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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Quadratic approximation</h3>
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<h2 class="hd hd-2 unit-title">2. Quadratic approximation</h2>
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<p><span style="color:#99182C"><b class="bf">Quadratic approximations.</b></span><br/><span style="color:#99182C"><b class="bf">Degree 2 polynomial approximation.</b></span><br/><span style="color:#99182C"><b class="bf">Error</b></span>. </p><p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Understand why a <span style="color:#27408C"><b class="bf">quadratic approximation</b></span> is the best fit <span style="color:#27408C"><b class="bf">parabola</b></span> to a function near a given point. </p></li><li><p><span style="color:#27408C"><b class="bf">Approximate functions</b></span> near a given point from the value of the first and second derivatives at that point. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">big “O" notation</b></span> to describe order of magnitude of the error functions of linear and quadratic approximations.. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
7 videos (41 minutes at 1x speed) 19 questions </p>
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Compare the function [mathjaxinline]\ f(x)[/mathjaxinline] and this tangent line through the point [mathjaxinline]x=a[/mathjaxinline]: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle L(x) = f(a) + f'(a)(x-a)[/mathjax]</td>
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<h2 class="hd hd-2 unit-title">4. Geometric intuition</h2>
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The best fit quadratic to a function [mathjaxinline]\ f(x)[/mathjaxinline] at the point [mathjaxinline]x=0[/mathjaxinline] is the quadratic function [mathjaxinline]q(x)[/mathjaxinline] whose value agrees 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="a0000000539" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000540"><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">(5.35)</td></tr><tr id="a0000000541"><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">(5.36)</td></tr><tr id="a0000000542"><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">(5.37)</td></tr></table>
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Suppose that you have a function [mathjaxinline]\ f(x)[/mathjaxinline]. The best fit quadratic function near [mathjaxinline]x=0[/mathjaxinline] is given by [mathjaxinline]q(x) = ax^2 + bx + c[/mathjaxinline]. </p>
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(Express [mathjaxinline]\ f(0), \ f^{\prime }(0)[/mathjaxinline], and [mathjaxinline]\ f^{\prime \prime }(0)[/mathjaxinline] in terms of [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], and [mathjaxinline]c[/mathjaxinline].) </p>
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Find the quadratic approximation of [mathjaxinline]e^ x[/mathjaxinline] and [mathjaxinline](1+x)^ r[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. (Note that the last approximation will depend on both [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]r[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]e^ x \approx[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline](1+x)^ r\approx[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">7. Quadratic approximation</h2>
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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="a0000000547" 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">(5.41)</td></tr></table><p>
When [mathjaxinline]a=0[/mathjaxinline], this quadratic approximation becomes </p><table id="a0000000548" 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">(5.42)</td></tr></table><p>
In the following problems, identify the point [mathjaxinline]x=a[/mathjaxinline] and use a quadratic approximation to approximate the values of the functions below. </p>
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<h2 class="hd hd-2 unit-title">8. Quadratic approximations of products</h2>
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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>
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<h2 class="hd hd-2 unit-title">10. Car Loan</h2>
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Suppose that you take out a $15,000 loan to purchase a car. The loan has an interest rate of 3% per year, compounded monthly. </p>
<p>
The formula for the amount of interest accrued over a time period [mathjaxinline]T[/mathjaxinline] measured in number of months is </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000557" style="table-layout:auto" width="100%">
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<td class="equation" style="width:80%; border:none">[mathjax]A = 15,000\left(1+\frac{.03}{12}\right)^ T-15,000.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Use a quadratic approximation to approximate the amount of <em>interest</em> accrued over 4 years. </p>
<p>
(Enter answer to 2 decimal places.) </p>
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Magnitude of linear term
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What is the order of magnitude of the linear term? </p>
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(Enter your answer as a power of 10; e.g. 10[mathjaxinline]\wedge[/mathjaxinline]6 for [mathjaxinline]10^6[/mathjaxinline].) </p>
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Magnitude of quadratic term
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What is the order of magnitude of the quadratic term? </p>
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(Enter your answer as a power of 10; e.g. 10[mathjaxinline]\wedge[/mathjaxinline]6 for [mathjaxinline]10^6[/mathjaxinline].) </p>
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Error term
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The error in this approximation is [mathjaxinline]O(x^3)[/mathjaxinline]. Find the order of magnitude of the [mathjaxinline]x^3[/mathjaxinline] term in our problem. </p>
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(Enter your answer as a power of 10; e.g. 10[mathjaxinline]\wedge[/mathjaxinline]6 for [mathjaxinline]10^6[/mathjaxinline].) </p>
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<p style="display:inline">Order of magnitude of [mathjaxinline](1/400)^3[/mathjaxinline]:</p>
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Use a calculator to compute the exact error term in the quadratic approximation. </p>
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(Enter as a decimal number to 2 decimal places.) </p>
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<p style="display:inline">What is the absolute value of the error?</p>
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Use the error to identify the order of magnitude of [mathjaxinline]k[/mathjaxinline] since the error is known to be [mathjaxinline]kx^3[/mathjaxinline] (where [mathjaxinline]x=1/400[/mathjaxinline]). </p>
<p>
(Enter your answer as a power of 10; e.g. 10[mathjaxinline]\wedge[/mathjaxinline]6 for [mathjaxinline]10^6[/mathjaxinline].) </p>
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<p style="display:inline">Order of magnitude of [mathjaxinline]k[/mathjaxinline]:</p>
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<h2 class="hd hd-2 unit-title">11. Complicated functions</h2>
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<h2 class="hd hd-2 unit-title">12. Practice combining approximations</h2>
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<p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Library of quadratic approximations<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
Near zero: [mathjaxinline]\displaystyle {f(x) \approx f(0) + f'(0)x + \frac{f^{\prime \prime }(0)}{2}x^2}[/mathjaxinline]. </p><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle {e^ x = 1 + x + \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\sin (x) = x + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\cos (x) = 1- \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\ln (1+x) = x- \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {(1+x)^ r = 1 + rx + \frac{r(r-1)}{2}x^2 + O\left(x^3\right)}[/mathjaxinline] </p></li></ul></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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What is the quadratic approximation of [mathjaxinline]e^{-x}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]? </p>
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Use a quadratic approximation for [mathjaxinline]e^ u[/mathjaxinline] to find an approximation to the function [mathjaxinline]e^{x^3}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. Find the magnitude of the error. </p>
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<p style="display:inline">[mathjaxinline]e^{x^3}\approx[/mathjaxinline]</p>
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<p style="display:inline">Error term is:</p>
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Practice with products 1
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Consider [mathjaxinline]\displaystyle \frac{\ln (1-x)}{(1+x)^2}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
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What is the quadratic approximation near [mathjaxinline]x=0[/mathjaxinline]? </p>
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(Do not enter the big-O term.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{\ln (1-x)}{(1+x)^2} \approx[/mathjaxinline]</p>
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Practice with products 2
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Find the quadratic approximation of [mathjaxinline]\ \displaystyle \frac{\sin (x)}{\cos ^2(x)}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] using the known approximations of [mathjaxinline]\sin (x)[/mathjaxinline] and [mathjaxinline]\cos (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
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(Do not enter the big-O term.) </p>
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Practice with products 3
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Find the quadratic approximation for [mathjaxinline]\ln (x)\sin (x-1)[/mathjaxinline] near [mathjaxinline]x=1[/mathjaxinline]. </p>
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(Do not enter the big-O term.) </p>
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<h2 class="hd hd-2 unit-title">13. Cubics</h2>
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Linear approximation of cubic
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What is the linear approximation for the function [mathjaxinline]\ f(x) = x^3[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline]? </p>
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(Do not enter the big-O term.) </p>
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Quadratic approximation of cubic
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What is the quadratic approximation for the function [mathjaxinline]\ f(x) = x^3[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline]? </p>
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(Do not enter the big-O term.) </p>
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<p><b class="bfseries">Advertisement</b></p><p>
To understand how to better approximate a cubic function, we'll be learning about Taylor polynomials and Taylor series in 18.01.3x! Stay tuned for this. </p>
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<h2 class="hd hd-2 unit-title">14. Summary</h2>
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<p><b class="bfseries">Best fit quadratic</b></p><p>
The best fit quadratic to a function [mathjaxinline]\ f(x)[/mathjaxinline] at the point [mathjaxinline]x=0[/mathjaxinline] is the quadratic function [mathjaxinline]q(x)[/mathjaxinline] whose value agrees 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="a0000000574" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000575"><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:center; border:none">
[mathjaxinline]\displaystyle =[/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">(5.49)</td></tr><tr id="a0000000576"><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:center; border:none">
[mathjaxinline]\displaystyle =[/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">(5.50)</td></tr><tr id="a0000000577"><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:center; border:none">
[mathjaxinline]\displaystyle =[/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">(5.51)</td></tr></table><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="a0000000578" 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">(5.52)</td></tr></table><p>
When [mathjaxinline]a=0[/mathjaxinline], this quadratic approximation becomes </p><table id="a0000000579" 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">(5.53)</td></tr></table><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><p><b class="bfseries">Library of quadratic approximations</b></p><p>
Near zero: [mathjaxinline]\ \displaystyle f(x) \approx f(0) + f'(0)x + \frac{f^{\prime \prime }(0)}{2}x^2[/mathjaxinline]. </p><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle {e^ x = 1 + x + \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\sin (x) = x + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\cos (x) = 1- \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {\ln (1+x) = x- \frac{x^2}{2} + O\left(x^3\right)}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\displaystyle {(1+x)^ r = 1 + rx + \frac{r(r-1)}{2}x^2 + O\left(x^3\right)}[/mathjaxinline] </p></li></ul>
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