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\centerline{\bf Math 120, Final}
\centerline{May 7, 2001}
\vskip10pt

\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own. 
\itemitem{$\bullet$} Be sure that you have all 25 questions (6 points each). 
%\itemitem{$\bullet$} No calculators are to be used.
\itemitem{$\bullet$} The exam lasts for two hours.
\itemitem{$\bullet$} You are to hand in just the front page.
\vskip5pt
\centerline{Good Luck!}
\vskip10pt
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Score: \ul{1in}
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\long\def\Exam{%
\coversheet{ }%

\vfill\eject\headline{\hfil}

%% 1 1 
\five
\PROB
The area of the region enclosed by the curves $y=x+1$ and
$y=(x-1)^2$ between $x=0$ and $x=2$ is
\ANS
(a) $\frac{10}{3}$
(b) $\frac{-10}{3}$
(c) $\frac{9}{2}$
(d) $\frac{26}{3}$
(e) $\frac{14}{3}$
\CORR31245.


%% 2 2
\two 
\PROB
Let $R$ be the region between the graph of $y=\frac{1}{x^2}$ and the
$x$-axis for $x$ between $1$ and $3$.
Consider the solid obtained by rotating $R$
about the line
$y=-1$. Which of the following integrals gives the volume of this solid?
\ANS
(a) $\DSP \pi \int_{1}^{3} \left(\frac{1}{x^2}+1\right)^2-1\, dx$
(b) $\DSP \pi \int_{1}^{3}  \frac{1}{x^4}-1\, dx$
(c) $\DSP \pi \int_{1}^{3}  \left(\frac{1}{x^2}+1\right)^2+2 \,dx$
(d) $\DSP \pi \int_{1}^{3}  \frac{1}{x^4} \,dx$
(e) $\DSP \pi \int_{1}^{3} \frac{1}{y}\,dy$
\CORR12354.

\newpage
%% 3 3
\two
\PROB
Let $R$ be the region between the graph of $y=\sin x$ and the $x$-axis
for $x$ between $0$ and~$\pi$. Consider the solid obtained by rotating
$R$ about the line $x=-1$. Which of the following integrals gives the
volume of this solid?
\ANS
(a) $\DSP\int_0^\pi 2\pi(x+1)\sin x\,dx$
(b) $\DSP\int_{-1}^\pi 2\pi x\sin x\,dx$
(c) $\DSP\int_0^\pi \pi (x\sin x)^2\,dx$
(d) $\DSP\int_{-1}^\pi \pi\sin^2x\,dx$
(e) $\DSP\int_0^\pi2\pi x(1-\sin x)\,dx$
\CORR21543.

%% 4 4
\five
\PROB
The work done in stretching a large spring from its natural length of 2 meters
to 5 meters equals 6 joules. Find the work done in stretching it from
5 meters to 8 meters.
\ANS
(a) 18 joules
(b) 6 joules
(c) 2 joules
(d) 36 joules
(e) 3 joules
\CORR15432.

\newpage
%% 5 5
\PROB
The average value of the function 
$f(x)= 3-3x^2$ on the interval $[0,2]$ is
\ANS
(a) $-1$
(b) $-4$
(c) $2$
(d) $ 1$
(e) $0$
\CORR51432.

%% 6 6
\PROB
Let $$f(x)= \sqrt{ x^3+x^2+x+1}$$ for $x\ge0$, and let $g$ be the inverse
function for $f$. Observe that $f(1)=2$. What is $g'(2)$?
\ANS
(a) $\frac{2}{3}$
(b) $\frac{3}{2}$
(c) $\frac{1}{2}$
(d) $\frac{1}{3}$
(e) $4$
\CORR14532.

\newpage
%%% 8 7 
%%Suppose that $f(t) = (2+e^t)^{1/2}$ Find $ f'(t)$.
%%(a) $\frac{e^t}{2(2+e^t)^{1/2}}$
%%(b) $(2\3)(2+e^t)^{3/2}$
%%(c) $(2+e^t)^{-1/2}$
%%(d) $e^t(2+e^t)^{1/2}$
%%(e) $2/(2+e^t)$

%% 9 8
\PROB
The limit $$\DSP\lim_{x\rightarrow 2^-} \ln (2-x) $$ is
\ANS
(a) $-\infty$
(b) $\infty$
(c) $0$
(d) $1$
(e) $-1$
\CORR31245.

%10
\three
\PROB
Solve the equation $$\DSP\ln\left(\frac{x+1}{x-1}\right)=2$$
\ANS
(a) $x=\frac{e^2+1}{e^2-1}$
(b) $x=\frac{e-1}{e+1}$
(c) $x=\frac{\ln 2}{1-\ln 2}$
(d) $x=\frac{e^2}2$
(e) $x=\frac 2{e^2}$
\CORR13245.

\newpage
%% 11 9
\three
\PROB
The equation of the line tangent to the curve $y=x\ln(x)$ at the point
$(e,e)$ is
\ANS
(a) $y=2x -e$
(b) $y=e$
(c) $y=2x+1$
(d) $y=ex-1$
(e) $y=x+2$
\CORR43215.


%%%12 10
%%2. Let $ x^y=y^x$. Find $y'$. (Hint: use implicit differentiation.)
%%a) $y'= \frac{\ln y -\frac{y}{x}}{\ln x -\frac{x}{y}}$
%%b) $y'= \frac{\ln y +\frac{x}{y} -\frac{y}{x}}{\ln x}$
%%c) $y'= \ln y -\ln y +\frac{x}{y} -\frac{y}{x}$
%%d) $y'= \frac{\frac{x}{y}-\frac{y}{x}}{\ln x} $
%%e) $y'= \frac{\ln y}{\ln x} -1$

%% 12 10
\two
\PROB
Find the derivative of $x^{\sin x}$.
\ANS
(a) $(\cos x \ln x +\frac{1}{x} \sin x) x^{\sin x}$
(b) $\frac {x^{\sin x }\cos x}{x}$
(c) $x^{\sin x} (\ln x)( \cos x)$
(d) $x^{\cos x}$
(e) $x^{\sin x} \cos x$
\CORR41253.

\newpage
%% 12 11 
\five
\PROB
Calculate $\DSP\int_e^{e^2} \frac{dx}{x\ln(x)}$.
\ANS
(a) $\ln 2$
(b) $1/2$
(c) $e/2 - 1$
(d) $3$
(e) $1 - \ln 2$
\CORR14325.

% 14
\two
\PROB
Find $\DSP\lim_{x\to\infty}\frac{x}{\ln(1+e^{2x})}$, if it exists.
\ANS
(a) $1/2$
(b) $1$
(c) $0$
(d) $1/4$
(e) limit does not exist
\CORR \43215.

\newpage
%% 15 12
\three
\PROB
Evaluate the integral $\DSP\int_1^3 x^2\ln x \,dx$.
\ANS
(a) $9\ln 3 -\frac{26}{9}$
(b) $9\ln 3 -9$
(c) $9\ln 3 -\frac{82}{9}$
(d) $9\ln 3 -\frac{79}{9}$
(e) $9\ln 3 -3$
\CORR25341.

%% 16 13
\five
\PROB
Calculate $\DSP \int_0^{\pi/2}\sin^6x\cos^3x\,dx$.
\ANS
(a) $2/63$
(b) $1/32$
(c) $1/37$
(d) $3/127$
(e) $\pi/130$
\CORR41325.

\newpage
% 17
\three
\PROB In order to evaluate the integral $\DSP\int\frac{dx}{x^2+9}$,
what substitution should you make?
\ANS
(a) $x=3\tan u$
(b) $x=\frac{\tan u }3$
(c) $u=3+ \tan x$
(d) $u=3\tan^{-1}x$
(e) $u=x^2$
\CORR31245.


% 23
\two
\PROB A bank account pays $6\%$ interest, compounded continuously. If you
deposit \$1,000 into the account, how  long will it be before the
account grows to \$2,000?
\ANS
(a) $\frac{\ln 2}{0.06}\approx 11.5$ years
(b) $\frac{1}{0.06}\approx 16.66$ years
(c) $\frac{\ln 2}{e^{0.06}-1}\approx 11.2$ years
(d) $\frac{e^{0.06}}{0.06}\approx 17.7$ years
(e) $\frac{1}{0.12}\approx8.3$ years
\CORR24351.


\newpage
%% 18 14
\one
\PROB
Which of the following is the correct partial fraction representation of
$$\frac{2x^3-x+1}{(x-2)^3(x^2+1)^2(x^2-1)}$$
\ANS
(a) $\frac{A}{x-2}+\frac{B}{(x-2)^2}+
\frac{C}{(x-2)^3}+\frac{Dx+E}{x^2+1}+
\frac{Fx+G}{(x^2+1)^2}+\frac{H}{x-1}+\frac{I}{x+1}$
(b) $\frac{A}{(x-2)^3}+\frac{B}{x^2+1}+\frac{C}{x^2-1}$
(c) $\frac{A}{x-2}+\frac{B}{(x-2)^2}+
\frac{C}{(x-2)^3}+\frac{Dx+E}{(x^2+1)^2}+
\frac{Fx+G}{x^2-1}$
(d) $\frac{A}{x-2}+\frac{B}{(x-2)^2}+
\frac{C}{(x-2)^3}+\frac{D}{x^2+1}+
\frac{E}{(x^2+1)^2}+\frac{H}{x-1}+\frac{I}{x+1}$
(e) $\frac{A}{x-2}+\frac{B}{(x-2)^2}+
\frac{C}{(x-2)^3}+\frac{D}{x^2+1}+
\frac{F}{(x^2+1)^2}+\frac{H}{x^2-1}$
\CORR51432.

\newpage
% 20
\five
\PROB
Suppose that the function $f(x)$ is given by the formula
$$f(x)=\int_0^x\sqrt{t^2+2t}\,dt$$ Find the length of the graph of
$f(x)$ for
$x$ between $0$ and~$2$.
\ANS
(a) $4$
(b) $3$
(c) $2$
(d) $1$
(e) $\frac32$
\CORR13245.

% 21
\PROB
Find the centroid of the region above the $x$-axis and below the graph
of $y=x^2$ for $x$ between $-1$ and~$1$.
\ANS
(a) $(0,3/10)$
(b) $(0,2/3)$
(c) $(0,2/15)$
(d) $(0, 3/2)$
(e) $(0, 4/5)$
\CORR21345.

\newpage
%% 22 15
\one
\PROB
For which real numbers $A$ and $B$ is the function  $$y=Ae^{-x}+Bxe^{-x}$$
a solution to the differential equation $y''+2y'+y =0$?
\ANS
(a) For all real numbers $A$ and $B$
(b) Only for positive real numbers $A$ and $B$
(c) Only for negative real numbers $A$ and $B$
(d) Only for $A=0$ and $B=1$
(e) For {\bf no} real numbers $A$ and $B$
\CORR31245.

%% 22 15 
%%Consider the differential equation 
%%y'\sin x - y\cos x = e^x \sin x - e^x \cos x
%%Which of the following (if any) are solutions to this equation?
%%(I) y = e^x + \sin x
%%(II) y = e^x

%%(a) (I) only
%%(b) (II) only
%%(c) (I) and (II)
%%(d) Neither is a solution.
%%(e) It cannot be determined with the given information.


\newpage
% 25
\five
\PROB
What number results from substituting $x=1$ into the {\bf second degree}
Taylor polynomial of $f(x)=(1+x)^{1/3}$?
\ANS
(a) $\frac{11}{9}$
(b) $\frac{2}3$
(c) $\frac{14}{9}$
(d) $\frac{5}3$
(e) $2$
\CORR43215.

%% 27 16 
\PROB
Calculate the sum $$\sum_{k=0}^{10} \Binomial {10}k 2^k$$
\ANS
(a) $3^{10}$
(b) $2^{11}$
(c) $\DSP\Binomial{20}{10}$
(d) $e^{20}$
(e) $\pi^{10}$
\CORR12354.

\newpage
% 28
\two
\PROB
An ordinary cubical die has six sides, numbered one to six. Suppose
that you throw  a fair die 12 times in a row, and record for each throw
the number which comes out on top. What is the probability of having
exactly two fours appear in this list of twelve numbers?
\ANS
(a) $\DSP\Binomial{12}{2}\Frac{1}{6}^2\Frac{5}{6}^{10}$
(b) $\DSP\Binomial{12}{2}\left(\Frac{5}{6}^{10}+\Frac{1}{6}^2\right)$
(c) $\DSP\Binomial{2}{1}\Binomial{10}{5}$
(d) $\DSP\Frac{5}{6}^{10}$
(e) $\DSP\Binomial{12}{2}\Binomial{6}{4}$
\CORR23145.
\vskip-0.5in\relax
%% 32 17
\three
\PROB
The infinite sum $$\sum_{k=1}^\infty \frac{3^k}{k!}=3+\frac{9}{2!} + \frac{27}{3!}
+\frac{81}{4!} +\cdots$$ is equal to
\ANS
(a) $e^3-1$
(b) $3e^2$
(c) $\DSP \frac{1}{2}$
(d) $3e$
(e) $ \sin 3 + \cos 3$
\CORR21435.

\newpage
%% 32 18
\two
\PROB
Which of the following power series is the Taylor series of the
function $f(x)=x^2\sin 2x$?
\ANS
(a) $\DSP \sum_{k=0}^\infty (-1)^{k} \frac{2^{2k+1}x^{2k+3}}{(2k+1)!}$
(b) $\DSP \sum_{k=0}^\infty (-1)^{k} \frac{2^{2k+3}x^{2k+3}}{(2k+3)!}$
(c) $\DSP \sum_{k=0}^\infty (-1)^{k} \frac{2^{2k+3}x^{2k+3}}{(2k+1)!}$
(d) $\DSP \sum_{k=0}^\infty (-1)^{2k+1} \frac{2^kx^{k}}{k!}$
(e) $\DSP \sum_{k=0}^\infty (-1)^{k} \frac{2^{2k+3}x^{2k+1}}{(2k+3)!}$
\CORR54321.

\newpage
}

\def\DSP{\displaystyle}
\def\Binomial#1#2{{{#1}\choose{#2}}}
\def\Frac#1#2{\left(\frac{#1}{#2}\right)}
\def\PROB#1\ANS(a)#2(b)#3(c)#4(d)#5(e)#6\CORR#7.{\newp%
#1%
\AN{#2}{#3}{#4}{#5}{#6}
\Pra
}
\def\NPROB#1\ANS#2\CORR#3.{\relax}


\Exam %%Dwyer, Cholak, Taylor, Wong, Cao, Jarre, {Nollet}
\eject
\marked{}
\end
