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\preface
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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl March 19, 1996}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Hand in this answer page only.  Record your answers 
to the multiple choice problems by  placing an $\times$ through one letter for
each problem on this answer sheet. There are  19 multiple choice questions, worth 5
points each.  An additional 5 points will be given for your correct tutorial
section number.  \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}


\note %1
\problem  Find the slope of the tangent line to the graph of $y = e^{3x}$ at
$x=1$.  

\correct $3e^3$
\wrong $e^3$
\wrong 1
\wrong 3
\wrong $3e$

\note %2
\problem Simplify the following expression:
$\displaystyle \left ( {1 \over {x^{5/2}}} \right )^{-3/5}$

\correct $\displaystyle x^{3/2}$
\wrong $\displaystyle {1 \over x^{3/2}}$
\wrong $x^{15}$
\wrong $\displaystyle {1 \over x^{2/3}}$
\wrong $\displaystyle x^{2/3}$

\note%3
\problem Find $\displaystyle \lim_{x \rightarrow 0} {{1-\cos x} \over {x^2}}$

\correct $\displaystyle {1 \over 2}$
\wrong $\infty$
\wrong 1
\wrong 0
\wrong 2

\note%4
\problem Find $\displaystyle {d \over {dx}} \ln (e^x +x)$

\correct $\displaystyle {{e^x +1} \over {e^x +x}}$
\wrong $\displaystyle 1 + {1 \over x}$
\wrong $\displaystyle {{1} \over {e^x +x}}$
\wrong $\displaystyle {{1} \over {e^x +1}}$
\wrong $\displaystyle {1 \over x}\cdot (e^x +x) + \ln (e^x +1)$

\note %5
\problem What are the critical numbers, if any,  for the function $f(x) = x +
e^x$?

\correct There are no critical numbers
\wrong $x=0$
\wrong $\displaystyle x = {1 \over e}$
\wrong $x = e$ 
\wrong $x = 1$

\note %6
\problem Find $\displaystyle \int_0^{\pi /2} (\cos x) e^{\sin x} dx$

\correct $e-1$
\wrong $e$
\wrong $e^{\pi/2} -1$
\wrong $e^{\pi/2}$
\wrong 0

\note %7
\problem Find $\displaystyle \log_{16} (1/4)$

\correct $\displaystyle -{1 \over 2}$
\wrong $-2$
\wrong $\displaystyle {1 \over 2}$
\wrong 2
\wrong $\displaystyle {1 \over 4}$

\note %8
\problem Solve for $x$: $\displaystyle \ln (2x+1) = 1$

\correct $\displaystyle x = {{e-1} \over 2}$
\wrong $\displaystyle x = 0$
\wrong $\displaystyle x = -{1 \over 2}$
\wrong $\displaystyle x = {1 \over 2}$
\wrong $\displaystyle x = {e \over 2} -1$

\note %9
\problem Find $\displaystyle {d \over {dx}} x^{\hbox{$x^2$}}$

\correct $\displaystyle x^{\hbox{$x^2$}} \left [ x + 2x\ln x \right ]$
\wrong $\displaystyle x^{\hbox{$x^2$}} \left [ 2x\ln x \right ]$
\wrong $\displaystyle x^{\hbox{$x^2$}}$
\wrong $\displaystyle x^{\hbox{$x^2$}} \left [ 2x \right ]$
\wrong $\displaystyle x^{\hbox{$2x$}} $

\note %10
\problem Use logarithmic differentiation to find the derivative $\displaystyle
{{dy}
\over {dx}}$ for  $\displaystyle y = (2x+5)^8 (4x-3)^5$.

\correct $\displaystyle {{dy} \over {dx}} = (2x+5)^8 (4x-3)^5 \left [ {{16} \over
{2x+5}} + {{20} \over {4x-3}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = (2x+5)^8 (4x-3)^5 \left [ {{8} \over
{2x+5}} + {{5} \over {4x-3}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = (2x+5)^8 (4x-3)^5 \left [ {{2} \over
{2x+5}} + {{4} \over {4x-3}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} =  \left [ {{16} \over
{2x+5}} + {{20} \over {4x-3}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} =  \left [ {{8} \over
{2x+5}} + {{5} \over {4x-3}} \right ]$

\note %11
\problem You invest a certain amount of money in an account which has an annual
interest rate of  10\%, compounded continuously.  After that you do not make any
deposits or withdraw any money from the account.  Which of the following is true
about the amount of money in the account at the end of 10 years?  ({\bf You do not
need a calculator for this problem!}  Note that ``annual interest rate'' is what
we talked about in class.  I do {\it not} mean ``effective annual yield.'')

\correct It is between 2 1/2 and 3 times the original amount.
\wrong It is between 1 1/2 and 2 times the original amount.
\wrong It is between 2 and 2 1/2 times the original amount.
\wrong It is between 3 and 3 1/2 times the original amount.
\wrong It is between 3 1/2 and 4 times the original amount.

\note %12
\problem A certain radioactive substance has a half-life of 100 days.  You begin
with 75 lbs.\ of the substance.  Find the general formula for the amount of the
substance that remains after $t$ days.

\correct $\displaystyle 75 \left ( {1 \over 2}\right )^{t/100}$
\wrong $\displaystyle 75 e^{100t}$
\wrong $\displaystyle 75 \left ( {1 \over 2}\right )^{t-99}$
\wrong $\displaystyle 75 \left ( {1 \over 2}\right )^{t}$
\wrong $\displaystyle 75 \left ( {1 \over 2}\right )^{100t}$

\note %13
\problem Find $\displaystyle \lim_{x \rightarrow 0} {x\over {e^x}}$
\correct 0
\wrong 1
\wrong $e$
\wrong ${1 \over e}$
\wrong $\infty$

\note %14
\problem Find $\displaystyle \lim_{x \rightarrow 0} {{\ln (1+2x)} \over x}$

\correct 2
\wrong 1
\wrong ${1 \over 2}$
\wrong 0
\wrong $-\infty$

\note %15
\problem Find $\displaystyle \int_1^e {{\ln x} \over x} dx$

\correct ${1 \over 2}$
\wrong 1
\wrong $e-1$
\wrong ${1 \over 2}e^2 - {1 \over 2}$
\wrong ${1 \over e}$

\note %16
\problem Find the equation of the tangent line to the curve $y=x^3 e^{-x}$ at the
point $\displaystyle (1,{1 \over e})$.

\correct $\displaystyle y - {1 \over e} = {2 \over e} (x-1)$
\wrong $\displaystyle y - {1 \over e} = {3 \over e} (x-1)$
\wrong $\displaystyle y - {1 \over e} = {1 \over e} (x-1)$
\wrong $\displaystyle y - {1 \over e} = {e} (x-1)$
\wrong $\displaystyle y - {1 \over e} = {2e} (x-1)$

\note %17
\problem A certain object cools at a rate (in $^\circ$C/min) equal to 3 times
the difference between its temperature and that of the surrounding air.  If a room
is kept at $20^\circ$C and the initial temperature of the object is $50^\circ$C,
find an expression for the temperature of the object $t$ minutes later.  (Hint:
if you let $y(t)$ be the temperature of the object $t$ minutes later, then
what is $y(0)$?)

\correct $\displaystyle 30 e^{\hbox{$3t$}} +20$
\wrong $\displaystyle 50 e^{\hbox{$3t$}} +20$
\wrong $\displaystyle 50 e^{\hbox{$3t$}}$
\wrong $\displaystyle 30 e^{\hbox{$3t$}}$
\wrong $\displaystyle 50 e^{\hbox{$3t$}} -20$

\note %18 
\problem Find the slope of the tangent line to the curve $y = 3^x$ at the point
$(2,9)$.  

\correct $9\ln 3$
\wrong $\ln 3$
\wrong $\ln 9$
\wrong $\log_3 2$
\wrong $9$

\note %19
\problem On what interval(s) is the function $f(x) = xe^x$ increasing?

\correct $(-1 ,\infty)$
\wrong $(-\infty, -1)$
\wrong $(1,\infty)$
\wrong $(-\infty, 1)$
\wrong $(-\infty, -1)$ and $(1,\infty)$

\appendix
%That's all folks!

\end
