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Your name:

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Final Exam., Math. 120, Fall, 1999



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\textbf{Instructions}:  The exam. consists of 22 problems.  There is
space to work beside each problem.  Most of the problems offer
multiple-choice answers.  You should use the answers only to check
yourself, not as a means of \emph{finding} an answer.  It is important to
show your work, and make your steps clear and readable.  Partial credit
may be given for solutions that are set up correctly, even though the
final answer is incorrect, and full credit may not be given for a correct
answer that is not supported by correct work.


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Please remember your obligations under the honor code.

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The spaces below are for recording scores, \emph{not} for answers.

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 1. \hspace{2.5 in} 12.

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 2. \hspace{2.5 in}  13.

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 3. \hspace{2.5 in}  14.

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 4. \hspace{2.5 in}  15.

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 5. \hspace{2.5 in}  16.

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 6. \hspace{2.5 in}  17.

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 7. \hspace{2.5 in}  18.

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 8. \hspace{2.5 in}  19.

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 9. \hspace{2.5 in}  20.

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10.\hspace{2.5 in}  21.

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11.\hspace{2.5 in}  22.


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1.  If $f$ is continuous on the interval $[1,3]$ and $2\leq f(x)\leq 4$,
what can you say about $\int_{1}^{3}f(x)\,dx$?\\

(a)  $1\leq \int_{1}^{3}f(x)\,dx\leq 5$ \hspace{.5in}
(b)  $2\leq \int_{1}^{3}f(x)\,dx\leq 6$ \hspace{.5in}\\

(c)  $3\leq\int_{1}^{3}f(x)\,dx\leq 7$ \hspace{.5in}
(d)  $4\leq \int_{1}^{3}f(x)\,dx\leq 8$\\

(e)  it is impossible to say anything without knowing the function

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2.  If $F(x) = \int_{0}^{\sin x}e^{t^{2}}\,dt$, what is $F'(x)$?\\

(a) $e^{x^{2}}$\hspace{.3in}
(b) $e^{\sin^{2}x}\cos x$\hspace{.3in}
(c) $e^{x}\cos x$\hspace{.3in}
(d) $e^{x^{2}}\cos x$\hspace{.3in}
(e) $e^{\cos x}\sin x$

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3.  Find $\int_{0}^{\frac{\pi}{2}}\sin^{2} x\cos x\,dx$.\\

(a)  $0$ \hspace{.5in}
(b)  $\frac{1}{12}$ \hspace{.5in}
(c)  $\frac{1}{6}$ \hspace{.5in}
(d)  $\frac{1}{4}$ \hspace{.5in}
(e)  $\frac{1}{3}$

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4. What is the area of the region between the curves $y = e^{x}$ and
$y = x$, for $0\leq x\leq 1$?\\

(a)  $e-\frac{3}{2}$ \hspace{.3in}
(b)  $\frac{2e}{3}$ \hspace{.3in}
(c)  $1-\frac{1}{e}$ \hspace{.3in}
(d)  $e+\frac{1}{2}$ \hspace{.3in}
(e)  $1+\frac{e}{2}$

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5.  The region bounded by $y = e^{x}$, $y = x$, $x = 0$, and $x = 1$ is
rotated about the $x$-axis.  What is the volume of the resulting solid of
revolution?\\

(a)  $2\pi e^{2}$\hspace{.5in}
(b)  $\frac{1}{2}(3e^{2}+5)$\hspace{.5in}
(c)  $\frac{\pi}{6}(3e^{2}-5)$\hspace{.5in}\\

(d)  $\frac{\pi}{3}(e^{2}+1)$\hspace{.8in}
(e)  $\frac{3}{2} (e + 1)$

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6.  If it requires a force of $2$ lbs. to hold a spring at $1$
ft. beyond its natural length, what is the work (in ft.-lbs.)
done in stretching the spring from its natural length to $3$
ft. beyond that length?\\

(a)  $5$ \hspace{.5in}
(b)  $6$ \hspace{.5in}
(c)  $7$ \hspace{.5in}
(d)  $8$ \hspace{.5in}
(e)  $9$

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7.  Complete the statement of the theorem below.\\

\textbf{Mean-value Theorem for Integrals}:  Let $f$ be continuous on the
closed interval $[a,b]$.  Then there exists $c$ in the open interval
$(a,b)$ such that\\

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$\int_{a}^{b}f(x)\,dx = $

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8.  What is $\frac{d}{dx}x^{2x}$?\\

(a)  $x^{2x-1}$\hspace{.5in}
(b)  $x^{2x}$ \hspace{.5in}
(c)  $x^{2x}(2 + 2\ln x)$\hspace{.5in}\\

(d)  $2x(1-\ln x)$\hspace{.8in}
(e)  $2\,x^{2x-1}$

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9.  What is $\int_{1}^{e}\frac{1}{x}\,dx$?\\

(a)  $0$ \hspace{.5in}
(b)  $1$ \hspace{.5in}
(c) $e$ \hspace{.5in}
(d) $1-e$ \hspace{.5in}
(e) $e-1$

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10.  If $y = \frac{e^{x}(x+1)^{6}}{(x-3)^{2}}$, what is $y'$?\\

(a)  $\frac{e^{x}(x+1)^{6}}{(x-3)^{2}}
(1+\frac{6}{x+1}-\frac{2}{x-3})$ \hspace{.5in}
(b)  $(1+\frac{6}{x+1}-\frac{2}{x-3})$ \hspace{.5in}
(c)  $\frac{3e^{x}(x+1)^{5}}{x-3}$ \hspace{.5in}\\

(d)  $e^{x}+6(x+1)^{5}-2(x-3)$ \hspace{.5in}
(e)  $e^{x}+6\ln(x+1)-2\ln(x-3)$

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11.  A radioactive substance has a half-life of
$10$ years.  How long (in years) will it take for a sample of $10$ g.\ to
decay to
$1$ g.?\\

(a)  $\frac{\ln 2}{10}$ \hspace{.3in}
(b)  $\frac{10\ln 10}{\ln 2}$ \hspace{.3in}
(c)  $\frac{2}{\ln 10}$ \hspace{.3in}
(d)  $10\ln 2$ \hspace{.3in}
(e)  $2\ln 10$

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12.  What is $\lim_{x\rightarrow 0}\frac{\sin 2x}{x}$?\\

(a)  $0$ \hspace{.5in}
(b)  $\frac{\pi}{2}$ \hspace{.5in}
(c)  $1$ \hspace{.5in}
(d)  $\frac{2}{\pi}$ \hspace{.5in}
(e)  $2$

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13.  Find $\int_{0}^{1}xe^{x}\,dx$.\\

(a)  $1$ \hspace{.5in}
(b)  $2$ \hspace{.5in}
(c)  $e$ \hspace{.5in}
(d)  $2e$ \hspace{.5in}
(e) $2e-1$

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14.  Find $\int_{4}^{5}\frac{x}{(x-2)(x-3)}\,dx$.\\

(a)  $5\ln 2 - 2\ln 3$ \hspace{.5in}
(b)  $7\ln 3 + 4\ln 2$ \hspace{.5in}
(c)  $3\ln 2 + 2\ln 3$\\

(d)  $5\ln 3 - 2\ln 2$ \hspace{.8in}
(e)  $7\ln 2 - 2\ln 3$

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15.  What is $\int_{0}^{\pi}\sin^{2}x\,dx$?\\

(a)  $\frac{\pi}{2}$ \hspace{.5in}
(b)  $\frac{\pi}{8}$ \hspace{.5in}
(c)  $\frac{\pi}{12}$ \hspace{.5in}
(d)  $\frac{\pi}{24}$ \hspace{.5in}  (e) $2\pi$

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16.  What is the centroid of the region bounded by $y = x^{2}$
and $y = x^{3}$?\\

(a)  $(\frac{4}{5},\frac{1}{7})$ \hspace{.3in}
(b)  $(\frac{3}{5},\frac{12}{35})$ \hspace{.3in}
(c)  $(\frac{5}{4},\frac{7}{3})$ \hspace{.3in}
(d)  $(\frac{1}{20},\frac{1}{35})$ \hspace{.3in}
(e)  $(\frac{5}{3},\frac{5}{9})$

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17.  Which of the following integrals represents the length of
the curve $y = x^{2}$, for $0\leq x\leq \frac{1}{2}$?\\

(a)  $\int_{0}^{\frac{1}{2}}(1 + 4x^{2})\,dx$ \hspace{.5in}
(b)  $\int_{0}^{\frac{1}{2}}\sqrt{1+2x}\,dx$ \hspace{.5in}
(c)  $\int_{0}^{\frac{1}{2}}\sqrt{1+4x^{2}}\,dx$\\

(d)  $\int_{0}^{\frac{1}{2}}(1+2x)\,dx$ \hspace{.8in}
(e)  $\int_{0}^{\frac{1}{2}}\sqrt{1+x^{4}}\,dx$

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18.  An appropriate substitution transforms the integral
$\int_{0}^{\frac{1}{2}}\sqrt{1+4x^{2}}\,dx$ into which of the
following?\\

(a)  $\int_{0}^{\frac{\pi}{4}}\frac{1}{2}\sec^{3}\theta\,d\theta$
\hspace{.5in}
(b)  $\int_{0}^{\frac{1}{2}}\sec\theta\,d\theta$ \hspace{.5in}
(c) $\int_{0}^{\frac{1}{2}}\sin^{3} \theta\,d\theta$\\

(d)  $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}
\sec^{3}\theta\,d\theta$ \hspace{.8in}
(e)
$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\frac{1}{2}\sec\theta\,d\theta$

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19.  If a fair die is rolled four times, what is the probability
that it will come up $1$ exactly twice?\\

(a)  $\frac{23}{216}$ \hspace{.5in}
(b)  $\frac{1}{9}$ \hspace{.5in}
(c)  $\frac{25}{216}$ \hspace{.5in}
(d)  $\frac{13}{108}$ \hspace{.5in}
(e)  $\frac{1}{8}$

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20.  What is $1+2+2^{2}+\ldots+2^{56}$?\\

(a)  $2^{57}-1$ \hspace{.5in}
(b)  $2^{56}+1$ \hspace{.5in}
(c)  $2^{55}-1$ \hspace{.5in}
(d)  $2^{56}+1$\\

(e)  there is no sum, since $|2|\geq 1$

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21.  For the function $f(x) = \cos x$, find the $5^{th}$ Taylor
polynomial $P_{5}(x)$.\\

(a)  $1+x+x^{2}+x^{3}+x^{4}+x^{5}$ \hspace{.3in}
(b)  $1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+
\frac{1}{24}x^{4}+\frac{1}{120}x^{5}$\\

(c)  $1-\frac{1}{2}x^{2}+\frac{1}{24}x^{4}$\hspace{.3in}
(d)  $x-\frac{1}{6}x^{3}+\frac{1}{120}x^{5}$\hspace{.3in}
(e)  $1+x+\frac{1}{2}x^{2}+\frac{1}{3}x^{3}+
\frac{1}{4}x^{4}+\frac{1}{5}x^{5}$

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22.  Find the Taylor series for $f(x) = e^{x^{2}}$.\\

(a)  $\sum_{k=0}^{\infty}\frac{x^{2k}}{k!}$\hspace{.5in}
(b)  $\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$\hspace{.5in}
(c)  $\sum_{k=0}^{\infty}\frac{x^{2k}}{(2k)!}$\hspace{.5in}
(d)  $\sum_{k=0}^{\infty}\frac{x^{2k+1}}{(2k+1)!}$\\

(e)  the function has no Taylor series

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