https://cdn.kastatic.org/googleusercontent/qoY4fIGSlPFLX3i61ama3uehSPpmWGLa_ksk11D-W8hFG_tezdylgH68l_kyjNwEa0CMaN4iIVkeMjMAPPmcK8LFgw"></p><section class="note-block block"><p>If <span inline="" math-inline="">\(f(-x) = f(x)\)</span> then <span inline="" math-inline="">\(f\)</span> is even and will be symmetric about the y-axis.</p></section><section class="note-block block"><p>
If <span inline="" math-inline="">\(f(x) = f(x + p)\)</span> where <span inline="" math-inline="">\(p\)</span> is a positive constant, then <span inline="" math-inline="">\(f\)</span> is a periodic function, and it's graph will be repeated every <span inline="" math-inline="">\(p\)</span> units (i.e. the period).
</p><section class="note-block block"><p><span inline="" math-inline="">\(p\)</span> is the period I believe, such as <span inline="" math-inline="">\(\pi\)</span></p></section></section></section><section no-title="" class="note-block block"><h4 id="asymptotes">Asymptotes</h4><section class="note-block block"><p>Asymptotes of e.g. rational functions can be found using the first and second derivatives, instead of using vertical and horizontal limits.</p></section><p>Steps:</p><ol>
<li>Horizontal Asymptotes</li>
<li>Vertical Asymptotes</li>
<li>Slant Asymptotes</li>
</ol><section no-title="" class="note-block block"><h5 id="vertical-asymptotes">Vertical Asymptotes</h5><p>
See <a href="https://colbyn.github.io/school-notes/calc/chapter1/index.html#vertical-asymptote-def">§1.5 | The Limit of a Function</a>.
</p><section class="note-block block"><p>Usually (but not always) this will involve checking to see where the denominator equals zero.</p></section><p>The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared).</p><p>For example, the following function has vertical asymptotes at <span inline="" math-inline="">\(x = 0\)</span>, and <span inline="" math-inline="">\(x = 1\)</span>, but not at <span inline="" math-inline="">\(x = 2\)</span>.</p><p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7135afaabb8d69e5fd041757ef7938d45609a6f7"></p></section><section no-title="" class="note-block block"><h5 id="horizontal-asymptotes">Horizontal Asymptotes</h5><p>
See <a href="#horizontal-asymptotes-section">3.4 | Limits at Infinity; Horizontal Asymptotes</a>.
</p><section class="note-block block"><h6 id="to-find%20horizontal%20asymptotes">To find horizontal asymptotes</h6><ul>
<li>If the degree of the <b>denominator</b> is <b>bigger</b> than the degree of the <b>numerator</b>, the horizontal asymptote is the x-axis (i.e. <span inline="" math-inline="">\(y = 0\)</span>).</li>
<li>If the degree of the <b>numerator</b> is <b>bigger</b> than the <b>denominator</b>, there is <b>no horizontal asymptote</b>.</li>
</ul></section></section><section no-title="" class="note-block block"><h5 id="slant-asymptotes">Slant Asymptotes</h5><section class="note-block block"><p>A slant (oblique) asymptote occurs when the polynomial in the <b>numerator</b> is a <b>higher degree</b> than the polynomial in the denominator.</p></section><p>
Can be found using polynomial long division, where the quotient <span inline="" math-inline="">\(mx + b\)</span> is your slant asymptotes, then set <span inline="" math-inline="">\(y = mx + b\)</span>.
</p><p>The above result can be confirmed using limits. Given some <span inline="" math-inline="">\(y = mx + b\)</span>, if:</p><div block="" math-block="">$$\lim_{x \to \infty} \left( f(x) - (mx + b)\right) = 0$$</div></section></section><section no-title="" class="note-block block"><h4 id="the-first%20derivative">The First Derivative</h4><section no-title="" class="note-block block"><h5 id="intervals-of%20increase%20or%20decrease">Intervals of Increase or Decrease</h5><p>
Find the intervals where:
</p><ul>
<li><span inline="" math-inline="">\(f^\prime(x)\)</span> is positive (<span inline="" math-inline="">\(f\)</span> is increasing)</li>
<li><span inline="" math-inline="">\(f^\prime(x)\)</span> is negative (<span inline="" math-inline="">\(f\)</span> is decreasing)</li>
</ul><section class="note-block block"><p>Use the I/D Test.</p></section></section><section no-title="" class="note-block block"><h5 id="local-maximum%20and%20minimum%20values:">Local Maximum and Minimum Values:</h5><ol>
<li>Find all critical numbers (numbers <span inline="" math-inline="">\(c\)</span> where <span inline="" math-inline="">\(f^\prime(c) = 0\)</span> or <span inline="" math-inline="">\(f^\prime(c)\)</span> does not exist)</li>
<li>Then use the First Derivative Test to check if these are local maxima or local minima.</li>
</ol></section></section><section no-title="" class="note-block block"><h4 id="the-second%20derivative">The Second Derivative</h4><section no-title="" class="note-block block"><h5 id="concavity-and%20points%20of%20inflection">Concavity and Points of Inflection</h5><p>
Find the intervals where:
</p><ul>
<li><span inline="" math-inline="">\(f^{\prime\prime}\)</span> is positive (<span inline="" math-inline="">\(f\)</span> is concave up)</li>
<li><span inline="" math-inline="">\(f^{\prime\prime}\)</span> is negative (f is concave down).</li>
</ul><p>
Check for points of inflection(points where <span inline="" math-inline="">\(f^{\prime\prime}\)</span> changes sign).
</p></section><section class="note-block block"><p>In some cases you can also use Second Derivative Test to test for local maximum and minimum values.</p></section></section><section no-title="" class="note-block block"><h4 id="sketch-the%20curve">Sketch the Curve</h4><p>
Sketch asymptotes as dashed lines. Plot intercepts, critical points, and inflection points. Draw a curve through these points which is consistent with the information found in the previous parts.
</p></section></section>
<h3 id="miscellaneous">Miscellaneous</h3>
<section class="note-block block"><h4 id="first-and%20second%20derivative%20for%20curve%20shape">First and second derivative for curve shape</h4><section class="note-block block"><p>For instance, given <span inline="" math-inline="">\(y = x^3\)</span>, the number line for the first and second derivatives will look like:</p><img block="" src="https://understandingcalculus.com/chapters/08/understanding_derivative_files/image121.gif"><p>On the negative (left) side, since <span inline="" math-inline="">\(f^\prime\)</span> is positive, and <span inline="" math-inline="">\(f^{\prime\prime}\)</span> is negative, it is increasing at a decreasing rate. So the curve will look like this</p><img block="" width="100px" src="https://colbyn.github.io/school-notes/image--002.370bad78.png" style="min-width: 0; max-width: 100px;"><p>Whereas on the positive (right) side of x, since the first derivative is positive and the second derivative is likewise positive, it will be increasing at an increasing rate. So therefore, the curve will look like this:</p><img block="" width="100px" src="https://colbyn.github.io/school-notes/image--000.dd643015.png" style="min-width: 0; max-width: 100px;"><p>Which matches the (compressed) curve of:</p><div width="200px" height="200px" lock="true" controls="false" block="" desmos="" style="
width: 100%;
max-width: unset;
margin: 0;
"><div></div>
<div id="des_423879a4_14aa_21f7_cb91_245b542ec462" style="width: 200px; height: 200px; margin: 0 auto;"></div>
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</div><section class="note-block block"><p>This process works for asymptotes as well, such as those from rational functions.</p></section></section></section>
<h2 id="3.6-%7C%20calculus%20with%20graphing%20calculators%20(skipped)">3.6 | Calculus with Graphing Calculators (Skipped)</h2>
<h2 id="3.7-%7C%20optimization%20problems">3.7 | Optimization Problems</h2>
<section class="note-block block"><h3 id="first-derivative%20test%20for%20absolute%20extreme%20values">First Derivative Test for Absolute Extreme Values</h3><p>
Suppose that <span inline="" math-inline="">\(c\)</span> is a critical number of a continuous function <span inline="" math-inline="">\(f\)</span> defined on an interval.
</p><section class="note-block block"><p>
If:
</p><ul>
<li>If <span inline="" math-inline="">\(f^\prime(x) > 0\)</span> for all <span inline="" math-inline="">\(x < c\)</span></li>
<li>If <span inline="" math-inline="">\(f^\prime(x) < 0\)</span> for all <span inline="" math-inline="">\(x > c\)</span></li>
</ul><p>
Then <span inline="" math-inline="">\(f(x)\)</span> is the absolute maximum value of <span inline="" math-inline="">\(f\)</span>.
</p></section><section class="note-block block"><p>
If:
</p><ul>
<li>If <span inline="" math-inline="">\(f^\prime(x) < 0\)</span> for all <span inline="" math-inline="">\(x < c\)</span></li>
<li>If <span inline="" math-inline="">\(f^\prime(x) > 0\)</span> for all <span inline="" math-inline="">\(x > c\)</span></li>
</ul><p>
Then <span inline="" math-inline="">\(f(x)\)</span> is the absolute minimum value of <span inline="" math-inline="">\(f\)</span>.
</p></section><section class="note-block block"><p>An alternative method for solving optimization problems is to use implicit differentiation.</p></section></section>
<h2 id="3.8-%7C%20newtons%20method">3.8 | Newtons Method</h2>
<h2 id="3.9-%7C%20anti-derivatives">3.9 | Anti-Derivatives</h2>
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