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\begin{center}
\bf{MATH 120\\FINAL EXAM\\December 17, 2002}
\end{center}
\vskip.5in


1.  Find $(f^{-1})^{'}(a)$ for $f(x) = \sqrt{x^3 + 2x^2 + 1}, a=2$
\vskip.3in

A)\ \ 2\hskip.8in B)\ \  1\hskip.8in  C) \ \ 1/2\hskip.8in   D)\ \
0 \hskip.8in  E)\ \  -1

\pagebreak


2.  Find $f^" (o)$ of \ \  $f(x) = x^2 e^x$

\vskip.2in


A)\ \ 0 \hskip.8in B)\ \ 1 \hskip.8in  C) \ \ 2 \hskip.8in   D)\ \  3
\hskip.8in  E)\ \  4


\pagebreak

3.  Differentiate the function $f(x) = e^{cos 5x}$

\vskip.2in

A)\ \  $5e^{cos 5x}sin 5x$ \hskip.5in B) $5e^{cos 5x}cos5x$
\hskip.5in C) $5e^{sin 5x}{cos 5x}$  \hskip.5in  D)  $-5e^{cos
5x}{sin5x}$ \hskip.5in E) $-5e ^{cos 5x} cos 5x$

\pagebreak



4.  Solve the equation for $x$$$e^{x^2 + 2x} = 1$$
\vskip.2in

$A)\ \  0,2 \hskip.4in B)\ \  1,2  \hskip.4in C) \ \  \pm 1
\hskip.4in D)\ \ 0,-2  \hskip.4in  E)\ \  -1,2$

\pagebreak

5.  Find $cos(sin^{-1} (\frac{\sqrt3}{2}))$ \vskip.3in

A)  \ \ $\frac{\sqrt3}{2}$ \hskip.5in B)\ \ 1 \hskip.5in  C)\ \
$\frac{1}{2}$\hskip.5in D)\ \ $0$\hskip.5in  E)\ \ $2$

\pagebreak

6.  Evaluate\ \ $\lim\limits_{x\to -\infty} \frac{\sinh
x}{e^{-x}}$ \vskip.3in


A)  \ \ $1$ \hskip.5in B)\ \ $\frac{1}{2}$ \hskip.5in  C)\ \ $0$
\hskip.5in D)\ \ $-1$ \hskip.5in  E)\ \ $0$

\pagebreak

7.  Find the limit\ \ \  $\lim \limits_{x\to 0}  \frac{e^x-1-x}{x^2}$
\vskip.3in

A)  \ \ $1$ \hskip.8in B)\ \ $\frac{1}{2}$ \hskip.8in  C)\ \ $\infty$
\hskip.8in D)\ \ \hskip.8in  E)\
\ $0$

\pagebreak

8.  Evaluate\ \ $\int \ln x dx$ \vskip.3in

A)\ \ $x\ln x - x + C$\hskip.4in     B)\ \ $\ln x + C$ \hskip.4in
C)\ \ $\frac{\ln x}{x} + x + C$\hskip.4in    D)\ \
$ \frac{\ln x}{x} - x + C$\hskip.4in    E)\ \ $x\ln x + x + C$

\pagebreak

9.  Evaluate \ \ \ \ $\int x e^x dx$

\vskip.2in

A)\ \ $e^x + C$  \hskip.4in  B)\ \ $x e^x + C$   \hskip.4in C)\ \
$xe^x - e^x + C$\hskip.4in    D)\ \ $xe^x + e^x + C$
\hskip.4in  E)\ \ $0$

\pagebreak

10.  Find the value of the integral\ \ \ $I= \int^{\pi}_0(sin x)^3
dx$ \vskip.2in

A)\ \ $1$ \hskip.8in B)\ \ $\frac{4}{3}$  \hskip.8in  C)\ \ $0$
\hskip.8in D)\ \ $8\pi$ \hskip.8in E)\ \ $\frac{2\pi}{3}$

\pagebreak

11.  Find the value of the integral \ \ \ $I=\int^1_0 \sqrt{x^2 + 1}
dx$  \vskip.3in

A)\ \ $1$ \hskip.8in B)\ \ $\ln (\sqrt{2} - 1)$  \hskip.8in  C)\ \
$\ln(\sqrt{2} + 1)$ \hskip.8in D)\ \ $\ln 2$ \hskip.8in
E)$0$

\pagebreak

12.  Find the value of the integral\ \ \ $I = \int^{\frac{1}{2}}_0
\frac{x}{x^2 - 1} dx$

\vskip.3in

A)\ \ $\frac{1}{2} \ln (\frac{3}{4})$ \hskip.8in    B)\ \ $\ln
(\frac{3}{4})$  \hskip.8in     C)\ \ $\frac{1}{2} \ln
(\frac{4}{3})$\hskip.8in    D)\ \ $\frac{1}{2} \ln (\frac{2}{3})$
\hskip.8in    E)\ \ $\frac{1}{2} \ln 2$

\pagebreak

13.  Solve the differential equation $ y' = y$ subject to the
initial condition $y(0) = 0$.  From your solution find $y(e)$.
\vskip.3in

A)\ \ $e^e$ \hskip.8in B)\ \ $e$  \hskip.8in  C)\ \ $e^e-1$
\hskip.8in D)\ \ $0$ \hskip.8in E)\ \ $e^e-e$

\pagebreak

14.  Find power series representation for the following function and
determine the interval on which it is valid.

$$f(x) = sin (x^2)$$  \vskip.3in

A)\ \ $\displaystyle {\sum^\infty_{k = 0}} (-1)^k
\frac{x^{4k+2}}{(2k+1)!}$, valid for all x. \vskip.2in

B)\ \ $\displaystyle {\sum^\infty_{k = 0}} \frac{(-1)^k
x^{2k+1}}{(2k+1)!}$, valid for all x. \vskip.2in

C)\ \ $\displaystyle {\sum^\infty_{k = 0}}\frac{(-1)^k
x^{4k+2}}{(2k+1)!}$, valid for all x.  \vskip.2in

D)\ \ $\displaystyle {\sum^\infty_{k = 0}} x^k,$ valid for $|x| <1
$ \vskip.2in

E)\ \ $x^2 + \frac{x^4}{2} + \frac{x^4}{6} +...,$ valid for
$|x|<2$.

\pagebreak

15.  A bridge hand consists of 13 cards chosen at random from a deck
of 52.  You are dealt a hand.  What is the
probability that it has at least one spade?  {\it Hint:  39 of the
cards in the deck are NOT spades.}  \vskip.2in


A)\ \ $1- \binom{39}{13}/\binom{52}{13}$\hskip.5in  B) \ \
$\binom{39}{13}/\binom{52}{13}$\hskip.5in  C)\ \
$\binom{39}{13}$\hskip.5in  D) \ \  $\binom{52}{13}$ \hskip.5in  E)\
\ $\binom{13}{13}$

\pagebreak

16.  Find the taylor polynomial of degree 4 for the function\ \ \ \
$f (x) = \frac{1}{1-x}$.\vskip.3in

A) $1-x+x^2-x^3 + x^4$ \hskip.3in  B) $1 + x + x^2 + x^3 + x^4$
\hskip.3in  C)$1 + x + x^2 + x^3$ \hskip.3in

  D)  $1 +\frac{x}{2} + \frac{x^3}{3} + \frac{x^4}{4}$  \hskip.3in
E)$\ \ \sin x$

\pagebreak

17.  Suppose an initial amount of \$500 is invested at an annual
rate of 6.5\% compounded continuously.  How much will there be at
the end of two years?\vskip.3in

A)   $500 (1+\frac{0.065}{12})^{(20)12}$\hskip.4in  B)  $500
e^{(20)65}$ \hskip.4in C) $500 e^{(20)0.065}$\hskip.4in D)
$501$\hskip.4in  E)  1000

\pagebreak

18.  Calculate $\binom{15}{2}$.\vskip.3in

A)  30\hskip.8in  B)  96\hskip.8in  C)  210\hskip.8in  D)
-7\hskip.8in  E)  105

\pagebreak

19.  Use the linear approximation and estimate the size of the error.
$f(x) = \sin x$ where $x = 0.02$\vskip.3in

A) $1.02, |error| \leq \frac{(0.02)^2}{2}$\hskip.2in B)   $0.02,
|error| \le \frac{(0.02)^2}{2}$\hskip.2in  C)  $-0.02, |error|
\leq \frac{(0.02)^2}{2}$\hskip.2in

D)  $0.02, |error| \leq  \frac{(0.02)^3}{2000}$\hskip.2in  E)  None
of the above.

\pagebreak

20.  Find the power series representation of the following function
and determine the interval on which it is valid:
$t(x) = xe^x.$\vskip.3in

A) $\displaystyle {\sum^\infty_{k = 0}} \frac{x^{k + 1}}{k!}$,
valid for all $x$ \vskip.2in

B)  $\displaystyle {\sum^\infty_{k = 0}}\frac{x^k}{k!}$, valid for
all $x$.  \vskip.2in

C)  $x + x^2 + \frac{x^3}{2!}$,  valid for all $x$.  \vskip.2in

D)  $\displaystyle {\sum^\infty_{k = 0}}\frac{(-1)^kx^{k+1}}{k!}$,
valid for all $x$.\vskip.2in

E)  None of the above.







\end{document}


