\long\def\hide#1{}
\def\zexp#1{{\hbox to 0pt{$\scriptstyle #1$\hss}}}
\def\tT{Math126, Final}
\def\ddd{May 6, 1999}
\def\rightanswers{
\ll{1} 2
\ll{2} 3
\ll{3} 4
\ll{4} 1
\ll{5} 5
\ll{6} 2
\ll{7} 2
\ll{8} 5
\ll{9} 4
\ll{10}5
\ll{11}3
\ll{12}1
\ll{13}5
\ll{14}4
\ll{15}2
\ll{16}3
\ll{17}3
\ll{18}1
\ll{19}4
\ll{20}3
\ll{21}2
\LL{22}1
\ll{23}4
\ll{24}3
\ll{25}2
}
\def\isum{\displaystyle\sum_{n=1}^{\infty}}
\def\jsum{\displaystyle\sum_{n=2}^{\infty}}
\input 126FinalHeader.tex

\exam{
\pr(6pts)
Which equation below is the equation of an ellipse with its
major axis the $x${--}axis?
\ans{ $\frac{x^2}{25}+\frac{y^2}{36}=1$}
{$\frac{x^2}{36}+\frac{y^2}{25}=1$}
{$\frac{x^2}{25}-\frac{y^2}{36}=1$}
{$\frac{x^2}{36}-\frac{y^2}{25}=1$}
{$\frac{x^2}{36}+\frac{y^2}{25}=0$}
%%1(b)

\pr(6pts)
Which equation below is that of a hyperbola with foci $(\pm 4,0)$?
\ans
{$\frac{x^2}{5}-\frac{y^2}{3}=1$}
{$-\frac{x^2}{5}+\frac{y^2}{3}=1$}
{$\frac{x^2}{14}-\frac{y^2}{2}=1$}
{$\frac{x^2}{9}-\frac{y^2}{7}=0$}
{$-\frac{x^2}{14}+\frac{y^2}{2}=1$}
%%2(c)

\pr(6pts)
The graph of $x=\frac{y^2}{20}$ is a parabola with directrix the line
\ans{$x=5$}{$y=5$}{$x=0$}{$x=-5$}{$y=-5$}
%%3(d)

\pr(6pts)
Find the slope of the tangent line to the parameterized curve
$x=t^2+3t+1$, $y=t^3-2t$ when $t=2$.
\ans{$\frac{10}{7}$}{$\frac{7}{10}$}{$1$}{$\frac{-2}{7}$}{$\frac{-7}{2}$}
%%4(a)

\pr(6pts)
Which integral below represents the arclength of the cycloid
$x=a(t-\sin t)$, $y=a(1-\cos t)$; $0\leq t\leq 2\pi$?
\twotwoone
{$\sqrt{2a^2}\Int^{2\pi}_{0} \sqrt{1-\cos t+\sin t\>}\>dt$}
{$\sqrt{2a^2}\Int^{2\pi}_{0} \sqrt{t-\sin t+1-\cos t\>}\>dt$}
{$\sqrt{2a^2}\Int^{2\pi}_{0} \sqrt{1-\cos t\>}\>dt$}
{$\sqrt{2a^2}\Int^{2\pi}_{0} \sqrt{1+\cos t\>}\>dt$}
{$\sqrt{2a^2}\Int^{2\pi}_{0} \sqrt{1-\sin t\>}\>dt$}
%%5(e)

\pr(6pts)
The parameterized curve $x=t^3+2t$, $y=\cos t$; $-\infty<t<\infty$
is also the graph of a function $y=f(x)$.
What is the coefficient of $x^2$ in the Mclaurin series expansion
for $f(x)$?
\ans{$\frac{-1}{2}$}{$\frac{-1}{8}$}{$0$}{$\frac{1}{8}$}{$\frac{1}{2}$}
%%6(b)

%%6 from each of the other tests
%%% Test 1 %%%
\pr(6pts)
The function $f(x)=x \sin x$ has one critical point for 
$-\smfrac{\pi}2<x<\smfrac{\pi}2$.
Where is it and determine whether it is a
local minima, maxima or neither.
\threetwo{$x=0$, local max}{$x=0$, local min}{$x=0$ neither}
{$x=\frac{\pi}4$, local max}{$x=\frac{\pi}4$, local min}
%%7(b)

\pr(6pts)
Which number below is equal to $\log_3(81)$\ ?
\ans{$-4$}{$-2$}{$0$}{$2$}{$4$}
%%8(e)

\pr(6pts) Let $f(x)=x+\ln x$ for $x>0$.
Find $\frac{df^{-1}}{dx}(e+1)$.\vskip10pt
\ans
{$f$ is not one to one}
{$e$}
{$\frac 1{e+1}$}
{$\frac e{e+1}$}
{$1+\frac 1e$}
%%9(d)

\pr(6pts)
A certain bacteria culture, undergoing natural growth, doubles
in size after 4 minutes.
If there were 100 specimens at time $t=0$, when will the number have
increased to 1600 specimens?
\ans{$2$ weeks}
{$1$ day}
{$3$ hours, $20$ minutes}
{$2$ hours}
{$16$ minutes}
%%10(e)

\pr(6pts)
Let $f(x)=\Int^{x}_0e^{-t^2}\>dt$.
Find the Mclaurin series for $f(x)$.
\threetwo
{$\sum^\infty_{n=0}\frac{(-1)^nx^{n+1}}{(n+1)!}$}
{$\sum^\infty_{n=0}\frac{x^{2n+1}}{n!}$}
{$\sum^\infty_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)n!}$}
{$\sum^\infty_{n=0}\frac{(-1)^{n+1}x^{3n}}{(3n)n!}$}
{$\sum^\infty_{n=0}\frac{(-1)^nx^{2n+1}}{(2n)!}$}
%%11(c)

\pr(6pts)
Calculate $\Lim_{u\to\infty}\frac{u^3+5u^2-2u+10}{3u^2+7u-8}$.
\ans{$\infty$}
{$\frac{5}{3}$}
{$\frac{-2}{7}$}
{$\frac{-5}{4}$}
{$\frac{1}{3}$}
%%12(a)

%%% Test 2 %%
\pr(6pts)
The solution to the initial value problem
$xy^\prime=y + x^3$, $y(1)=1$ is
\ans{$y=x^3$}
{$y=e^{x-1}+1$}
{$y^2+y=x^2+x$}
{$y=\frac{x^2+2}{3}$}
{$y=\frac{x^3}{4}+\frac{3}{4x}$}
%%13(e)

\def\stick{\vrule height0pt depth 10pt width 0pt}
\pr(6pts)
The improper integral $\Int^\infty_1\frac 1{x^{1.01}}\> dx$
\threetwo{\stick converges to $\smfrac1{1.01}$}
{converges to $1.01$}
{converges to $.01$}
{converges to $100$}
{diverges}
%%14(d)

\pr(6pts)
The partial fraction expansion of
$\frac{x+7}{x^2+4x+3}$ is
\threetwo{\stick$\frac 2{x-1}+\frac3{x-3}$}{$\frac 3{x+1}-\frac2{x+3}$}
{$\frac 5{x+1}+\frac4{x+3}$}{$\frac 4{x+1}-\frac3{x+3}$}
{$\frac{x+7}{x^2+4x+3}$}
%%15(b)

\pr(6pts)
$\Int^3_0 xe^x\>dx$
\ans{$3e^3$}{$3e^3-3$}{$2e^3+1$}{$2e^3$}{$3e^2$}
%%16(c)

\pr(6pts)
Which expression below is equal to
$\Int^{\smfrac{1}{2}}_0 \sqrt{1-x^2\ }\>dx$?
\ans
{$\Int^{\smfrac{\pi}6}_0 \cos\theta\>d\theta$}
{$\Int^{\smfrac{\pi}6}_0 \cos^3\theta\>d\theta$}
{$\Int^{\smfrac{\pi}6}_0 \cos^2\theta\>d\theta$}
{$\Int^{\smfrac{\pi}6}_0 \sin^2\theta\>d\theta$}
{$\Int^{\smfrac{\pi}6}_0 \sin\theta\>d\theta$}
%%17(c)

\pr(6pts)
{\sl According to the limit comparison test\/} for definite integrals,
what can we say about the two improper integrals
$\Int_0^\infty \frac{x^2-3x+10}{x^4+2x^2+4}\>dx$ and
$\Int_0^\infty \frac{x^4-3x^2+10}{x^6+x^4+9}\>dx$?
\onetwooneone{Either they both converge or they both diverge.}
{They both converge.}
{They both diverge.}
{The first diverges but the second converges.}
{The first converges but the second diverges.}
%%18(a) but (b) OK

%%%%%%%%%%
\pr(6pts) Determine whether the following series converge
or diverge.
$$1) \sum_{n=1}^{\infty} \frac{(-1)^n}{n}\>, \hskip 16pt
2) \sum_{n=1}^{\infty} \frac{1}{n!}\>,\hskip 16pt
3) \sum_{n=2}^{\infty} \frac{\root 3\of{n^3-1}}{3n -1}.$$

\fivelines
{1) 2) and 3) converge}
{1) absolutely converges, 2) and 3) diverge}
{1) conditionally converges, 2) and 3) diverge}
{1) conditionally converges, 2) absolutely converges and 3) diverge}
{1) 2) and 3) diverge}
%%19(d)


\def\stick{\vrule height0pt depth 10pt width 0pt}

\pr(6pts)
Find the radius $R$ of convergence of the following power series
$$\sum_{n=1}^{\infty} \frac{(n! )^2}{(2n)!}(x-5)^n\ .$$
\ans{$R=0$}{$R=\infty$}{$R=4$}{$R=5$}{$R=\sqrt{5}$}
%%20(c)

\pr(6pts)
All of the series below have radius of convergence $1$.
Which one conditionally converges at both endpoints
of its interval of convergence?

\threetwo{$\sum_{n=1}^{\infty} (-1)^n\> \frac{x^\zexp{n-1}}{n}$}
{$\sum_{n=1}^{\infty} (-1)^n \>\frac{x^\zexp{2n-1}}{n}$}
{$\sum_{n=1}^{\infty} (-1)^n \>\frac{x^\zexp{2n-1}}{n^2}$}
{$\sum_{n=1}^{\infty} (-1)^n \>\frac{x^\zexp{3n}}{n}$}
{$\sum_{n=1}^{\infty} (-1)^n \>\frac{x^\zexp{3n}}{n^3}$}
%%21(b)

\pr(6pts)
What is the behavior of the series
 $\isum \frac1{(\arctan n)^2(1+n^2)}$\ ?
\Three{It converges absolutely.}
{It converges conditionally.}
{It diverges.}
{}{}%%22(a)

\pr(6pts)
Give the first three nonzero terms of the Maclaurin  series
expansion of $e^{x^2} \sin x$.
\threetwo
{$x-\smfrac13x^3-\smfrac{1}{120}x^5$}
{$1+3x^2+\smfrac{1}{6}x^4$}
{$x-x^3+x^7$}
{$x+\smfrac56x^3+\smfrac{41}{120}x^5$}
{$x+x^2-x^3$}
%%23(d)

\pr(6pts)
Which series conditionally converges?
\ans{$\jsum \frac1{n^2\ln n}$}
{$\jsum \frac1{n\ln n}$}
{$\jsum \frac{(-1)^{n+1}}{n\ln n}$}
{$\jsum \frac{(-1)^{n+1}}{n^2\ln n}$}
{$\jsum \frac1{n^2+n\ln n}$}
%%24(c)

\pr(5pts)
Find the sum of the following series
$$\sum_{n=0}^{\infty}  \frac{2^{n-2}}{3^n}\ .$$
\ans{diverges}{$\frac 34$}{$e^{2/3}$}{$1$}{$3$}
%%25(b)

}

\print{}
%\print{Cao}
%\print{Alber}
%\print{Taylor}
%\print{Nollet}
%\print{Patrizio \quad{\sl Section \#:}}
\marked{}
\end

