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<h2 class="hd hd-2 unit-title">1. Differentiating sine and cosine</h2>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<p>
polynomials<br/>powers of [mathjaxinline]x[/mathjaxinline] </p><p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Differentiate <span style="color:#99182C"><b class="bf">sine</b></span> and <span style="color:#99182C"><b class="bf">cosine</b></span>. </p></li><li><p>
Use sine and cosine to model oscillations. </p></li></ul><p><b class="bfseries">Contents: 12 pages</b></p><p>
6 videos (27 minutes 1x speed) 11 questions </p>
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<h2 class="hd hd-2 unit-title">3. Graphical motivation</h2>
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Graphical motivation
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Use the tangent line approximation tool to approximate the graph of [mathjaxinline]\frac{d}{dx}\sin (x)[/mathjaxinline]. </p>
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What does your graph for [mathjaxinline]\ f'(x)[/mathjaxinline] most resemble? </p>
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<h4 onclick="hideshow(this);" style="margin: 0px">Graph of cos(x)<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">The graph of [mathjaxinline]\cos x,[/mathjaxinline] in case you have forgotten, looks like this: <center><img alt="Graph of cosine of x from negative pi over 2 to 2 pi. Cosine is zero when x equals negative pi over 2, increases to a value of 1 at x equals 0, decreases to a value of negative 1 at x equals pi, and increases to a value of 1 at x equals 2 pi. Cosine equals zero when x is equal to any odd multiple of pi over 2, for example, negative pi over 2, pi over 2, and 3 pi over 2. Cosine has horizontal tangents when x equals 0, pi, and 2 pi." src="/assets/courseware/v1/8089681bbcd62c84db3a9067a57f5eb3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_cosine.svg" style="margin: 10px 25px 25px 25px" width="400px"/></center> </div>
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<text> [mathjaxinline]k\sin x[/mathjaxinline], for some positive [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\sin x[/mathjaxinline], for some negative [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\cos x[/mathjaxinline], for some positive [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\cos x[/mathjaxinline], for some negative [mathjaxinline]k[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">4. Intro continued</h2>
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The derivative of [mathjaxinline]\sin x[/mathjaxinline] sure looks a lot like some sort of cosine function. It's positive when [mathjaxinline]x = 0[/mathjaxinline], becomes negative past [mathjaxinline]x=\pi /2[/mathjaxinline], and then becomes positive again. Since the sine function repeats every [mathjaxinline]2\pi[/mathjaxinline], we would expect its derivative to repeat every [mathjaxinline]2\pi[/mathjaxinline] as well. Now let's try the cosine function. </p>
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Cosine
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Here's the graph of the cosine function. </p>
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<img alt="Graph of cosine of x from negative pi over 2 to 2 pi. Cosine is zero when x equals negative pi over 2, increases to a value of 1 at x equals 0, decreases to a value of negative 1 at x equals pi, and increases to a value of 1 at x equals 2 pi. Cosine equals zero when x is equal to any odd multiple of pi over 2, for example, negative pi over 2, pi over 2, and 3 pi over 2. Cosine has horizontal tangents when x equals 0, pi, and 2 pi." src="/assets/courseware/v1/8089681bbcd62c84db3a9067a57f5eb3/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_cosine.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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Which of these pictures would the graph for its derivative most resemble? </p>
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<img alt="Function starts at a negative value when x equals negative pi over 2. The function increases when x is between negative pi over 2 and pi over 2, decreases when x is between pi over 2 and pi, and increases from pi to 2 pi. Function has value 0 when x is equal to 0, pi, and 2 pi. Function has horizontal tangents when x equals negative pi over 2, pi over 2, and 3 pi over 2." src="/assets/courseware/v1/f35cb8e432479e3a5a54c52390590800/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_decoy1.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
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<img alt="Function starts at a positive value when x equals negative pi over 2. The function decreases when x is between negative pi over 2 and pi over 2, increases when x is between pi over 2 and pi, and decreases from pi to 2 pi. Function has value 0 when x is equal to 0, pi, and 2 pi. Function has horizontal tangents when x equals negative pi over 2, pi over 2, and 3 pi over 2." src="/assets/courseware/v1/10282e3f8f3f580a8fe2bc3afd04d567/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_decoy2.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
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<img alt="Function starts with value zero when x equals negative pi over 2. The function increases when x is between negative pi over 2 and 0, decreases when x is between 0 and pi, and increases from pi to 2 pi. Function has value 0 when x is equal to negative pi over 2, pi over 2, and 3 pi over 2. Function has horizontal tangents when x equals 0, pi, and 2 pi." src="/assets/courseware/v1/b85d64dccca85c72b6dd73f32d38a682/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_decoy3.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
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<img alt="Function starts with value zero when x equals negative pi over 2. The function decreases when x is between negative pi over 2 and 0, increases when x is between 0 and pi, and decreases from pi to 2 pi. Function has value 0 when x is equal to negative pi over 2, pi over 2, and 3 pi over 2. Function has horizontal tangents when x equals 0, pi, and 2 pi." src="/assets/courseware/v1/d7eb6ab9496d6321f1880e48757e1e81/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der6_decoy4.svg" style="margin: 10px 25px 25px 25px" width="200px"/>
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Cosine&#39;s derivative
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The correct graph above most resembles which of the following functions? </p>
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<text> [mathjaxinline]k\cos x[/mathjaxinline], for some positive [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\cos x[/mathjaxinline], for some negative [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\sin x[/mathjaxinline], for some positive [mathjaxinline]k[/mathjaxinline]</text>
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<text> [mathjaxinline]k\sin x[/mathjaxinline], for some negative [mathjaxinline]k[/mathjaxinline]</text>
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Derivative definition
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Let's start to do this mathematically. We don't have any tools other than the definition of the derivative, so we'll use that. Which expression is the best approximation for the derivative of [mathjaxinline]\sin x[/mathjaxinline] (where [mathjaxinline]\Delta x[/mathjaxinline] is small)? </p>
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<text> [mathjaxinline]\displaystyle \frac{\sin (x-\Delta x) - \sin x}{\Delta x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{\sin (x+\Delta x) - \sin x}{\Delta x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{\sin (x-\Delta x) + \sin (\Delta x)}{\Delta x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{\sin (x+\Delta x) + \sin (\Delta x)}{\Delta x}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">5. Derivative of sine</h2>
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Identify derivative of sine at zero
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Identify the expression that best approximates [mathjaxinline]\displaystyle \left.\frac{d \sin (x)}{dx}\right|_{x=0}[/mathjaxinline]. </p>
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<text> [mathjaxinline]\displaystyle \frac{1-\sin (\Delta x)}{\Delta x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{\sin (\Delta x) -1}{\Delta x}[/mathjaxinline]</text>
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Identify derivative of cosine at zero
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Identify the expression that best approximates [mathjaxinline]\displaystyle \left.\frac{d \cos (x)}{dx}\right|_{x=0}[/mathjaxinline]. </p>
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<text> [mathjaxinline]\displaystyle \frac{1-\cos (\Delta x)}{\Delta x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{\cos (\Delta x) -1}{\Delta x}[/mathjaxinline]</text>
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<p>
We found that </p><table id="a0000000115" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000116"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d \ \sin (x)}{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\quad \displaystyle \lim _{\Delta x \to 0} \left( \sin (x)\left(\frac{\cos (\Delta x) - 1}{\Delta x}\right) + \cos (x)\left( \frac{\sin (\Delta x)}{\Delta x}\right) \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000117"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad \sin (x)\left( \lim _{\Delta x \to 0} \frac{\cos (\Delta x) - 1}{\Delta x}\right) + \cos (x)\left(\lim _{\Delta x \to 0} \frac{\sin (\Delta x)}{\Delta x}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
We've identified these limits as the derivatives of [mathjaxinline]\cos[/mathjaxinline] and [mathjaxinline]\sin[/mathjaxinline] at zero respectively. This reduces our problem! To find the derivative of [mathjaxinline]\sin[/mathjaxinline] everywhere, it is enough to find the derivatives [mathjaxinline]\sin '(0)[/mathjaxinline] and [mathjaxinline]\cos '(0)[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">6. Derivative of sine at zero</h2>
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Identify limit
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What can we conclude about the limit [mathjaxinline]\displaystyle \lim _{\Delta x \rightarrow 0} \frac{\sin (\Delta x)}{\Delta x}[/mathjaxinline]? </p>
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<h3 class="hd hd-2">Derivative of sine at 0</h3>
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The main geometric idea is that when [mathjaxinline]\theta[/mathjaxinline] is small, [mathjaxinline]\sin (\theta )[/mathjaxinline] is approximately: </p>
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<h2 class="hd hd-2 unit-title">7. Derivative of cosine at zero</h2>
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Identify limit
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What can we conclude about the limit [mathjaxinline]\displaystyle \lim _{\Delta x \rightarrow 0} \frac{\cos (\Delta x)-1}{\Delta x}[/mathjaxinline]? </p>
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<p>
For an alternate proof that uses the value of the derivative of [mathjaxinline]\sin (\theta )[/mathjaxinline] at [mathjaxinline]\theta = 0[/mathjaxinline], see the proof below. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Alternate algebraic proof<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
We want to find the limit [mathjaxinline]\displaystyle \left.\frac{d}{d\theta }\cos \theta \right|_{\theta =0} = \lim _{\theta \rightarrow 0} \frac{\cos \theta -1}{\theta }[/mathjaxinline]. In the video, we actually compute the negative of this limit [mathjaxinline]\displaystyle \lim _{\theta \rightarrow 0} \frac{1-\cos \theta }{\theta }[/mathjaxinline]. Let's see another way to compute this limit algebraically using the limit of [mathjaxinline]\sin (\theta )[/mathjaxinline] at [mathjaxinline]\theta =0[/mathjaxinline] that we just found. </p><p>
By algebraic manipulation, we can write this argument of this limit in the following way: </p><table id="a0000000118" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000119"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{1- \cos \theta }{\theta }[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1- \cos \theta }{\theta } \cdot \frac{1+ \cos \theta }{1+ \cos \theta }[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.1)</td></tr><tr id="a0000000120"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1-\cos ^2 \theta }{\theta (1+ \cos \theta )}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.2)</td></tr><tr id="a0000000121"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\sin ^2 \theta }{\theta (1+ \cos \theta )}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.3)</td></tr><tr id="a0000000122"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\sin \theta }{\theta }\frac{\sin \theta }{1+ \cos \theta }[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.4)</td></tr></table><p>
Now take the limit as [mathjaxinline]\theta[/mathjaxinline] approaches [mathjaxinline]0[/mathjaxinline]: </p><table id="a0000000123" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000124"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{\theta \rightarrow 0} \left(\frac{\sin \theta }{\theta }\right)\left(\frac{\sin \theta }{1+ \cos \theta }\right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (1)\left(\frac{0}{2}\right) = 0.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.5)</td></tr></table><p>
Note that we computed [mathjaxinline]\displaystyle \lim _{\theta \rightarrow 0} \frac{\sin \theta }{\theta } =1[/mathjaxinline] using a geometric argument. And now we use the fact that [mathjaxinline]\displaystyle \lim _{\theta \rightarrow 0} \frac{\sin \theta }{1+\cos \theta } = \frac{0}{2} = 0[/mathjaxinline]. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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<h2 class="hd hd-2 unit-title">8. Derivative definition of cosine</h2>
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Which of these expressions is the derivative of [mathjaxinline]\cos x[/mathjaxinline]? (Hint: Apply the cosine sum angle formula to [mathjaxinline]\cos (x + \Delta x)[/mathjaxinline].) </p>
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<text> [mathjaxinline]\sin (x)\sin '(0)-\cos (x)\cos '(0)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">9. Practice derivatives</h2>
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<p>
The number of <em>significant figures</em> is the count of those digits that carry meaning with regards to precision. </p><p><b class="bfseries">Examples</b></p><ul class="itemize"><li><p>
All non-zero digits are significant – 1235 has 4 significant digits. </p></li><li><p>
Zeros appearing between nonzero digits are significant – 101 has 3 significant digits. </p></li><li><p>
Trailing zeros in a number containing a decimal are significant – 32.000 has 5 significant figures. </p></li></ul><p><b class="bfseries">Non-examples</b></p><ul class="itemize"><li><p>
Trailing zeros in a number with no decimal are <em>not</em> significant – 5400 has 2 significant figures. </p></li><li><p>
Leading zeros in a decimal number are not significant – 0.0003 has 1 significant figure. </p></li><li><p>
Extraneous digits introduced in a computation to greater precision than measured data are <em>not</em> significant – if .25 and .50 are measurements accurate to [mathjaxinline]\pm[/mathjaxinline].01, then in the product (.25)(.50) = 0.125 the last 5 is <em>not</em> significant. </p></li></ul>
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<h2 class="hd hd-2 unit-title">10. Second derivatives</h2>
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What is the second derivative of [mathjaxinline]\ f(x) = \sin x[/mathjaxinline]? </p>
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Second derivative of sine and concavity
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<p><b class="bf">A.</b> Given that [mathjaxinline]\ f^{\prime \prime }(x) = -\sin x[/mathjaxinline], in which of these intervals is the graph of [mathjaxinline]\ f(x)[/mathjaxinline] concave up? </p>
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<text>[mathjaxinline]\left(0, \frac{\pi }{2} \right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\left(\pi , 2\pi \right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\left(-\frac{\pi }{2}, \frac{\pi }{2} \right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\left(-\pi , -\frac{\pi }{2} \right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\left(3\pi , 4\pi \right)[/mathjaxinline]</text>
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<p><b class="bf">B.</b> True or False: When [mathjaxinline]\sin (x)&gt;0[/mathjaxinline], the graph of [mathjaxinline]\ f[/mathjaxinline] is concave down. </p>
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<h2 class="hd hd-2 unit-title">11. Summary</h2>
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<p><b class="bfseries">Definition of significant figures</b></p><p>
The number of <em>significant figures</em> is the count of those digits that carry meaning with regards to precision. </p><p><b class="bfseries">Examples</b></p><ul class="itemize"><li><p>
All non-zero digits are significant – 1235 has 4 significant digits. </p></li><li><p>
Zeros appearing between nonzero digits are significant – 101 has 3 significant digits. </p></li><li><p>
Trailing zeros in a number containing a decimal are significant – 32.000 has 5 significant figures. </p></li></ul><p><b class="bfseries">Non-examples</b></p><ul class="itemize"><li><p>
Trailing zeros in a number with no decimal are <em>not</em> significant – 5400 has 2 significant figures. </p></li><li><p>
Leading zeros in a decimal number are not significant – 0.0003 has 1 significant figure. </p></li><li><p>
Extraneous digits introduced in a computation to greater precision than measured data are <em>not</em> significant – if .25 and .50 are measurements accurate to [mathjaxinline]\pm[/mathjaxinline].01, then in the product (.25)(.50) = 0.125 the last 5 is <em>not</em> significant. </p></li></ul><p><b class="bfseries">Derivative of sine and cosine</b></p><p>
The derivative of of the trig functions are: </p><table id="a0000000129" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000130"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \sin (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \cos (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.6)</td></tr><tr id="a0000000131"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \cos (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\sin (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.7)</td></tr><tr id="a0000000132"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d^2}{dx^2} \sin (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\sin (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.8)</td></tr><tr id="a0000000133"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d^2}{dx^2} \cos (x)[/mathjaxinline]
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[mathjaxinline]\displaystyle = -\cos (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(3.9)</td></tr></table>
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