<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">1. Study Guide</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab1-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab1-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Typical Mistakes</b></p><p>
Take a moment to remind yourself why each statement below is false. </p><ul class="itemize"><li><p>
[mathjaxinline]f^{\prime \prime }(x) = 0 \iff \left( x, f(x) \right)[/mathjaxinline] is an inflection point. </p></li><li><p>
[mathjaxinline]f(x)[/mathjaxinline] is a local max (min) [mathjaxinline]\iff f'(x) = 0[/mathjaxinline]. </p></li><li><p>
Average rate of change of [mathjaxinline]f[/mathjaxinline] on [mathjaxinline]\left[ a, b\right][/mathjaxinline] is [mathjaxinline]\displaystyle \frac{f'(a) + f'(b)}{2}[/mathjaxinline]. </p></li><li><p>
[mathjaxinline]\displaystyle \frac{d}{dx}f(y) = f'(y)[/mathjaxinline] and other chain rule errors. </p></li></ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">2. Review Unit 0</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of One-Sided Limit</b></p><center><img src="/assets/courseware/v1/c0083509b7babe8df8e6e4d68de08c37/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim1_leftright.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
Suppose [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]R[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the right. Then we say [mathjaxinline]R[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">right-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the right. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow R[/mathjaxinline] as [mathjaxinline]x \rightarrow a^+[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^+}} f(x) = R}[/mathjaxinline]. </td></tr></table><p>
If [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]L[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the left, we say that [mathjaxinline]L[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">left-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the left. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow L[/mathjaxinline] as [mathjaxinline]x \rightarrow a^-[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^-}} f(x) = L}.[/mathjaxinline] </td></tr></table>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of the Limit</b></p><p><b class="bfseries">The Limit in Words</b></p><p>
If a function [mathjaxinline]f(x)[/mathjaxinline] approaches some value [mathjaxinline]L[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from <em>both the right and the left</em>, then <span style="color:#27408C"><b class="bf">the limit</b></span> of [mathjaxinline]f(x)[/mathjaxinline] exists and equals [mathjaxinline]L[/mathjaxinline]. </p><p><b class="bfseries">The Limit in Symbols</b></p><p>
If </p><table id="a0000000875" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {\lim _{x\rightarrow a^+} f(x)} = \displaystyle {\lim _{x\rightarrow a^-} f(x)} = L[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then </p><table id="a0000000876" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {\lim _{x\rightarrow a} f(x) = L}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Alternatively, </p><table id="a0000000877" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(x) \rightarrow L \quad \mathrm{as} \quad x\rightarrow a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Remember that [mathjaxinline]x[/mathjaxinline] is approaching [mathjaxinline]a[/mathjaxinline] but does not equal [mathjaxinline]a[/mathjaxinline]. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Formal definition of limit<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
Formally, the statement [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] is defined as: </p><p><br/></p><p>
For all [mathjaxinline]\varepsilon >0,[/mathjaxinline] there exists some [mathjaxinline]\delta > 0[/mathjaxinline] such that if [mathjaxinline]0 < |x-a| < \delta ,[/mathjaxinline] then [mathjaxinline]|f(x)-L| < \varepsilon .[/mathjaxinline] </p><p><br/></p><p>
As is traditional, we use the Greek letters [mathjaxinline]\varepsilon[/mathjaxinline] and [mathjaxinline]\delta[/mathjaxinline]. </p><p>
Here is how one might understand that statement. The distance between two numbers [mathjaxinline]y[/mathjaxinline] and [mathjaxinline]z[/mathjaxinline] is given by [mathjaxinline]|y-z|[/mathjaxinline]. Thus, the very last part of the definition is saying that the distance from [mathjaxinline]f(x)[/mathjaxinline] to [mathjaxinline]L[/mathjaxinline] is less than [mathjaxinline]\varepsilon[/mathjaxinline]; one should think of [mathjaxinline]\varepsilon[/mathjaxinline] as representing a small distance. This close distance occurs if [mathjaxinline]0<|x-a|<\delta[/mathjaxinline]; that is, if [mathjaxinline]x[/mathjaxinline] is within some distance [mathjaxinline]\delta[/mathjaxinline] from [mathjaxinline]a[/mathjaxinline], but not necessarily if that distance is 0 (we don't care about [mathjaxinline]x = a[/mathjaxinline] itself). </p><p>
The "for all" and "there exists" clauses have to do with how small these distances need to get. We want [mathjaxinline]f(x)[/mathjaxinline] to eventually get arbitrarily close to [mathjaxinline]L[/mathjaxinline], so this statement needs to be satisfied no matter how small [mathjaxinline]\varepsilon[/mathjaxinline] gets. Given any choice of [mathjaxinline]\varepsilon[/mathjaxinline], we can satisfy the condition [mathjaxinline]|f(x) - L | < \varepsilon[/mathjaxinline] as long as [mathjaxinline]x[/mathjaxinline] gets close enough to [mathjaxinline]a[/mathjaxinline]; the proximity required is measured by [mathjaxinline]\delta[/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"/>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Limit Laws:</b></p><p>
Suppose [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}, \qquad \displaystyle {\lim _{x\rightarrow a} g(x) = M}.[/mathjaxinline] </p><p>
Then we get the following Limit Laws: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Limit Law for Addition: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)+g(x)\right] }[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L+M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Subtraction: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)-g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L-M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Multiplication: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)\cdot g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L\cdot M.[/mathjaxinline] </td></tr></table><p>
We also have part of the Limit Law for Division: <b class="bfseries">Limit Law for Division</b></p><p>
If [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} g(x) = M},[/mathjaxinline] then: </p><ul class="itemize"><li><p>
If [mathjaxinline]M \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle { \lim _{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}}[/mathjaxinline]. </p></li><li><p>
If [mathjaxinline]M = 0[/mathjaxinline] but [mathjaxinline]L \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] does not exist. </p></li><li><p>
If both [mathjaxinline]M = 0[/mathjaxinline] and [mathjaxinline]L = 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] might exist, or it might not exist. More work is necessary to determine whether the last type of limit exists, and what it is if it does exist. </p></li></ul>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of Continuous at a Point</b></p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if </p><table id="a0000000878" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {\lim _{x\rightarrow a} f(x) } = f(a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In particular, if either [mathjaxinline]\ f(a)[/mathjaxinline] or [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] fails to exist, then [mathjaxinline]\ f[/mathjaxinline] is discontinuous at [mathjaxinline]a[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">right-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) } = f(a)[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">left-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) } = f(a)[/mathjaxinline]. </p><p>
It is sometimes useful to classify certain types of discontinuities. </p><p>
If the left-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) }[/mathjaxinline] and the right-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) }[/mathjaxinline] both exist at a point [mathjaxinline]x=a[/mathjaxinline], but they are not equal, then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">jump discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/4c60a6f5760133fa9aad6bea60c814ac/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_jump.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
If the overall limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] exists (i.e., the left- and right-hand limits agree), but the overall limit does not equal [mathjaxinline]\ f(a)[/mathjaxinline], then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">removable discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/3367cd90f4550e53c7950aaaf93ff088/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_removable.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text5">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Basic Continuous Functions</b></p><p>
Note: we have not proven all of the following facts, but you should feel free to use them. </p><p>
The following functions are continuous at <em>all real numbers</em>: </p><ul class="itemize"><li><p>
all polynomials </p></li><li><p>
[mathjaxinline]\sqrt [3]{x}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]|x|[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\cos x[/mathjaxinline] and [mathjaxinline]\sin x[/mathjaxinline] </p></li><li><p>
exponential functions [mathjaxinline]a^ x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline] </p></li></ul><p>
The following functions are continuous <em>at the specified values</em> of [mathjaxinline]x[/mathjaxinline]: </p><ul class="itemize"><li><p>
[mathjaxinline]\sqrt {x}[/mathjaxinline], for [mathjaxinline]x >0[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\tan x[/mathjaxinline], at all [mathjaxinline]x[/mathjaxinline] where it is defined </p></li><li><p>
logarithmic functions [mathjaxinline]\log _{a} x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline], for [mathjaxinline]x > 0[/mathjaxinline] </p></li></ul>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text6">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab2-text6" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Intermediate Value Theorem</b></p><p>
If [mathjaxinline]\ f[/mathjaxinline] is a function which is continuous on the interval [mathjaxinline][a,b][/mathjaxinline], and [mathjaxinline]M[/mathjaxinline] lies between the values of [mathjaxinline]\ f(a)[/mathjaxinline] and [mathjaxinline]\ f(b)[/mathjaxinline], then there is at least one point [mathjaxinline]c[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] such that [mathjaxinline]\ f(c) = M.[/mathjaxinline] </p><p>
(A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous on a closed interval</b></span> [mathjaxinline][a,b][/mathjaxinline] if it is right-continuous at [mathjaxinline]a[/mathjaxinline], left-continuous at [mathjaxinline]b[/mathjaxinline], and continuous at all points between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline].) </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">3. Review Unit 1</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Definition of Average Rate of Change</b></p><p>
The <span style="color:#99182C"><b class="bf">average rate of change</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] over an interval [mathjaxinline]a \leq x \leq b[/mathjaxinline] is defined to be </p><table id="a0000000879" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{f(b) - f(a)}{b-a}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Geometrically</b></p><p>
Geometrically, the average rate of change is the slope of the secant line through the points [mathjaxinline](a, f(a))[/mathjaxinline] and [mathjaxinline](b, f(b))[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Definition of the Derivative</b></p><p>
The <span style="color:#99182C"><b class="bf">derivative</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] at a point [mathjaxinline]x = a[/mathjaxinline] is defined to be </p><table id="a0000000880" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f'(a) = \lim _{b\rightarrow a} \frac{f(b) - f(a)}{b-a}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Geometrically</b></p><p>
Geometrically, the derivative [mathjaxinline]\ f'(a)[/mathjaxinline] is the slope of the tangent line to the function [mathjaxinline]\ f[/mathjaxinline] through the point [mathjaxinline](a, f(a))[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Properties of derivatives</b></p><p>
The derivative of a function is itself a function, and satisfies the following linearity properties. </p><p><b class="bfseries">Derivatives of Constant Multiples</b></p><p>
If [mathjaxinline]g(x) = k\ f(x)[/mathjaxinline] for some constant [mathjaxinline]k[/mathjaxinline], then </p><table id="a0000000881" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle g'(x) = k\ f'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where [mathjaxinline]\ f[/mathjaxinline] is differentiable. </p><p><b class="bfseries">Derivatives of Sums</b></p><p>
If [mathjaxinline]h(x) = f(x) + g(x)[/mathjaxinline], then </p><table id="a0000000882" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle h'(x) = f'(x) + g'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable. </p><p><b class="bfseries">Derivatives of Differences</b></p><p>
Similarly, if [mathjaxinline]j(x) = f(x) - g(x)[/mathjaxinline], then </p><table id="a0000000883" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle j'(x) = f'(x) - g'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable. </p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Power Rule</b></p><p>
If [mathjaxinline]n[/mathjaxinline] is any fixed real number, and [mathjaxinline]\ f(x) = x^ n[/mathjaxinline], then [mathjaxinline]\ f'(x) = n x^{n-1}[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text5">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Properties of Leibniz notation</b></p><ul class="itemize"><li><p><b class="bf">Units:</b> If [mathjaxinline]P[/mathjaxinline] has units of pressure, and [mathjaxinline]t[/mathjaxinline] has units of [mathjaxinline]time[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{dP}{dt}[/mathjaxinline] has units of pressure per time. </p></li><li><p><b class="bf">Evaluating at points:</b> If we want to take the derivative at a particular point [mathjaxinline]x = 3[/mathjaxinline], then we use the notation [mathjaxinline]\left.\displaystyle \frac{df}{dx}\right|_{x=3}[/mathjaxinline]. The bar is read as “evaluated at". </p></li><li><p><b class="bf">Derivatives act on functions:</b></p><ul class="itemize"><li><p>
We can write [mathjaxinline]\displaystyle \frac{d(x^2)}{dx}[/mathjaxinline] for the derivative of [mathjaxinline]x^2[/mathjaxinline]. </p></li><li><p>
If a formula is long, we can write [mathjaxinline]\displaystyle \frac{d}{dy} \left(y^3 + 2y^2 \right).[/mathjaxinline] </p></li></ul></li></ul>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text6">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text6" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Second Derivative</b></p><p>
The second derivative of a function [mathjaxinline]\ f(x)[/mathjaxinline] is the first derivative of [mathjaxinline]\ f'(x)[/mathjaxinline], and is denoted by [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{d^2f}{dx^2}[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text7">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text7" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Second Derivative and Concavity Summary</b></p><p>
On intervals where [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave up. </p><center><img src="/assets/courseware/v1/51e4935d434a86fb05fcb5a7ee76c350/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concaveupgraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
On intervals where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is concave down. </p><center><img src="/assets/courseware/v1/ce792b66460f76e31c504bb3b8169028/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der5_concavedowngraph.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text8">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text8" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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>
</div>
</div>
<div class="vert vert-8" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text9">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab3-text9" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Derivative of sine and cosine</b></p><p>
The first and second derivatives of sine and cosine: </p><table id="a0000000884" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000885"><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">(7.1)</td></tr><tr id="a0000000886"><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">(7.2)</td></tr><tr id="a0000000887"><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">(7.3)</td></tr><tr id="a0000000888"><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]
</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">(7.4)</td></tr></table>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">4. Review Unit 2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Linear Approximation</b></p><p>
The linear approximation for a function [mathjaxinline]\ f[/mathjaxinline] near a point [mathjaxinline]x=a[/mathjaxinline] is given by the following equivalent formulas: </p><table id="a0000000889" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000890"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \approx \left. \frac{df}{dx} \right|_{x=a} \cdot \Delta x \qquad \ \mathrm{for} \ \Delta x \ \mathrm{near} \ 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000891"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \approx f'(a) (x-a) + f(a) \ \ \ \mathrm{for} \ x \ \mathrm{near} \ a[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Product Rule</b></p><p>
If [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline], then </p><table id="a0000000892" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]h'(x) = f(x)g'(x)+ g(x)f'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(x)[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Quotient Rule</b></p><p>
If [mathjaxinline]\displaystyle h(x) = \frac{f(x)}{g(x)}[/mathjaxinline] for all [mathjaxinline]x[/mathjaxinline], then </p><table id="a0000000893" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle h'(x) = \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable and [mathjaxinline]g(x) \ne 0[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Chain Rule</b></p><p>
If [mathjaxinline]h(x) = f(g(x))[/mathjaxinline], then </p><table id="a0000000894" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]h'(x) = f'\left(g(x)\right) g'(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all points where the derivatives [mathjaxinline]\ f'(g(x))[/mathjaxinline] and [mathjaxinline]g'(x)[/mathjaxinline] are defined. </p><p>
Alternatively, if [mathjaxinline]y = f(u)[/mathjaxinline], and [mathjaxinline]u=g(x)[/mathjaxinline], then </p><table id="a0000000895" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\left. \frac{dy}{dx} \right|_{x = a} = \left. \frac{dy}{du} \right|_{u=g(a)} \left. \frac{du}{dx} \right|_{x=a}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at any point [mathjaxinline]x=a[/mathjaxinline] where the derivatives on the right hand side are defined. </p>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text5">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Implicit Differentiation</b></p><p>
To implicitly differentiate a function [mathjaxinline]\ f(x)g(y) = 1[/mathjaxinline] with respect to [mathjaxinline]x[/mathjaxinline]: </p><table id="a0000000896" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000897"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \left( f(x)g(y)\right.[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\left. 1 \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.5)</td></tr><tr id="a0000000898"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \left( f(x)g(y) \right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{d}{dx} 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.6)</td></tr><tr id="a0000000899"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \left( f(x)g(y) \right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0 \quad \textrm{(derivatives of constant functions are 0)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.7)</td></tr><tr id="a0000000900"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(x)g(y) + f(x) \frac{d}{dx}g(y)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0 \quad \textrm{(product rule)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.8)</td></tr><tr id="a0000000901"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(x)g(y) + f(x)g'(y) \frac{dy}{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0 \quad \textrm{(chain rule)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.9)</td></tr></table>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text6">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text6" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of Inverse Function</b></p><p>
If functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] satisfy [mathjaxinline]g\left(f(x)\right) = x[/mathjaxinline] and [mathjaxinline]\ f\left(g(y)\right) = y[/mathjaxinline], then we say [mathjaxinline]g[/mathjaxinline] is the inverse of [mathjaxinline]\ f[/mathjaxinline], and denote it by [mathjaxinline]\ f^{-1}[/mathjaxinline]. (Similarly, [mathjaxinline]\ f = g^{-1}[/mathjaxinline].) </p><p>
If a function [mathjaxinline]\ f[/mathjaxinline] has an inverse function [mathjaxinline]\ f^{-1}[/mathjaxinline], then [mathjaxinline]\ f^{-1}(b)= a[/mathjaxinline] if and only if [mathjaxinline]\ f(a) = b[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text7">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text7" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of One-to-One</b></p><p>
A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#99182C"><b class="bf">one-to-one</b></span> if [mathjaxinline]\ f(a) \ne f(b)[/mathjaxinline] whenever [mathjaxinline]a\ne b[/mathjaxinline]. It is one-to-one if and only if its graph satisfies the horizontal line test (no horizontal line intersects its graph at more than one place). </p>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text8">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text8" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Domain and Range, Interval Notation</b></p><p>
Recall that the <span style="color:#99182C"><b class="bf">domain</b></span> of a function [mathjaxinline]\ f[/mathjaxinline] is the set of allowable input values. For instance, the domain of the function [mathjaxinline]\ f(x) = 1/x[/mathjaxinline] is the set of all non-zero real numbers. </p><p>
The <span style="color:#99182C"><b class="bf">range</b></span> of [mathjaxinline]\ f[/mathjaxinline] is the set of all possible output values. For instance, the range of the function [mathjaxinline]g(x) = x^2[/mathjaxinline] is the set of all real numbers that are non-negative. </p><p>
We often use interval notation to express sets of numbers like domains and ranges. A <span style="color:#99182C"><b class="bf">closed interval</b></span>, denoted [mathjaxinline][a,b][/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a \le x \le b[/mathjaxinline]. </p><p>
An <span style="color:#99182C"><b class="bf">open interval</b></span>, denoted [mathjaxinline](a,b)[/mathjaxinline], is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]a < x < b[/mathjaxinline]. </p><p>
One can have a half-open, half-closed interval. For instance, [mathjaxinline][-1, 3)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-1 \le x <3[/mathjaxinline]. One can also use [mathjaxinline]\pm \infty[/mathjaxinline] as endpoints: [mathjaxinline](-\infty , 0)[/mathjaxinline] is the set of numbers [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]-\infty <x < 0[/mathjaxinline] (the set of negative numbers, in other words). </p><p>
This notation using round parentheses for open intervals is not universal; many mathematicians use reversed square brackets instead. For instance, they would denote the interval [mathjaxinline]3 < x < 7[/mathjaxinline] as [mathjaxinline]]3,7[[/mathjaxinline] rather than [mathjaxinline](3,7)[/mathjaxinline]. In this course, however, we will stick to round parentheses for open intervals. </p>
</div>
</div>
<div class="vert vert-8" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text9">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text9" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Inverse Trig Functions</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arcsin x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left[-\pi /2,\pi /2\right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\sin \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arccos x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left[0,\pi \right][/mathjaxinline]</td><td style="text-align:left; border:none">
such that [mathjaxinline]\cos \theta = x.[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\arctan x} =[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\theta[/mathjaxinline] in [mathjaxinline]\displaystyle \left(-\pi /2,\pi /2 \right)[/mathjaxinline] </td><td style="text-align:left; border:none">
such that [mathjaxinline]\tan \theta = x.[/mathjaxinline] </td></tr></table>
</div>
</div>
<div class="vert vert-9" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text10">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text10" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Derivatives of Inverse Functions</b></p><p>
If [mathjaxinline]g[/mathjaxinline] is a (full or partial) inverse of a function [mathjaxinline]\ f[/mathjaxinline], then </p><table id="a0000000902" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle {g'(x) = \frac{1}{f'\left(g(x)\right)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
at all [mathjaxinline]x[/mathjaxinline] where [mathjaxinline]\displaystyle {f'\left(g(x)\right)}[/mathjaxinline] exists and is non-zero. </p>
</div>
</div>
<div class="vert vert-10" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text11">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text11" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Derivatives of the Inverse Trig Functions</b></p><p>
We now have more basic functions that we can differentiate. </p><table id="a0000000903" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000904"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arcsin x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000905"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arccos x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -\frac{1}{\sqrt {1-x^2}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000906"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} \arctan x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{1+x^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
</div>
</div>
<div class="vert vert-11" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text12">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text12" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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>
</div>
</div>
<div class="vert vert-12" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text13">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text13" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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>
</div>
</div>
<div class="vert vert-13" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text14">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text14" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The derivative of an exponential function</b></p><p>
The derivative of the exponential function is </p><table id="a0000000907" 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">(7.10)</td></tr></table><p>
where the mystery number [mathjaxinline]M(a)[/mathjaxinline] is the slope of the tangent line at zero: </p><table id="a0000000908" 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>
</div>
</div>
<div class="vert vert-14" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text15">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text15" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of [mathjaxinline]e[/mathjaxinline]</b></p><p>
The base [mathjaxinline]e[/mathjaxinline] is the unique real number so that </p><table id="a0000000909" 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">(7.11)</td></tr></table>
</div>
</div>
<div class="vert vert-15" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text16">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text16" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Differentiating Exponential Functions with other Bases</b></p><p>
We can finally identify our mystery number, and differentiate exponential functions with any base. </p><p>
For any positive constant [mathjaxinline]a[/mathjaxinline], </p><table id="a0000000910" 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 = a^ x \ln a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
</div>
</div>
<div class="vert vert-16" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text17">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text17" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Properties of [mathjaxinline]x[/mathjaxinline]</b></p><p>
[mathjaxinline]\log _{10}(x)[/mathjaxinline] is the inverse function of [mathjaxinline]10^ x[/mathjaxinline]. </p><p>
The natural log, denoted [mathjaxinline]\ln (x)[/mathjaxinline], is the inverse function of [mathjaxinline]e^ x[/mathjaxinline]. </p><ul class="itemize"><li><p>
[mathjaxinline]\ln e^ x =x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]e^{\ln x} = x[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (ab) = \ln (a) + \ln (b)[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\ln (a^ b) = b\ln (a)[/mathjaxinline] </p></li></ul>
</div>
</div>
<div class="vert vert-17" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text18">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab4-text18" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The derivative of the natural logarithm</b></p><p>
Our collection of basic functions that we can differentiate now has one more entry: </p><table id="a0000000911" 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} \ln x = \frac{1}{x}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">5. Review Unit 3</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Linear Approximations of basic functions near 0</b></p><p>
[mathjaxinline]\displaystyle (1+x)^ r \approx 1 + rx[/mathjaxinline]<br/>[mathjaxinline]\displaystyle \sin (x) \approx \sin (0) + \cos (0)x = x[/mathjaxinline]<br/>[mathjaxinline]\displaystyle \cos (x) \approx \cos (0) - \sin (0)x = 1[/mathjaxinline]<br/>[mathjaxinline]\displaystyle e^ x \approx e^0 + e^0 x = 1 + x[/mathjaxinline]<br/>[mathjaxinline]\displaystyle \ln (1+x) \approx \ln (1+0) + \frac{1}{1+0}x = x[/mathjaxinline] </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Best fit quadratic</b></p><p>
The best fit quadratic or best fit parabola to a function [mathjaxinline]\ f(x)[/mathjaxinline] at the point [mathjaxinline]x=0[/mathjaxinline] is the quadratic function [mathjaxinline]q(x)[/mathjaxinline] whose value agree with the value of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline], and whose first and second derivatives agree with the first and second derivatives of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline], i.e.: </p><table id="a0000000912" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000913"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = q(0)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.12)</td></tr><tr id="a0000000914"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f'(0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = q'(0)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.13)</td></tr><tr id="a0000000915"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f^{\prime \prime }(0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = q^{\prime \prime }(0)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(7.14)</td></tr></table>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Quadratic Approximation</b></p><p>
The <b class="bfseries"><span style="color:#27408C">quadratic approximation</span></b> near [mathjaxinline]x=a[/mathjaxinline] is the <b class="bfseries"><span style="color:#27408C">best fit parabola</span></b> to [mathjaxinline]\ f[/mathjaxinline] at the point [mathjaxinline]x=a[/mathjaxinline]. </p><p>
The formula for the quadratic approximation of a function [mathjaxinline]\ f[/mathjaxinline] near a point [mathjaxinline]x=a[/mathjaxinline] is </p><table id="a0000000916" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f(x) \approx f(a) + f'(a)(x-a) + \frac{f^{\prime \prime }(a)}{2}(x-a)^2.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.15)</td></tr></table><p>
When [mathjaxinline]a=0[/mathjaxinline], this quadratic approximation becomes </p><table id="a0000000917" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle f(x) \approx f(0) + f'(0)x + \frac{f^{\prime \prime }(0)}{2}x^2.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.16)</td></tr></table>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Big-O notation</b></p><p>
A function [mathjaxinline]\ f(x)[/mathjaxinline] is on the order [mathjaxinline]x^ n[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], which is denoted using big “O" notation as [mathjaxinline]\ f(x) = O(x^ n)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], if [mathjaxinline]\left|f(x)\right| \leq k x^ n[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text5">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab5-text5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Newton's Method</b></p><p>
Given a function [mathjaxinline]\ f(x)[/mathjaxinline], find [mathjaxinline]x[/mathjaxinline] such that [mathjaxinline]\ f(x) = 0[/mathjaxinline]. </p><ol class="enumerate"><li value="1"><p>
Make a good guess [mathjaxinline]x_0[/mathjaxinline]. </p></li><li value="2"><p>
Call [mathjaxinline]x_1[/mathjaxinline] the [mathjaxinline]x-[/mathjaxinline]intercept of the tangent line through [mathjaxinline](x_0, f(x_0))[/mathjaxinline]. It has the formula </p><table id="a0000000918" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\boxed { \displaystyle x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}.}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.17)</td></tr></table></li><li value="3"><p>
Repeat. The general formula is </p><table id="a0000000919" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\boxed { \displaystyle x_{n+1} = x_ n - \frac{f(x_ n)}{f'(x_ n)}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.18)</td></tr></table><p>
for [mathjaxinline]n=0,1,2, \cdots[/mathjaxinline]. </p></li></ol>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@study-tab6" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">6. Review Unit 4</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of Critical Points</b></p><p>
The critical points of a function [mathjaxinline]\ f(x)[/mathjaxinline] to be all points [mathjaxinline]x[/mathjaxinline] in the domain of [mathjaxinline]f(x)[/mathjaxinline] such that </p><ul class="itemize"><li><p>
[mathjaxinline]f'(x) = 0[/mathjaxinline], or </p></li><li><p>
[mathjaxinline]f'(x)[/mathjaxinline] does not exist. </p></li></ul>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The First Derivative Test</b></p><p><b class="bfseries">Finding Local Maxima and Minima</b></p><p>
Suppose the function [mathjaxinline]\ f(x)[/mathjaxinline] is continuous at [mathjaxinline]x=a[/mathjaxinline] and has a critical point at [mathjaxinline]x=a[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/1e03b0de4b279c259dca7ea419ddd2c5/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test2.svg" width="100 px" style="margin: 5px 5px 5px 5px; border:0px"/></td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline] if [mathjaxinline]\ f'(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]\ f'(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/c777088002a0e851a6d08b7b0bb00da2/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test1.svg" width="100 px" style="margin: 5px 5px 5px 5px; border:0px"/></td></tr><tr><td style="text-align:left; border:none">
The point [mathjaxinline]x=a[/mathjaxinline] is neither a local minimum nor a local maximum of [mathjaxinline]\ f[/mathjaxinline] if [mathjaxinline]\ f'(x)[/mathjaxinline] has the same sign just to the left of [mathjaxinline]a[/mathjaxinline] and just to the right of [mathjaxinline]a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/d1be24440af85083dcb862e2dd21c119/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test3.svg" width="100 px" style="margin: 5px 5px 5px 5px; border:0px"/></td></tr></table><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Just to the left or right<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
When we use the phrase “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the left of [mathjaxinline]a[/mathjaxinline]," we mean that there is some open interval [mathjaxinline](a-c,a)[/mathjaxinline] of positive width [mathjaxinline]c[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. This interval does not have to be very big, as long as it has some size! </p><p>
Similarly, “[mathjaxinline]\ f'(x)>0[/mathjaxinline] just to the right of [mathjaxinline]a[/mathjaxinline]" means that there is some open interval [mathjaxinline](a, a+d)[/mathjaxinline] of positive width [mathjaxinline]d[/mathjaxinline] on which [mathjaxinline]\ f'[/mathjaxinline] is positive. </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"/>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Second Derivative Test</b></p><p>
Suppose that [mathjaxinline]x=a[/mathjaxinline] is a critical point of [mathjaxinline]\ f[/mathjaxinline], with [mathjaxinline]\ f'(a) = 0[/mathjaxinline]. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) > 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local minimum at [mathjaxinline]x=a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/a8b74dac86ee612f810857ee664cd3f1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test4.svg" width="100 px" style="margin: 5px 5px 5px 5px; border:0px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) < 0[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] has a local maximum at [mathjaxinline]x=a[/mathjaxinline]. </td><td style="text-align:center; border:none"><img src="/assets/courseware/v1/0986206230460827ea0d348145be62de/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app1_test5.svg" width="100 px" style="margin: 5px 5px 5px 5px; border:0px"/></td></tr><tr><td style="text-align:left; border:none">
If [mathjaxinline]\ f^{\prime \prime }(a) = 0[/mathjaxinline], or does not exist, then the test is inconclusive — there might be a local maximum, or a local minimum, or neither. </td><td style="text-align:center; border:none"> </td></tr></table>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Definition of Inflection Point</b></p><p>
An inflection point is a point where the concavity of the function changes. That is the second derivative [mathjaxinline]f^{\prime \prime }(x)[/mathjaxinline] changes sign— [mathjaxinline]f^{\prime \prime }(x) > 0[/mathjaxinline] just to the left of [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]f^{\prime \prime }(x) < 0[/mathjaxinline] just to the right of [mathjaxinline]x[/mathjaxinline] (or vice versa). </p>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text5">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text5" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">General strategy for sketching functions</b></p><ol class="enumerate"><li value="1"><p>
Plot </p><ul class="itemize"><li><p>
discontinuities (especially infinite ones) </p></li><li><p>
end points (or [mathjaxinline]x \rightarrow \pm \infty[/mathjaxinline]) </p></li><li><p>
easy points ([mathjaxinline]x = 0[/mathjaxinline], or [mathjaxinline]y=0[/mathjaxinline]) (This is optional.) </p></li></ul></li><li value="2"><p>
Plot critical points and values. (Solve [mathjaxinline]\ f'(x) = 0[/mathjaxinline] or undefined.) </p></li><li value="3"><p>
Decide whether [mathjaxinline]\ f' < 0[/mathjaxinline] or [mathjaxinline]\ f' >0[/mathjaxinline] on each interval between endpoints, critical points, and discontinuities. (Valuable double check) </p></li><li value="4"><p>
Identify where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline] (concave down and concave up). </p><p>
Identify inflection points. (Makes graph look nice. Can be used to double check.) </p></li><li value="5"><p>
Combine into graph. </p></li></ol>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text6">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text6" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Indeterminate Forms</b></p><p>
We call [mathjaxinline]\displaystyle \frac00[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline] <b class="bf"><span style="color:#27408C">indeterminate forms,</span></b> because when we run into them in a limit, they require further analysis to determine whether the numerator or denominator wins in the race to 0 or [mathjaxinline]\infty[/mathjaxinline] respectively, or whether they balance out and reach some other finite limit. </p>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text7">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text7" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">L'Hospital's Rule Version 1: Indeterminate form 0/0 </b></p><p>
If </p><table id="a0000000920" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow 0\\ \displaystyle g(x) \rightarrow 0 \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000921" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.19)</td></tr></table><p>
provided that and the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p>
If </p><table id="a0000000922" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow \pm \infty \\ \displaystyle g(x) \rightarrow \pm \infty \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000923" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(7.20)</td></tr></table><p>
provided that and the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p>
Note that </p><ul class="itemize"><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]a^+[/mathjaxinline] or [mathjaxinline]a^-[/mathjaxinline] and the results (versions 1 and 2) still hold. </p></li><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]\pm \infty[/mathjaxinline], and the results (versions 1 and 2) still hold. </p></li></ul><p><b class="bfseries">Other indeterminate forms</b></p><p>
Other indeterminate forms [mathjaxinline]0\cdot \infty[/mathjaxinline], [mathjaxinline]\infty - \infty[/mathjaxinline], [mathjaxinline]0^0[/mathjaxinline], [mathjaxinline]1^{\infty }[/mathjaxinline], and [mathjaxinline]\infty ^{0}[/mathjaxinline] should be rearranged to be of the form [mathjaxinline]0/0[/mathjaxinline] or [mathjaxinline]\infty /\infty[/mathjaxinline] in order to apply l'Hôpital's rule. </p>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text8">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text8" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Extreme Value Theorem</b></p><p>
If [mathjaxinline]\ f[/mathjaxinline] is continuous on a closed interval [mathjaxinline][a,b][/mathjaxinline], then there are points at which [mathjaxinline]\ f[/mathjaxinline] attains its maximum and its minimum on [mathjaxinline][a,b][/mathjaxinline]. </p><p><b class="bfseries">Maxima and Minima</b></p><p>
The maxima and minima will be attained at either a critical point or an end point. </p>
</div>
</div>
<div class="vert vert-8" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text9">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="a4c0e89aea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@study-tab6-text9" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Related Rates Strategy</b></p><p>
To solve related rates problems, it is useful to follow this strategy: </p><ol class="enumerate"><li value="1"><p>
Start with a good picture! </p></li><li value="2"><p>
Identify the relevant variables and rates. </p></li><li value="3"><p>
Find an equation relating the relevant variables that always holds. </p></li><li value="4"><p>
Differentiate implicitly. </p></li><li value="5"><p>
Plug in and solve! </p></li></ol>
</div>
</div>
</div>
</div>
© All Rights Reserved