\def\whichexam{Final Exam}
\def\examdate{May 7, 2003}
\def\whichcourse{Math 126}
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\itemitem{$\bullet$} Be sure that you have all \numberpages\ pages of the test. 
\itemitem{$\bullet$} No calculators are to be used.
\itemitem{$\bullet$} The exam lasts for two hours.
\itemitem{$\bullet$} {\impno When told to begin, remove this answer sheet 
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The Honor Code is in effect for this examination, including keeping your answer
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\pageno=2 %
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%%%%%%%     START OF TEST
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\Newp TO1.
Solve the following equation for $x$:
$$\ln(x+4) - \ln x = 1\ .$$
�2
(a)$x = \frac{4}{e-1}$
(b)$x=e+2$ and $x=e-2$
(c)$x=\frac{4}{e-1}$ and $x=\frac{4}{e+1}$
(d)$x = \frac{4}{1-e}$
(e)There is no solution.
\endp

\Newp TO2.
Find the derivative of $(x^2+1)^{x^2+1}$.
�2
(a)$(x^2+1)^{x^2+1}\ 2x\bigl(\ln(x^2+1) + 1\bigr)$
(b)$2x(x^2+1)^{x^2}$
(c)$(x^2+1)^{x^2+1}$
(d)$(x^2+1)^{x^2+1}\bigl(2x\ln(x^2+1)\bigr)$
(e)This function is not defined and hence has no derivative.\hss
\endp


\Newp TO3.
Find $\Lim_{x\to \infty} \frac{3^x}{x^5-6x^4+3x^3+8x^2-117x+10}$.
�4
(a)$+\infty$
(b)$-\infty$
(c)$81$
(d)$\frac{1}{81}$
(e)Does not exist but is not $\infty$ or $-\infty$.\hss
\endp

\Newp TO4.
Evaluate $\Int_0^{\pi/3} \sec\theta\ d\theta$.
�3
(a)$\ln\bigl(2 + \sqrt 3\ \bigr)$
(b)$\ln2 - \frac{\ln3}{2}$
(c)$\ln \bigl(1 + \sqrt3\ \bigr)$
(d)$\ln 3 - \ln 2$
(e)Diverges
\endp

\Newp D1.
$\displaystyle\int_0^{\pi\over2}\sin^3 x\cos^2 x \ dx=$
�5
(a) $\frac{2}{15}$
(b) $\frac{4}{21}$
(c) $\frac{3}{70}$
(d) $\frac{5}{12}$
(e) $\frac{2}{45}$
\endp

\Newp D2. 
In the partial fraction expansion of $\frac{x}{(x+1)(x+2)^2}$\ , 
the term with denominator $(x+2)$ has what numerator?
�5
(a) $1$
(b) $-2$
(c) $3$
(d) $x$
(e) $x+1$
\endp

\Newp D3. 
$\displaystyle\int_{-2}^1\frac{x^3+4x^2+13x+3}{x^2+4x+13}dx=$
�5
(a) $-\frac{3}{2}+\frac{\pi}{4}$
(b) $\frac{5}{2}+\frac{3\pi}{4}$
(c) $-1+\frac{3\pi}{4}$
(d) $-1+\frac{\pi}{6}$
(e) $\frac{1}{2}+\frac{\pi}{3}$
\endp

\Newp D4. 
$\displaystyle\int_1^\infty \frac{dx}{x^2\ \sqrt{1+x^2}}=$
�5
(a) $-1+\sqrt 2$
(b) $\sqrt 2$
(c) Divergent
(d) $-1+2\sqrt 2$
(e) $2$
\endp

\Newp D5.
$\displaystyle\int_0^1 \frac{dx}{x^2-1}=$
�5
(a) Divergent
(b) $\ln\frac{\sqrt{3}+1}{2}$
(c) $\frac{\ln2}{4}$
(d) $\frac{1}{\sqrt 3}$
(e) $2\ln 2$
\endp

\Newp D6.
Find the $y$-coordinate of the center of mass of the region 
bounded by $y=1-x^2$ and the $x$-axis.
The area of this region is $\frac{4}{3}$.
�5
(a) $\frac{2}{5}$
(b) $\frac{1}{3}$
(c) $\frac{1}{5}$
(d) $\frac{1}{4}$
(e) $\frac{3}{8}$
\endp

\Newp D7. 
Sugar dissolves in water at a rate proportional to the amount still
undissolved.  If it takes 4 hours to reduce 30 pounds of sugar to 10 
pounds, how many hours does it take to reduce 30 pounds of sugar 
to 3 pounds?
�4
(a) $\frac{4\ln 10}{\ln 3}$
(b) $12$
(c) $\frac{4\ln 20}{\ln 3}$
(d) $0.9(\ln 30-\ln 10)$
(e) $10.8$
\endp

\Newp D8. 
If $x\frac{dy}{dx}+3y=\frac{4}{x}$, and $y(1)=10$, find $y(2)$.
�5
(a) $2$
(b) $\frac{4}{3}$
(c) $\frac{1}{2}$
(d) $7$
(e) $0$
\endp

\Newp D9. 
The curve given by $x=\cos^2 t, y=\sin t, 0\le t\le {\pi\over3}$, 
is rotated about the $x$-axis.  
The area of the resulting surface is given by which 
one of the following integrals?
�2
(a) $\displaystyle2\pi\int_0^{\pi\over3}\sin t \cos  t\sqrt{4\sin^2 t+1}\ dt$
(b) $\displaystyle2\pi\int_0^{\pi\over3}\sin t \sqrt{\cos^4 t+\sin^2 t}\ dt$
(c) $\displaystyle2\pi\int_0^{\pi\over3} \sin t \cos  t\ dt$
(d) $\displaystyle2\pi\int_0^{\pi\over3} \sin  t\sqrt{4\sin^2 t+1}\ dt$
(e) $\displaystyle2\pi\int_0^{\pi\over3} \cos^2  t\sqrt{\cos^4 t+\sin^2 t}\ dt$
\endp

\Newp A1. 
The area inside the curve  $r=3 +  2 \cos\theta$ 
and outside the circle $r=2$ is given by which integral below?

\vskip 1in\vtop{
\localbox{polarregion}\vss}
\vskip-1in
�2
(a)$\Int_0^{\smfrac{2\pi}{3}}\ \bigl(7 + 12\cos\theta +2\cos(2\theta)\bigr)\, d\theta$
(b)$\Int_0^{\smfrac{\pi}{3}}\ \bigl(7 + 12\cos\theta +2\cos(2\theta)\bigr)\, d\theta$
(c)$\Int_0^{\smfrac{2\pi}{3}}\ \bigl(12 - 7\cos\theta +2\cos(2\theta)\bigr)\, d\theta$
(d)$\Int_0^{\smfrac{\pi}{3}}\ \bigl(12 - 7\cos\theta +2\cos(2\theta)\bigr)\, d\theta$
(e)$\Int_0^{\smfrac{7\pi}{6}}\ \bigl(12 - 7\cos\theta +2\cos(2\theta)\bigr)\, d\theta$
\endp

\Newp A2.
 Find the arc length of the curve with polar equation:
$r = 2 -2 \cos \theta, \hskip10pt 0 \leq \theta \leq 2\pi$.
\par\noindent{\bf Hint:} 
$\sin^2\bigl(\smfrac{\theta}{2}\bigr) = \frac{1}{2} - \frac{1}{2}\cos\theta$.
�5
(a)$16$
(b)$12$
(c)$34$
(d)$8$
(e)$32$
\endp


\Newp A3. 
Find the slope of the tangent line to the curve 
$r = 3 \sin \theta$ at $\theta =0$.
�5
(a)$0$
(b)$-1$
(c)$1$
(d)$\pi$
(e)$2$
\endp

\Newp A4. 
Find an equation of the hyperbola with center at
$(-2,0)$, vertices at $(-4,0)$ and $(0,0)$ and 
foci at $(-5,0)$ and $(1,0)$.
�2
(a)$\displaystyle\frac{(x+2)^2}{4} -\frac{y^2}{5} =1$
(b)$\displaystyle\frac{(x-2)^2}{5} -\frac{y^2}{4} =1$
(c)$\displaystyle\frac{(x+4)^2}{2} -\frac{y^2}{5} =1$
(d)$\displaystyle\frac{(x-5)^2}{2} -\frac{y^2}{4} =1$
(e)$\displaystyle\frac{(x+2)^2}{5} -\frac{y^2}{4} =1$
\endp

\Newp A5. 
Determine the center and the foci of the ellipse 
$$\frac{x^2}{9} + \frac{(y+1)^2}{25} =1.$$
�1
(a)\vtop{\parindent=0pt\advance\hsize by -10pt%
center is at $(0,-1)$, the foci are at $(0,-5)$ and $(0,3)$.}
(b)\vtop{\parindent=0pt\advance\hsize by -10pt%
center is at $(0,-1)$, the foci are at $(0,-4)$ and $(0,2)$.}
(c)\vtop{\parindent=0pt\advance\hsize by -10pt%
center is at $(0,-1)$, the foci are at $(0,-6)$ and $(0,4)$.}
(d)\vtop{\parindent=0pt\advance\hsize by -10pt%
center is at $(0,+1)$, the foci are at $(0,-5)$ and $(0,3)$.}
(e)\vtop{\parindent=0pt\advance\hsize by -10pt%
center is at $(0,+1)$, the foci are at $(0,-4)$ and $(0,2)$.}
\endp

\Newp A6. 
Which of the following series are convergent and which are divergent?

$$1) \sum_{n=2}^{\infty} \frac{2}{n \ln n}; \hskip40pt 
2) \sum_{n=1}^{\infty} \frac{\sin n + 2}{n^2}; \hskip40pt 
3) \sum_{n=2}^{\infty} \frac{\ln n}{n}$$
�1
(a)\ads{diverges} {converges}{diverges}
(b)\ads{diverges} {diverges} {diverges}
(c)\ads{diverges} {diverges} {converges}
(d)\ads{converges}{converges}{converges}
(e)\ads{converges}{converges}{diverges}
\endp

\Newp A7. 
Determine if the series are absolutely convergent,
conditionally convergent or divergent
$$
{\bf 1)} \sum_{n=1}^{\infty} \frac{n^2}{e^n}; \hskip40pt
{\bf 2)} \sum_{n=2}^{\infty} \frac{(-1)^n}{\ln n}; \hskip40pt
{\bf 3)} \sum_{n=1}^{\infty} \frac{n!}{4^n}.$$
�1
(a)\Ads{absolutely convergent}{conditionally convergent}{divergent}
(b)\Ads{divergent}{conditionally convergent}{conditionally convergent}
(c)\Ads{absolutely convergent}{absolutely convergent}{divergent}
(d)\Ads{absolutely convergent}{conditionally convergent}{absolutely convergent}
(e)\Ads{divergent}{divergent}{conditionally convergent}
\endp

\Newp A8. 
Determine intervals of convergence for the following power series
$$%
{\bf 1)} \sum_{n=0}^{\infty} \frac{2^n}{n!} (x-2)^n, \hskip40pt 
{\bf 2)} \sum_{n=1}^{\infty} \frac{1}{n} (x-1)^n.$$
�2
(a)\tds{$(-\infty, \infty)$}{$[0,2)$}
(b)\tds{$(-\infty, \infty)$}{$(0,1)$}
(c)\tds{$(0, 2)$}{$[-1,2)$}
(d)\tds{$[-1,0)$}{$[0,2)$}
(e)\tds{$(-\infty, \infty)$}{$(0,2)$}
\endp




\Newp TN1. 
Compute $\quad\displaystyle
\lim_{x\to 0}\ \frac{\arctan x\ - \ \sin x}{e^{x^3}-1}$.
\par\noindent{\bf Hint:} Use series.
�5
(a)$-\frac{1}{6}$
(b)$-\frac{5}{12}$
(c)$\frac{\pi}{4}$
(d)$\frac{4}{\pi}$
(e)$\frac{1}{3}$
\endp


\Newp TN2. 
What is the coefficient of $x^3$ in the Maclaurin series
expansion of $(1+x)^{^\smfrac{3}{4}}$?
�5
(a)$\frac{5}{128}$
(b)$\frac{15}{128}$
(c)$\frac{1}{27}$
(d)$\frac{128}{27}$
(e)$0$
\endp

\Newp TN3. 
Which series below evaluates to the length of the curve
$y = \frac{x^3}{3}$ from $x=0$ to $x=1$?
{\bf Hint:}
First write an integral which gives the length.
Then use power series methods to evaluate it.
�3
(a)$\Sum n0\ \frac{1}{4n+1}\cdot \displaystyle{{\smfrac{1}{2}}\choose n}$
(b)$\Sum n0\ \frac{1}{n+1}\cdot \displaystyle{{\smfrac{1}{2}}\choose n}$
(c)$\Sum n0\ \frac{1}{4n+1}\cdot \displaystyle{{\smfrac{1}{4}}\choose n}$
(d)$\Sum n0\ \frac{1}{n+1}\cdot \displaystyle{{\smfrac{1}{4}}\choose n}$
(e)$\Sum n0\ \frac{1}{6n+1}\cdot \displaystyle{{\smfrac{1}{3}}\choose n}$
\endp


\Newp TN4.  
Find the coefficient of $x^{4n+1}$ in the Maclaurin series for $x\cos(x^2)$.
�5
(a)$\frac{(-1)^n}{(2n)\,!}$
(b)$\frac{1}{(2n)\,!}$
(c)$\frac{(-1)^n}{(4n)\,!}$
(d)$\frac{1}{(4n)\,!}$
(e)$\frac{(-1)^n}{(4n+1)\,!}$
\endp
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