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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 28, 1995}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators {\it are} allowed on this exam.  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
Let $\displaystyle f(x) = x^{(x^2 )}$.  What is $f'(x)$?
\correct $\displaystyle x^{(x^2 )} \left [ x + 2x \ln x \right ]$
\wrong $\displaystyle x^{2x} \left [ 2 + 2 \ln x \right ]$
\wrong $\displaystyle x^2 \cdot x^{(x^2 - 1)}$
\wrong $\displaystyle x^{(x^2 )} \cdot 2x$
\wrong $\displaystyle x^{(x^2 )} \left [ 2 + 2 \ln x \right ]$

\note %2
\problem
Let $\displaystyle f(x) = x^x$.  What is $f'(x)$?
\correct $\displaystyle x^x \left [ 1 + \ln x \right ]$
\wrong $\displaystyle x \cdot x^{x-1}$
\wrong $\displaystyle x^x \cdot \left [ 1 + {1 \over x} \right ]$
\wrong $\displaystyle x^x \cdot {1 \over x}$
\wrong $\displaystyle xe^x + e^x$

\note %3
\problem
Fred invests \$5000 in an account at an annual interest rate of 10\%, compounded
\break continuously.  How much will he have (in dollars) in the account after 20
years?
\correct $\displaystyle 5000 e^2$
\wrong $\displaystyle 500 e^{20}$
\wrong 10,000
\wrong 15,000
\wrong $\displaystyle 5000 \left ( 1 + .10 \right )^{20}$

\note %4
\problem
A certain radioactive substance is being studied.  It is observed that after 100
days, 75\% of the original amount remains.  What is the half-life of that
radioactive substance (in days)?
\correct $\displaystyle {{100 \ln (.5)} \over {\ln (.75)}} \approx 241$
\wrong 200
\wrong $\displaystyle {{200 \ln (.75)} \over {\ln (.5)}} \approx 83$
\wrong $\displaystyle 1000 \cdot |\ln (.75)| \approx 288$
\wrong $\displaystyle 100 \cdot (\ln (.75) - \ln (.5)) \approx 405$

\note %5
\problem
A bacteria culture starts with 600 bacteria and after 4 hours there are 6000
bacteria.  Find an expression for the number of bacteria after $t$ hours.
\correct $\displaystyle 600 \cdot 10^{t / 4}$
\wrong $\displaystyle 600 \cdot e^{t / 4}$
\wrong $\displaystyle (0.6) \cdot 10^t$
\wrong $\displaystyle 600 \cdot 10^t$
\wrong $\displaystyle 600 e^t$

\note %6
\problem
Find the following antiderivative: $\displaystyle \int {{e^x} \over {1 - e^{2x} }}
dx$.  $\bigg ($ Hint: Find a substitution, then use the fact that 
$\displaystyle \left.
{1 \over {1 - u^2}} = {{1/2} \over {1-u}} + {{1/2} \over {1+u}} \ 
\right )$
\correct $\displaystyle {1 \over 2} \ln \left | {{1+e^x} \over {1 - e^x}} \right
| + C$
\wrong $\displaystyle {1 \over 2} \ln \left | 1 - e^{2x} \right |+ C$
\wrong $\displaystyle {1 \over 2} \ln \left | {{1-e^x} \over {1 + e^x}} \right
|+ C$
\wrong $\displaystyle e^x + e^{-x} + C$
\wrong $\displaystyle -\ln \left | 1 - e^{2x} \right | + C$

\note %7
\problem
Evaluate the following definite integral:
$\displaystyle \int_0^\pi x \, \sin x \, dx$
\correct $\pi$
\wrong $4\pi -2$
\wrong $2\pi -2$
\wrong $4\pi - 4$
\wrong $\pi - 2$

\note %8
\problem 
Evaluate the following definite integral:
$\displaystyle \int_0^{\pi \over 2} \sin^2 x \cos x \, dx$
\correct $\displaystyle {1 \over 3}$
\wrong $\displaystyle {{\pi^2} \over 4}$
\wrong $\displaystyle {1 \over 3}\cdot {{\pi^3} \over 8}$
\wrong $0$
\wrong $\displaystyle -{{1} \over 3}$

\note %9
\problem
In the partial fraction decomposition $$\displaystyle {{x^2 +1} \over {x^3 - x}}
= {A \over x} + {B \over {x-1}} + {C \over {x+1}}$$
find the value of $A$.  (You don't have to find the values of $B$ or $C$.)
\correct$-1$
\wrong $1$
\wrong $1/2$
\wrong $-1/2$
\wrong $2$

\note %10
\problem
Use long division to divide
$\displaystyle {{x^3 + 2x^2 +2x+1} \over {x^2 +1}}$
\correct $\displaystyle x+2 + {{x-1} \over {x^2 +1}}$
\wrong $\displaystyle x +1 + {{2x} \over {x^2 +1}}$
\wrong $\displaystyle x +1 + {{x+1} \over {x^2 +1}}$
\wrong $\displaystyle x+2 + {{2x+1} \over {x^2 +1}}$
\wrong $\displaystyle x+2 + {{3x-1} \over {x^2 +1}}$

\note %11
\problem
It is a fact, which you don't have to verify, that 
$$
{{x^2 + 1} \over {x(2x-1)^2}} = {1 \over x} - {{3/2} \over {2x-1}} + {{5/2} \over
{(2x-1)^2}}
$$
Using this, find $\displaystyle \int {{x^2 + 1} \over {x(2x-1)^2}} dx$
\correct $\displaystyle \ln |x| - {3 \over 4} \ln |2x-1| - {{5 / 4} \over {2x-1}}
+C$
\wrong $\displaystyle \ln |x| -3 \ln |2x-1| - {{5 } \over {2x-1}}+C$
\wrong $\displaystyle \ln |x| + 3 \ln |2x-1| + {{5 } \over {2x-1}}+C$
\wrong $\displaystyle \ln |x| + {3 \over 4} \ln |2x-1| +{{5 / 4} \over {2x-1}}+C$
\wrong $\displaystyle \ln |x| -{3 \over 4} \ln |2x-1| - {5 \over 4} \ln
\left( (2x-1)^2 \right )+C$

\note %12
\problem
Evaluate the following definite integral:
$\displaystyle \int_0^\pi \cos^2 2x \, dx$
\correct $\displaystyle {{\pi} \over 2}$
\wrong  $\displaystyle \pi$
\wrong  $\displaystyle 0$
\wrong  $\displaystyle {{\pi} \over 4}$
\wrong  $\displaystyle 2\pi$

\note %13
\problem
Evaluate the following definite integral:
$\displaystyle \int_0^{{\pi} \over 2} \cos^3 x \, dx$ 
\correct $2/3$
\wrong $\displaystyle {{\pi} \over 2} - {{\pi^3} \over {24}}$
\wrong $\displaystyle {{\pi} \over 2}$
\wrong $\displaystyle 0$
\wrong $\displaystyle 1$

\note %14
\problem
Evaluate the following definite integral:
$\displaystyle \int_1^2 x \ln x \, dx$.  (Hint: $\ln 1 = 0$.)
\correct $\displaystyle 2 \ln 2 - {3 \over 4}$
\wrong $\displaystyle 2 \ln 2 - {3 \over 4} - {e \over 2}$
\wrong $\displaystyle 2 \ln 2 $
\wrong $\displaystyle 2 \ln 2 -1$
\wrong $\displaystyle 2 \ln 2 -e-1$

\note %15
\problem
Evaluate the following definite integral:
$\displaystyle \int_0^1 {{dx} \over {\sqrt{4-x^2}}}$
\correct $\pi /6$
\wrong $\pi /3$
\wrong $\pi /4$
\wrong $\pi$
\wrong $0$

\note %16
\problem In order to integrate
$$
\int {{x^2-10} \over {x(x+1)(3x-1)^2}} dx
$$
by partial fractions, which of the following is the correct decomposition to set
it up?
\correct 
$\displaystyle {{x^2-10} \over {x(x+1)(3x-1)^2}} = {A \over x} + {B \over
{x+1}} + {{C_1} \over {3x-1}} + {{C_2} \over {(3x-1)^2}}$
\wrong
$\displaystyle {{1} \over {x(x+1)(3x-1)^2}} = {A \over x} + {B \over
{x+1}} + {{C_1} \over {3x-1}} + {{C_2} \over {3x-1}}$
\wrong
$\displaystyle {{x^2-10} \over {x(x+1)(3x-1)^2}} = {A \over x} + {B \over
{x+1}} + {{C_1} \over {3x-1}} + {{C_2} \over {3x-1}}$
\wrong
$\displaystyle {{x^2-10} \over {x(x+1)(3x-1)^2}} = {A \over x} + {B \over
{x+1}} +  {{C} \over {(3x-1)^2}}$
\wrong
$\displaystyle {{1} \over {x(x+1)(3x-1)^2}} = {A \over x} + {B \over
{x+1}}  + {{C} \over {(3x-1)^2}}$

\note %17
\problem 
Suppose that $y$ is some quantity changing over time $t$, and that the rate at
which it changes \break is proportional to the quantity present at any given time. 
In other words, suppose that \break $\displaystyle {{dy} \over {dt}} = ky$.  What
is the
 solution for $y$?
\correct $y = Ce^{kt}$, where $C = y(0)$
\wrong $y = {k \over 2}y^2$
\wrong $y = e^{kt} + C$, where $C = y(0)$
\wrong $y = C_1 d^{kt} + C_2$, where $C_1 = y(0)$ and $C_2$ is an arbitrary
constant
\wrong $y = {k \over 2} t^2$

\note %18
\problem
Evaluate the following definite integral:
$\displaystyle \int_0^3 xe^{x^2} dx$
\correct $\displaystyle {1 \over 2} (e^9 -1)$
\wrong $\displaystyle {1 \over 2} (e^6 -1)$
\wrong $\displaystyle {1 \over 2} e^9$
\wrong $\displaystyle {1 \over 2} e^6$
\wrong $\displaystyle  e^6$

\note %19
\problem
In order to evaluate the following definite integral
$$ \int_4^8 {{dx} \over {x \sqrt{x^2 + 4}}}$$
what substitution should you make as a first step? 
\correct $x = 2 \tan \theta$
\wrong $x = 2 \sin \theta$
\wrong $u = x^2 + 4$
\wrong $u = \sqrt{x^2 + 4}$
\wrong $x=u$

\appendix
%That's all folks!

\end
