MATH 142 Fall 2015 Review Exam
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Compute the following antiderivatives:
(a) \( \displaystyle{\int 2x^3\, dx} \)
(b) \( \displaystyle{\int 2x\sqrt{1+x^2}\, dx} \)
(c) \( \displaystyle{x^{50} \sin \big( x^{43} \big)}\, dx \)
(d) \( \displaystyle{\int x^3 \ln x\, dx} \)
(e) \( \displaystyle{\int e^x \sin x\, dx} \)
(f) \( \displaystyle{\int \sec x \tan x \, dx} \)
(g) \( \displaystyle{\int \frac{dx}{x^2+4}} \)
(h) \( \displaystyle{\int \frac{x^2}{\sqrt{25-x^2}}\, dx} \)
(i) \( \displaystyle{\int \frac{5}{(x+3)(x-2)}\, dx} \)
(j) \( \displaystyle{\int \frac{21x}{(x+3)(x+2)}\, dx} \)
(k) \( \displaystyle{\int \frac{x^3+5x^2+3x+11}{(x^2+4x+ 5)^2}\, dx} \)
(l) \( \displaystyle{\int \frac{x^4+7x^3-2}{x-11}\, dx} \)
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Compute the derivative of the following functions:
(a) \( f(x) = \displaystyle{\int_3^{x^2} e^{t^4}\, dt} \).
(b) \( f(x) = \displaystyle{\int_{\frac{x^3-2x^2}{\tan^2 x}}^{\sec^2 x} \cos^3 \big( t^5 \big) \, dt} \).
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Compute the following:
(a) \( \displaystyle{\int_5^\infty \sqrt{x+2}\, dx} \).
(b) \( \displaystyle{\int_{-\infty}^\infty \frac{dx}{1+x^2} } \).
(c) \( \displaystyle{\int_1^4 \frac{dx}{x-1} } \).
(d) \( \displaystyle{\int_0^2 \frac{dx}{x(x-2)} } \).
(e) \( \displaystyle{\int_0^{2\pi} \csc x\, dx} \).
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Find the area of the region bounded by the curves \( y=7x^2-2 \) and \( y=3-x/2 \).
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Find the area of the region bounded by the curves \( y = \cos^2 x \) and \( y = \sin^2 x \) in the interval \( -\frac{\pi}{2} \leq x \leq \pi \).
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Find the volume of the solid obtained by rotating the graph of \( y = \sqrt{x} \) between \( x=0 \) and \( x=4 \) around the x-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( y = x \) and \( y = \sqrt{x} \) around the x-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( x= 4y^{1/3}, x=0, x=10 \) around the y-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( x=3, x=4+y^2/64, y=2, y=8 \), around the y-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( y = 1/x, x=1, x=2 \), around the y-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( x=y^3+3y^2, x=0 \), around the x-axis.
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Find the volume of the solid obtained by rotating the region bounded by the graphs of \( x=1, x=4y^2-4y^3 \), around the line \( y=6 \).
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Compute the next three terms of the following sequences:
(a) \( \big\{ 1, \dfrac{1}{3}, \dfrac{1}{5}, \dfrac{1}{9}, \dfrac{1}{11}, \dotsc, \big\} \)
(b) \( \big\{ 1, -\dfrac{3}{4}, \dfrac{9}{16}, -\dfrac{27}{64}, \dotsc, \big\} \)
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Compute the following limits:
(a) \( \lim_{n\to\infty} \dfrac{n}{2n+1} \)
(b) \( \lim_{n\to\infty} \dfrac{5+3n^2}{n+3n^2} \)
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Study the convergence of the following series:
(a) \( \displaystyle{\sum_{n=13}^\infty \sqrt{\frac{7n^3+5}{5n^5+3n^2+1}}} \).
(b) \( \displaystyle{\sum_{n=1}^\infty \frac{5+6\sin(n^2)}{10^n}} \).
(c) \( \displaystyle{\sum_{n=0}^\infty \frac{6}{5n^2+7n+3}} \).
(d) \( \displaystyle{\sum_{n=2}^\infty \frac{2}{2^n-2}} \).
(e) \( \displaystyle{\sum_{n=1}^\infty \frac{(-8)^n}{n!}} \).
(f) \( \displaystyle{\sum_{n=1}^\infty \frac{(2n)!}{2^n n!}} \).
(g) \( \displaystyle{\sum_{n=10}^\infty \frac{4n^3+5}{7n^2-11n^3}} \).
(h) \( \displaystyle{\sum_{n=0}^\infty n^2 e^{-n}} \).
(i) \( \displaystyle{\sum_{n=1}^\infty \frac{e^{2n}}{n^n}} \).
(j) \( \displaystyle{\sum_{n=1}^\infty \frac{e^n \ln n}{3^n n^3}} \).
(k) \( \displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}} \).
(l) \( \displaystyle{\sum_{n=0}^\infty \frac{\cos \pi n}{\sqrt{n}}} \).
(m) \( \displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}} \).
(n) \( \displaystyle{\sum_{n=0}^\infty \frac{(-1)^{n-1}}{2n+1}} \).
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Find the sum of the following series:
(a) \( \displaystyle{\sum_{n=1}^\infty 4682 \big( \tfrac{1}{2} \big)^n} \).
(b) \( \displaystyle{\sum_{n=0}^\infty 47 \big( \tfrac{5}{64} \big)^{n+3}} \).
(c) \( \displaystyle{\sum_{n=1}^\infty \frac{1}{4n^2-1}} \).
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Find a power series representation of the following functions:
(a) \( f(x) = \dfrac{1}{4+x^2} \).
(b) \( f(x) = \dfrac{5}{2-3x} \).
(c) \( f(x) = \dfrac{3}{(x-1)(x+2)} \).
(d) \( f(x) = \dfrac{\ln (3x)}{(x^2-2)(x+5)} \).
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Find the Taylor series of the following functions:
(a) \( f(x) = 10^x \) at \( x = 0 \).
(b) \( f(x) = \cos(10 x) \) at \( x = 0 \).
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Find the center and radius of convergence of the following series
(a) \( \displaystyle{\sum_{n=0}^\infty (-3)^n \sqrt{n+1} (x+1)^n } \).
(b) \( \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n}{n 10^n} (x-2)^n } \).
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Find the interval of convergence of the following series:
(a) \( \displaystyle{\sum_{n=0}^\infty \frac{x^n}{n!}} \).
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Find a power series representation for the function \( f(x) = \dfrac{\ln (x-5)}{(x+10)} \), and compute center, radius of convergence, and domain.
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Compute the area of one loop of the curve given in polar coordinates by \( r = \sin (5\theta) \).
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Compute the area of the region bounded by the graphs of the curves given in polar coordinates by \( r = 3 \cos \theta \) and \( r = 3 \sin \theta \).
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Express in rectangular coordinates the curves given in polar coordinates by \( r=2 \) and \( r=1-\sin\theta \).
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Express in polar coordinates the curves given in rectangular coordinates by \( x+y=5 \) and \( 7y=x^2 \).
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Find the length of the following polar curves:
(a) \( r=4 \) for \( 0 \leq \theta \leq 13\pi/20 \).
(b) \( r = e^{5\theta} \) for \( 0 \leq \theta \leq \pi \).
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Sketch the curve given in parametric equations by \( x = 2\sin t, y = \cos^2 t \) for \( 0 \leq t \leq 2\pi \).
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Sketch the curve given in parametric equations by \( x = 3\cos(5t), y = 1 + \cos(5t) \) for \( 0 \leq t \leq \pi/2 \).