%Taylor
\def\whichexam{Final}
\def\examdate{May 12, 2000}


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\centerline{\bf Math 126, \whichexam}
\centerline{\examdate}
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\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 \numberpages\ pages of the test. 
\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.
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Score: \ul{1in}
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\long\def\Exam{%
\coversheet{ }%

\vfill\eject\headline{\hfil}
\five
\newp 1.
Let $f(x)= 2x^2+1$ for $x\ge 0$.
Then $(f^{-1})^\prime(9)=$
\ans
$\frac18$€
$\frac19$€
$\frac 1{24}$€
$\frac16$€
$\frac29$€
\endp 1.

\five
\newp 2.
$\Int_e^{e^2}{1\over x\ln x}dx =$
\ans
$\ln 2$€
$\frac1{2e^2}-\frac1{e}$€
$2$€
$2e^2-e$€
$\Sum n0 e^{2n}$€
\endp 2.

\five
\newp 3.
If $y^\prime(t) = te^{-t^2}$ and $y(0)=1$, then $y(1)=$
\ans
$\frac32-\frac 1{2e}$€
$e^3-\frac16$€
$\frac1{e^2 }- \frac12$€
$e-\frac1{e+1}$€
$e-\frac13$€
\endp 3.

\five
\newp 4.
$\Lim_{x\to 0^+}\frac{\ln(e^x-1)}{\ln (2x)}=$
\medskip
\noindent{\bf Hint:} Apply l'H\^opital's rule, simplify, and apply l'H\^opital's rule again.

\ans
$1$€
$0$€
$\infty$€
$2$€
$-1$€
\endp 4.


\three
\newp 5. 
Which of the following five functions

1) $\log_2(x+1)$, \ 2) $\ln(x+1)$, \ 3) $\ln(x^{100})$, \ 4) $\sqrt{x^2+1}$,
\ 5) $\sqrt{\frac x{100}\ }$

\noindent
grow at the same rate? 
\ans
Only 1), 2), and 3)€
Only 1) and 2)€
Only 2) and 4)€
Only 4) and 5)€
2) and 3), as well as 4) and 5)\hss€
\endp 5.

\five
\newp 6.
$\Int^1_0\frac{dx}{(1+x^2)^{\smfrac32}}=$
\par\medskip\noindent{\bf Hint:} This problem is shorter if you change limits of 
integration when you do the substitution.
\ans
$\frac1{\sqrt{2\ }}$€
Diverges€
$1$€
$-\frac1{\sqrt{2\ }}$€
$\sqrt{2\ }$€
\endp 6.

\five
\newp 7.
$\Int_0^1(x+1)e^{x+1}dx=$
\ans
$e^2$€
$e$€
$2e^2$€
$2e$€
Diverges€
\endp 7.

\two
\newp 8.
The solution to the initial value problem
$$x\frac{dy}{dx}+x^2 y-x^2 = 0\qquad y(0) = 0$$
is
\def\xs{-{{\textstyle x^{\hbox to 0pt{$\scriptstyle2$\hss}}}\over{\textstyle2}}}
\ans
$y=1-e^{^{\xs}}$€
$y=e^{^{\xs}}-1$€
$y=e^{-x}-1$€
$y=1-e^{-x}$€
$y=xe^x$€
\endp 8.

\five
\newp 9.
$\sec\bigl(\arccos\smfrac12)=$
\ans
$2$€
$1$€
$0$€
$\frac12$€
$\infty$€
\endp 9.


\five
\newp 10. 
Find $\Int_{-1}^{1} \frac{dx}{x^3}$.\par{\bf Be careful.}
\ans
diverges€
$0$€
$1$€
$2$€
$ -1$€
\endp 10.

\five
\newp 11. 
Find 
$\Sum n1 \frac{2^{2n}}{5^{n+1}}$
\ans
$\frac{4}{5}$€
$\frac{5}{3}$€
$\frac{1}{3}$€
$\frac{1}{25}$€
$\frac{1}{2}$€
\endp 11.

\two
\newp 12. 
Which of the following series converge?

\noindent\null\hfil(1) $\Sum n0 \frac{4^n}{n!}$\hfil(2) $\Sum n2 \frac{n}{(\ln n)^2}$\hfil
(3) $\Sum n1\frac{n^2}{n^\smfrac{8}{3}}$\hfil\null
\ans
(1) converges, (2) and (3) diverge€
(1) and (2) converge, (3) diverges€
(1) and (3) converge, (2) diverges€
(2) and (3) converge, (1) diverges€
(3) converges, (1) and (2) diverge€
\endp 12.

\five\newp 13. 
Find $\Lim_{n \rightarrow \infty} (1-\frac{1}{2n})^n$
\ans
$\frac{1}{\sqrt{e}}$€
$\sqrt{e}$€
$e^2$€
$e^{-2}$€
$1$€
\endp 13.

\one
\newp 14. Test the following series for
absolute convergence, conditional convergence or divergence:

\noindent\null\hfil
(1) $\Sum n1 \frac{(-1)^n}{n^{0.9}}$\hfil
(2) $\Sum n0 \frac{(-1)^n}{(1.1)^n}$\hfil
(3) $\Sum n0 {n!}(-1)^n$\hfil\null
\ans
(1) converges conditionally, (2) converges absolutely, (3) diverges€
(1) and (2) converge conditionally, (3) diverges€
(1) and (2) converge absolutely, (3) diverges€
(1) converges absolutely, (2) converges conditionally, (3) diverges€
(1) diverges, (2) and (3) converge conditionally.€
\endp 14.

\three
\newp 15.
What is the MacLaurin series for $xe^{x^2}$?
\ans
$\Sum n0 \frac{x^{2n+1}}{n!}$€
$\Sum n0  x^{2n+1}$€
$\Sum n0 \frac{x^{2n}}{n}$€
$\Sum n0 \frac{(-1)^nx^{2n+1}}{n!}$€
$\Sum n0 (-1)^nx^{2n}$€
\endp 15.

\five
\newp 16.
Find the interval of convergence for 
$$\Sum n1 \frac{(-1)^nnx^n}{n^2+2}$$
\ans
$-1<x\leq 1$€
$-1<x < 1$€
$-1\leq x < 1$€
$-1\leq x\leq 1$€
all $x$€
\endp 16.

\three
\newp 17.
Find the first 3 non{--}zero terms of the MacLaurin series solution to
$$y^\prime + xy = 0 \qquad\qquad y(0)=1$$
\ans
$1-\smfrac12x^2+\smfrac18 x^4$€
$1+x+x^2$€
$1-\smfrac12 x+\frac18 x^2$€
$x-\smfrac12 x^3+\smfrac18 x^5$€
$x+x^3+x^5$€
\endp 17.

\three
\newp 18.
Which line below is the directrix of the parabola
$y^2=8x$.
\ans
$x=-2$€
$x=2$€
$y=-2$€
$y=2$€
Can not be determined from the given information\hss€
\endp 18.

\three
\newp 19.
What is the equation of the ellipse with vertices $(\pm 4,0)$ and eccentricity $e=\frac12$?
\ans
$\ellipse{16}{12}$€
$\ellipse{12}{16}$€
$\ellipse{16}{4}$€
$\ellipse{4}{16}$€
$\ellipse{12}{4}$€
\endp 19.

\five
\newp 20.
Find the slope of the parameterized curve
$$x^3+t^3+2t=8\qquad e^y=t+\cos t$$
when $t=0$.
\par\medskip\noindent{\bf Hint:} First find $x(0)$ and $y(0)$ and then use implicit differentiation.
\ans
$-6$€
$-3$€
$0$€
$3$€
$6$€
\endp 20.

\five
\newp 21.
Find the arclength of the parameterized curve
$$x(t)=1-\cos t\qquad y(t)=4-\sin t\qquad\qquad -2\leq t \leq 3\ .$$
\ans
$5$€
$4$€
$3$€
$2$€
$2\pi$€
\endp 21.

\three
\newp 22.
Which formula below yields the surface area of the surface obtained by rotating the paramterized curve
$$x(t)=\frac{t^2}{2}\quad y(t)=\frac{t^3}{3} \quad\quad 0\leq t\leq 1$$
about the $y${--}axis?
\ans
$2\pi\Int_0^1 \frac{t^3\sqrt{t^2+1\ }}{2}\>dt$€
$2\pi\Int_0^1 \frac{t^3\sqrt{t^2+1\ }}{3}\>dt$€
$2\pi\Int_0^1 \frac{t^2\sqrt{t^2+1\ }}{2}\>dt$€
$2\pi\Int_0^1 \frac{t^2\sqrt{t^2+1\ }}{3}\>dt$€
$2\pi\Int_0^1 \frac{t^4\sqrt{t^2+1\ }}{4}\>dt$€
\endp 22.

\two
\newp 23.
Which graph below is the graph of the polar curve $r=1-\sin(\theta)$?
\HideDisplacementBoxes
\pictboxdim=60pt
\ans
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{cartiodD.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{cartiodA.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{cartiodB.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{cartiodC.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{cartiodE.eps}€
\endp 23.

\five
\newp 24.
Polar area
Find the area inside the six{--}leafed rose $r^2=2\sin 3\theta$.
\par\medskip\noindent{\bf Warning:} Be careful with your limits of integration.
\par\pictboxdim=120pt
\pictboxstrut\ForceWidth{\pictboxdim}\vskip-100pt\BoxedEPSF{six.eps}
\par
\ans
$4$€
$3$€
$2$€
$1$€
$0$€
\endp 24.

\five
\newp 25.
Find the arclength of the curve given in polar coordinates by
$r=\cos\theta + \sin\theta$ for $0\leq\theta\leq\pi$.
\ans
$\sqrt{2}\pi$€
$\pi$€
$\frac{\pi}{2}$€
$\sqrt{3}\pi$€
$\frac{7\pi}{8}$€
\endp 25.
}

\Exam{}
\vfill\eject
\marked{}
\end
