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<h2 class="hd hd-2 unit-title">1. About the exam</h2>
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The exam you take will have 12 questions, some of which will have multiple parts. Expect it to take between 3 and 4 hours. WARNING: Each problem has only 2 attempts. </p><p>
DO NOT OPEN THE EXAM UNTIL YOU ARE READY TO TAKE IT. The exam is timed! Once you open the exam, you have 48 hours to finish the exam. (Note if you start the exam with less than 48 hours before the exam due date, you will not be granted the full 48 hours.) </p><p>
UNTIL THEN: Feel free to study using the practice problems provided here! You are free to discuss these practice problems in as much detail as you would like on the forum. You may NOT discuss EXAM problems on the forum at all. </p>
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<h2 class="hd hd-2 unit-title">2. Preparing for the exam</h2>
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Students typically find it most useful to do lots of practice problems when studying for the exam. There are links below to practice exam problems. Keep in mind that none of the practice exams will be identical in length to the exam you will be given. </p><p><h2>Practice problems: </h2> These problems and solutions are taken from the 18.01 OCW course. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Exam 3 (18.01SC OCW Scholar) </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/exam-3/materials-for-exam-3/MIT18_01SCF10_exam3.pdf" target="_blank">pdf</a></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/exam-3/materials-for-exam-3/MIT18_01SCF10_exam3sol.pdf" target="_blank">solutions</a></td></tr><tr><td style="text-align:left; border:none">
Practice Exam 3A (18.01 OCW) </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/prexam3a.pdf" target="_blank">pdf</a></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/prexam3asol.pdf" target="_blank">solutions</a></td></tr><tr><td style="text-align:left; border:none">
Practice Exam 3B (18.01 OCW) </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/prexam3b.pdf" target="_blank">pdf</a></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/prexam3bsol.pdf" target="_blank">solutions</a></td></tr><tr><td style="text-align:left; border:none">
Exam 3 (18.01 OCW) </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/exam3.pdf" target="_blank">pdf</a></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none"><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/exam3sol.pdf" target="_blank">solutions</a></td></tr></table><p><h2>More review problems from OCW:</h2></p><p>
These problems are taken from Exam 2 on 18.01 on OCW: </p><p><h3>Exam 2 #5b</h3> Find [mathjaxinline]y(x)[/mathjaxinline] such that [mathjaxinline]y' = \frac{1}{y^3}[/mathjaxinline] and [mathjaxinline]y(0) =1[/mathjaxinline]. </p><p><h3>Exam 2 #6</h3> Suppose that [mathjaxinline]\ \displaystyle f'(x) = e^{(x^2)}[/mathjaxinline], and [mathjaxinline]\ f(0)=10[/mathjaxinline]. One can conclude from the Mean Value Theorem that </p><table id="a0000002981" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle A < f(1) < B[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
for which numbers A and B? </p><p>
Solutions to these problems can be found here: <a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/exam2sol.pdf" target="_blank">pdf solutions</a> </p>
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<h2 class="hd hd-2 unit-title">3. Review problems from old AP exams</h2>
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These two problems from old AP exams are particularly nice at tying multiple concepts together. We will not offer solutions to these AP problems. Instead we encourage you to try the problems on your own, and discuss solutions on the forum. </p><p><h3>2011 AP BC Free Response Question #4:</h3></p><p>
The graph of a differentiable function [mathjaxinline]y=f(x)[/mathjaxinline] with domain [mathjaxinline]0 \leq x \leq 10[/mathjaxinline] is shown in the figure below. </p><center><img src="/assets/courseware/v1/03826db8dad911e1417e5bf3b9cafce3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_exam2prep_AP.svg" width="500 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><ul class="itemize"><li><p>
The area of the region enclosed between the graph of [mathjaxinline]\ f[/mathjaxinline] and the [mathjaxinline]x[/mathjaxinline]-axis for [mathjaxinline]0 \leq x \leq 5[/mathjaxinline] is [mathjaxinline]10[/mathjaxinline]. </p></li><li><p>
The area of the region enclosed between the graph of [mathjaxinline]\ f[/mathjaxinline] and the [mathjaxinline]x[/mathjaxinline]-axis for [mathjaxinline]5 \leq x \leq 10[/mathjaxinline] is [mathjaxinline]27[/mathjaxinline]. </p></li><li><p>
The arc length of the portion of the graph of [mathjaxinline]\ f[/mathjaxinline] between [mathjaxinline]x=0[/mathjaxinline] and [mathjaxinline]x=5[/mathjaxinline] is 11. </p></li><li><p>
The arc length of the portion of the graph of [mathjaxinline]\ f[/mathjaxinline] between [mathjaxinline]x=5[/mathjaxinline] and [mathjaxinline]x=10[/mathjaxinline] is 18. </p></li><li><p>
The function [mathjaxinline]\ f[/mathjaxinline] has exactly two critical points located at [mathjaxinline]x=3[/mathjaxinline] and [mathjaxinline]x=8[/mathjaxinline]. </p></li></ul><p>
(a) Find the average value of [mathjaxinline]\ f[/mathjaxinline] on the interval [mathjaxinline]0 \leq f \leq 5[/mathjaxinline]. </p><p>
(b) Evaluate [mathjaxinline]\displaystyle \int _0^{10}\left(3f(x)+2\right)\, dx.[/mathjaxinline] </p><p>
(c) Let [mathjaxinline]\ g(x) = \displaystyle \int _5^ x f(t) \, dt[/mathjaxinline]. On what intervals, if any, is the graph of [mathjaxinline]\ g[/mathjaxinline] both concave up and decreasing? </p><p>
(d) The function [mathjaxinline]h[/mathjaxinline] is defined by [mathjaxinline]\displaystyle \ h(x) = 2f\left(\frac{x}{2}\right)[/mathjaxinline]. The derivative of [mathjaxinline]h[/mathjaxinline] is [mathjaxinline]h'(x) = f'\left(\frac{x}{2}\right)[/mathjaxinline]. Find the arc length of the graph of [mathjaxinline]y=h(x)[/mathjaxinline] from [mathjaxinline]x=0[/mathjaxinline] to [mathjaxinline]x=20[/mathjaxinline]. </p><p><h3>2011 AP BC Free Response Question #5:</h3></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]t[/mathjaxinline] (seconds) </td><td style="text-align:center; border:none">
0 </td><td style="text-align:center; border:none">
10 </td><td style="text-align:center; border:none">
40 </td><td style="text-align:center; border:none">
60</td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]B(t)[/mathjaxinline] (meters) </td><td style="text-align:center; border:none">
100 </td><td style="text-align:center; border:none">
136 </td><td style="text-align:center; border:none">
9 </td><td style="text-align:center; border:none">
49</td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]v(t)[/mathjaxinline] (meters per second) </td><td style="text-align:center; border:none">
2.0 </td><td style="text-align:center; border:none">
2.3 </td><td style="text-align:center; border:none">
2.5 </td><td style="text-align:center; border:none">
4.6</td></tr></table><p>
Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function [mathjaxinline]B[/mathjaxinline] models Ben's position on the track, measured in meters from the western end of the track, at time [mathjaxinline]t[/mathjaxinline], measured in seconds from the start of the ride. The table above gives values for [mathjaxinline]\ B(t)[/mathjaxinline] and Ben's velocity, [mathjaxinline]\ v(t)[/mathjaxinline], measured in meters per second, at selected times [mathjaxinline]t[/mathjaxinline]. </p><p>
(a) Use the data in the table to approximate Ben's acceleration at time [mathjaxinline]t=5[/mathjaxinline] seconds. Indicate units of measure. </p><p>
(b) Using correct units, interpret the meaning of [mathjaxinline]\displaystyle \int _0^{60}\left| v(t) \right| \, dt[/mathjaxinline] in the context of this problem. Approximate [mathjaxinline]\displaystyle \int _0^{60}\left| v(t) \right| \, dt[/mathjaxinline] using a left Riemann sum with the subintervals indicated by the data in the table. </p><p>
(c) For [mathjaxinline]40 \leq t \leq 60[/mathjaxinline], must there be a time [mathjaxinline]t[/mathjaxinline] when Ben's velocity is [mathjaxinline]2[/mathjaxinline] meters per second? Justify. </p><p>
(d) A light is directly above the western end of the track. Ben rides so that at time [mathjaxinline]t[/mathjaxinline], the distance [mathjaxinline]L(t)[/mathjaxinline] between Ben and the light satisfies </p><table id="a0000002982" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left(L(t)\right)^2 = 12^2 + \left(B(t)\right)^2.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
At what rate is the distance between Ben and the light changing at time [mathjaxinline]t=40[/mathjaxinline]? </p>
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