\input ignore.tex %This line causes the mce macros to be ignored.

\preface
\magnification=1200

\def\picture #1 by #2 (#3){
  \vbox to #2{
    \hrule width #1 height 0pt depth 0pt
    \vfill
    \special{picture #3} % this is the low-level interface
    }
  }

\def\scaledpicture #1 by #2 (#3 scaled #4){{
  \dimen0=#1 \dimen1=#2
  \divide\dimen0 by 1000 \multiply\dimen0 by #4
  \divide\dimen1 by 1000 \multiply\dimen1 by #4
  \picture \dimen0 by \dimen1 (#3 scaled #4)}
  }

\def\cars{\scaledpicture .5in by 0in (cars scaled 700)}
\def\cube{\scaledpicture .5in by 0in (cube scaled 700)}


\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\hfill
MWF Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl March 6, 1997}\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  20 multiple choice questions, worth 5
points each. \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}

\note %1
\problem  Find $\displaystyle \lim_{x \rightarrow 0} {{2^x + 2x - 1} \over x}$
\correct $\ln (2) + 2$
\wrong $-1$
\wrong 3
\wrong $\displaystyle {1 \over {\ln (2)}} + 2$
\wrong $\log_2 (0) + 2$

\note %2
\problem  Find the slope of the tangent line to the graph of $\displaystyle y =
x^{x}$ at $x = e$.

\correct $\displaystyle 2 e^e$
\wrong $\displaystyle e \cdot e^{e-1}$
\wrong $\displaystyle e^e (e+1)$
\wrong $\displaystyle e^e$
\wrong $\displaystyle e \cdot e^e$

\note %3 
\problem Differentiate $\displaystyle g(t) = \sqrt{e^{t^2} + 1}$

\correct $\displaystyle {{t e^{t^2}} \over {\sqrt{e^{t^2} + 1}}}$
\wrong $\displaystyle {{t} \over {\sqrt{e^{t^2} + 1}}}$
\wrong $\displaystyle {{e^{t^2}} \over {\sqrt{e^{t^2} + 1}}}$
\wrong $\displaystyle {{e^{t^2}} \over {2\sqrt{e^{t^2} + 1}}}$
\wrong $\displaystyle {{1} \over {2\sqrt{e^{t^2} + 1}}}$

\note %4
\problem Differentiate $\displaystyle f(x) = {{e^x + 1} \over {e^x - 1}}$

\correct $\displaystyle {{-2e^x} \over {(e^x - 1 )^2}}$
\wrong 0
\wrong $\displaystyle {{2e^x} \over {(e^x - 1 )^2}}$
\wrong $\displaystyle {{2} \over {(e^x - 1 )^2}}$
\wrong $\displaystyle {{2e^{2x}} \over {(e^x - 1 )^2}}$

\note %5
\problem On what interval(s) is the function $\displaystyle {{e^x} \over x}$
decreasing?

\correct only $(-\infty, 0)$ and $(0,1)$
\wrong only $(1,\infty )$
\wrong only $(0,1)$
\wrong only $(-\infty, 0)$
\wrong it is never decreasing

\note %6
\problem Solve for $x$: \ \  $\displaystyle 2\log_5 x - \log_5 3 = 1$

\correct $x =  \sqrt{15}$
\wrong $x = 5^{3/2}$
\wrong $\displaystyle x = {{15} \over 2}$
\wrong $x = 30$
\wrong $x = 4$

\note %7
\problem Simplify $\displaystyle e^{ t^2 (\ln 3 ) / 4 }$

\correct $\displaystyle 3^{t^2 /4}$
\wrong $\displaystyle \left ( {{t^2} \over 4} \right )^3$
\wrong $\displaystyle \left ( {{3t^2} \over 4} \right )$
\wrong $\displaystyle 3 \ln \left ( {{t^2} \over 4} \right )$
\wrong $\displaystyle 3 e^{t^2 /4}$

\note %8
\problem Suppose that $A$, $B$ and $C$ are numbers satisfying
$$
\log_{10} \left ( {A \over C} \right ) = 3.5 \hskip 1cm \hbox{ and } \hskip 1cm
        \log_{10} \left ( {B \over C} \right ) = 2.1
$$
Find $\displaystyle {A \over B}$.

\correct $\displaystyle 10^{1.4}$
\wrong $\displaystyle {{3.5} \over {2.1}}$
\wrong $\displaystyle 10^{{3.5}/ {2.1}}$
\wrong $1.4$
\wrong $\displaystyle 10^{3.5} - 10^{2.1}$

\note %9
\problem Find $\displaystyle {d \over {dx}} \ln \left ( \sqrt{x^2 +1} \right )$

\correct $\displaystyle {x \over {x^2 + 1}}$
\wrong $\displaystyle {{2x} \over {x^2 + 1}}$
\wrong $\displaystyle {{1} \over {\sqrt{x^2 + 1}}}$
\wrong $\displaystyle {{x} \over {2(x^2 + 1)}}$
\wrong $\displaystyle {{2x} \over {\sqrt{x^2 + 1}}}$

\note %10
\problem A bacteria culture grows at a rate proportional to its size.  Suppose
that the initial size is 8000 bacteria, and after 2 hours there are 12,000
bacteria.  How many bacteria are there after 6 hours?

\correct 27,000
\wrong 20,000
\wrong 24,000
\wrong 36,000
\wrong 48,000

\note %11
\problem A certain radioactive substance is known to have a half-life of 10
years.  If you have a certain initial amount of this substance, how many years
will elapse before only 1/3 of it remains?  (The following answers are all in
years.)

\correct $\displaystyle {{10 \ln 3} \over {\ln 2}}$
\wrong $\displaystyle {{10 \ln 2} \over {\ln 3}}$
\wrong $\displaystyle 10 (\ln 3 - {\ln 2})$
\wrong $\displaystyle {{20} \over 3}$
\wrong It is impossible to tell without knowing the initial amount.

\note %12
\problem Fred plans to invest money into an account which pays an annual interest
rate of 10\%, compounded continuously.  How much should he invest if he does not
plan to make any further investments into this account, and he wants the balance
after two years to be \$20,000?

\correct $\displaystyle {{20,000} \over {e^{0.2}}}$
\wrong $\displaystyle 20,000 e^{0.2}$
\wrong 16,000
\wrong 16,200
\wrong $\displaystyle  {{40,000} \over {e^{0.1}}}$

\note %13
\problem Suppose that $y(t)$ is a solution to the differential equation
$\displaystyle {{dy} \over {dt}} = 5y$, and suppose that $y(1) = 100$.  (Note that
this is $y(1)$, not $y(0)$.)  Find $y(t)$.

\correct $\displaystyle 100 e^{5(t-1)}$
\wrong $\displaystyle 5 e^{100t}$
\wrong $\displaystyle 5 e^{100}$
\wrong $\displaystyle {{e^{5t}} \over {100}}$
\wrong $\displaystyle 100 e^5$

\note %14
\problem Evaluate $\displaystyle \int_1^4 \left ( \sqrt{x} + {1 \over {\sqrt{x}}}
\right )^2 dx$

\correct $\displaystyle {{27} \over 2} + \ln 4$
\wrong $\displaystyle {{33} \over 2} + \ln 4$
\wrong $\displaystyle {{37} \over 2} + \ln 4$
\wrong $\displaystyle {{41} \over 2} + \ln 4$
\wrong $\displaystyle {{51} \over 2} + \ln 4$

\note %15
\problem Evaluate $\displaystyle \int_0^1 2^x dx$

\correct $\displaystyle {1 \over {\ln 2}}$
\wrong $\displaystyle {2 \over {\ln 2}}$
\wrong $\displaystyle \ln 2$
\wrong $\displaystyle 2\ln 2$
\wrong $\displaystyle {3 \over {\ln 2}}$

\note %16
\problem Use logarithmic differentiation to find $\displaystyle {{dy} \over
{dx}}$ for $\displaystyle y = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}$

\correct $\displaystyle {{dy} \over {dx}} = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}
\left [ {5 \over x} + {x \over {x^2 +1}} - {{4 \cos x} \over {\sin x}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}
\left [ {5 \over x} + {1 \over {2(x^2 +1)}} - {{4 \cos x} \over {\sin x}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}
\left [ 5x^4 + x(x^2 +1)^{-1/2} -4 \sin^3 x \cos x \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}
\left [ {{(5x^4 )[ x(x^2 +1)^{-1/2}]} \over {4 \sin^3 x \cos x}} \right ]$
\wrong $\displaystyle {{dy} \over {dx}} = {{x^5 \sqrt{x^2 +1}} \over {\sin^4 x}}
\left [ {5 \over x} + {2 \over {x^2 +1}} + {{4 \cos x} \over {\sin x}} \right ]$

\note %17
\problem Find $\displaystyle \lim_{x \rightarrow 0^+} x \ln x$.

\correct 0
\wrong 1
\wrong $e$
\wrong $\infty$
\wrong $-\infty$

\note %18
\problem Evaluate $\displaystyle \int {x \over {e^{x^2}}} dx$.

\correct $\displaystyle {{-1} \over {2 e^{x^2}}} + C$
\wrong $\displaystyle {{1} \over {2 e^{x^2}}} + C$
\wrong $\displaystyle {{-1} \over {e^{x^2}}} + C$
\wrong $\displaystyle {{1} \over {e^{x^2}}} + C$
\wrong $\displaystyle {{x^2} \over {2e^{x^3}}} + C$

\note %19
\problem Simplify: $\displaystyle \log_{1 \over 2} 16$

\correct $-4$
\wrong 4
\wrong $\displaystyle {1 \over 4}$
\wrong $\displaystyle {{-1} \over 4}$
\wrong 8

\note %20
\problem Find the slope of the tangent line to the curve $y = \log_5 x$ at $x=2$.

\correct $\displaystyle {1 \over {2 \ln 5}}$
\wrong $\displaystyle {{2 \ln 5}}$
\wrong $\displaystyle {2 \over { \ln 5}}$
\wrong $\displaystyle {\ln 5 \over {2}}$
\wrong $\ln 5$

\appendix
%That's all folks!

\end
