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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Exponential Functions and their Derivatives</h3>
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<h2 class="hd hd-2 unit-title">2. Exponential functions and their derivatives</h2>
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<p><b class="bfseries">What functions can we differentiate so far?</b></p><p>
powers of [mathjaxinline]x[/mathjaxinline]<br/>polynomials<br/>trigonometric functions <br/>products of basic functions<br/>quotients of basic functions <br/>compositions of basic functions<br/>inverses of basic functions<br/></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 the <span style="color:#27408C"><b class="bf">exponential</b></span> function. </p></li><li><p>
Define the number [mathjaxinline]e[/mathjaxinline] in terms of the value of the derivative [mathjaxinline]\displaystyle \left.\frac{d}{dx}e^ x\right|_{x=0}[/mathjaxinline]. </p></li></ul><p><b class="bfseries">Contents: 9 pages</b></p><p>
5 videos (21 minutes 1x speed) 10 questions </p>
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<h2 class="hd hd-2 unit-title">3. Properties of exponents</h2>
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<p>
Let [mathjaxinline]a[/mathjaxinline] be a positive real number. </p><ul class="itemize"><li><p>
[mathjaxinline]a^0=1[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^1=a[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^{m}a^{n} = a^{m+n}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\left(a^ m\right)^{n} = a^{mn}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^{m/n} = \sqrt [n]{a^ m}[/mathjaxinline] </p></li></ul>
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<h3 class="hd hd-2">Graph of an exponential function</h3>
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Sketch an exponential function
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Sketch the graph of [mathjaxinline]\left(\frac12\right)^{x}.[/mathjaxinline] </p>
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<p><b class="bfseries">Properties of exponential functions</b></p><p>
The function [mathjaxinline]\ f(x) = a^ x[/mathjaxinline] has base [mathjaxinline]a[/mathjaxinline] for a positive real number [mathjaxinline]a[/mathjaxinline]. </p><ul class="itemize"><li><p>
The function [mathjaxinline]a^ x[/mathjaxinline] is a continuous function. </p></li><li><p>
The domain of [mathjaxinline]a^ x[/mathjaxinline] is all real numbers. </p></li><li><p>
The range of [mathjaxinline]a^ x[/mathjaxinline] is all positive real numbers. </p></li></ul>
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<h2 class="hd hd-2 unit-title">4. Finding the derivative</h2>
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Can we use the power rule?
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The function [mathjaxinline]\ f(x) = 2^ x[/mathjaxinline] is an example of an exponential function. The number [mathjaxinline]2[/mathjaxinline] is called the <em>base</em>. And the variable [mathjaxinline]x[/mathjaxinline] is the <em>exponent</em>. </p>
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Does the power rule apply to the function [mathjaxinline]\ f(x) = 2^ x[/mathjaxinline]? </p>
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<h3 class="hd hd-2">Finding the Derivative</h3>
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<h2 class="hd hd-2 unit-title">5. The derivative of an exponential function</h2>
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The derivative of the exponential function is </p><table id="a0000000361" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx}a^ x = M(a) a^ x,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(4.34)</td></tr></table><p>
where the mystery number [mathjaxinline]M(a)[/mathjaxinline] is the slope of the tangent line at zero: </p><table id="a0000000362" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle M(a) =\left. \frac{d}{dx}a^ x\right|_{x=0} = \lim _{\Delta x\rightarrow 0} \frac{a^{\Delta x}-1}{\Delta x}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Estimate the derivative at 0
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The graph of [mathjaxinline]\ f(x) = 2^ x[/mathjaxinline] is shown below. Use the mathlet to estimate the value of [mathjaxinline]\ f'(0)[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\ f'(0)\approx[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">6. Numerical results</h2>
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Another exponential
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We found that the derivative of [mathjaxinline]\ f(x) = 2^ x[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline] is about 0.69. That's a weird number if you haven't seen it before. </p>
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Let's try another exponential. Let [mathjaxinline]g(x) = 4^ x[/mathjaxinline]. How does [mathjaxinline]g(x)[/mathjaxinline] compare to [mathjaxinline]\ f(x)[/mathjaxinline]? </p>
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First let's compare [mathjaxinline]g(0)[/mathjaxinline] and [mathjaxinline]\ f(0)[/mathjaxinline]. What is the relationship? </p>
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What is the relationship between [mathjaxinline]g'(0)[/mathjaxinline] and [mathjaxinline]\ f'(0)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">7. Definition of e</h2>
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The base [mathjaxinline]e[/mathjaxinline] is the unique real number so that [mathjaxinline]M(e) = \displaystyle \left.\frac{d}{dx} e^ x\right|_{x=0} = 1[/mathjaxinline]. Then </p><table id="a0000000366" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{d}{dx}e^ x = e^ x.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(4.35)</td></tr></table>
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The derivative at 0
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True or false: If [mathjaxinline]p(x) = e^ x[/mathjaxinline], then [mathjaxinline]p'(0) = p(0)[/mathjaxinline]. </p>
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What is [mathjaxinline]\displaystyle {\left.\frac{d}{dx} e^ x \right|_{x=-1}}[/mathjaxinline]? Enter e for [mathjaxinline]e[/mathjaxinline]. </p>
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<font color="#0078b0">2520</font>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Higher Derivatives
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What is [mathjaxinline]\displaystyle {\left.\frac{d^{17}}{dx^{17}} e^ x \right|_{x=-1}}[/mathjaxinline]? Enter e for [mathjaxinline]e[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_7-tab7-problem3_2_1" class="inputtype formulaequationinput">
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<h2 class="hd hd-2 unit-title">8. Why e exists</h2>
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Numeric approximation
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<h2 class="hd hd-2 unit-title">9. Additional practice</h2>
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Estimation
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If you calculate [mathjaxinline]e^{4.5}[/mathjaxinline] you get a number extremely close to 90. Without using a calculator, estimate [mathjaxinline]e^{4.55}[/mathjaxinline]. </p>
<p>
(Enter to one decimal place.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diff_7-tab9-problem1_2_1" class="inputtype formulaequationinput">
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A related function
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True or false: If [mathjaxinline]q(x) = e^{5x}[/mathjaxinline], then [mathjaxinline]q'(x) = e^{5x}[/mathjaxinline]. </p>
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Another related function
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True or false: If [mathjaxinline]r(x) = e^{x+5}[/mathjaxinline], then [mathjaxinline]r'(x) = r(x)[/mathjaxinline]. </p>
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The shape of a cable that hangs from two points of equal height is called a catenary. </p>
<center>
<img alt="See text below." src="/assets/courseware/v1/2653ed93438bb46ab8e285e5fcbc9f0e/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_catenary.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
<br/>
<em>The catenary.</em>
</center>
<p>
This curve is the graph of [mathjaxinline]g(x) = 0.5a(e^{x/a} + e^{-x/a})[/mathjaxinline] for some constant [mathjaxinline]a[/mathjaxinline]. Find the derivative of [mathjaxinline]g[/mathjaxinline] if [mathjaxinline]a = 2[/mathjaxinline]. </p>
<p>
(You can type e for [mathjaxinline]e[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponents. Use parentheses to denote order of operations.) </p>
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<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
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<p><b class="bfseries">Properties of exponents</b></p><p>
Let [mathjaxinline]a[/mathjaxinline] be a positive real number. </p><ul class="itemize"><li><p>
[mathjaxinline]a^0=1[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^1=a[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^{m}a^{n} = a^{m+n}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\left(a^ m\right)^{n} = a^{mn}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]a^{m/n} = \sqrt [n]{a^ m}[/mathjaxinline] </p></li></ul><p><b class="bfseries">Properties of exponential functions</b></p><p>
The function [mathjaxinline]\ f(x) = a^ x[/mathjaxinline] has base [mathjaxinline]a[/mathjaxinline] for a positive real number [mathjaxinline]a[/mathjaxinline]. </p><ul class="itemize"><li><p>
The function [mathjaxinline]a^ x[/mathjaxinline] is a continuous function. </p></li><li><p>
The domain of [mathjaxinline]a^ x[/mathjaxinline] is all real numbers. </p></li><li><p>
The range of [mathjaxinline]a^ x[/mathjaxinline] is all positive real numbers. </p></li></ul><p><b class="bfseries">The derivative of an exponential function</b></p><p>
The derivative of the exponential function is </p><table id="a0000000375" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{d}{dx}a^ x = M(a) a^ x,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(4.42)</td></tr></table><p>
where the mystery number [mathjaxinline]M(a)[/mathjaxinline] is the slope of the tangent line at zero: </p><table id="a0000000376" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle M(a) =\left. \frac{d}{dx}a^ x\right|_{x=0} = \lim _{\Delta x\rightarrow 0} \frac{a^{\Delta x}-1}{\Delta x}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Definition of [mathjaxinline]e[/mathjaxinline]</b></p><p>
The base [mathjaxinline]e[/mathjaxinline] is the unique real number so that [mathjaxinline]M(e) = \displaystyle \left.\frac{d}{dx} e^ x\right|_{x=0} = 1[/mathjaxinline]. Then </p><table id="a0000000377" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{d}{dx}e^ x = e^ x.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(4.43)</td></tr></table>
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