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\preface
\magnification=1200

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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl May 8, 1996}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are {\it not} 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  29 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 coefficient of $x^{98}y^2$ in the expansion of $(x+y)^{100}$.
\correct 4950
\wrong 100
\wrong 5000
\wrong 98
\wrong 196

\note %2
\problem Find $\displaystyle {d \over {dx}} \int_{3x}^5 \sin({t^2})\ dt$
\correct $-3\sin(9x^2 )$
\wrong $3 \sin(9x^2 )$
\wrong $ \cos (9x^2) - \cos (25)$
\wrong $-3\cos (9x^2 )$
\wrong $\sin (9x^2 ) - \sin (25)$

\note %3
\problem  What is the slope of the tangent line to the curve $y = 2^x$ at $x=0$?
\correct $\ln 2$
\wrong 1
\wrong 2
\wrong $e^2$
\wrong 0

\note %4
\problem A particle is moving along a line, with acceleration function $a(t) = 6t$
(in $\hbox{m/sec}^2$) and initial velocity $v(0) = -3$.  Find the distance
traveled (not the displacement) during the time interval $0 \leq t \leq 2$.
\correct 6 m
\wrong 2 m
\wrong 8 m
\wrong 9 m
\wrong 12 m

\note %5
\problem Evaluate $\displaystyle \int {{\sin \sqrt{x}} \over {\sqrt{x}}} \ dx$
\correct $-2 \cos \sqrt{x} +C$
\wrong $2 \cos \sqrt{x} +C$
\wrong $\displaystyle {2 \over 3} \cos (x^{3/2}) +C$
\wrong $\displaystyle -{2 \over 3} \cos (x^{3/2}) +C$
\wrong $-\cos x +C$

\note %6
\problem Set up, but do not simplify or evaluate, an integral for the volume of the
solid obtained by rotating about the line $y = 3$ the region bounded by $y = e^x$,
$y = x$, $x=-1$ and $x=1$ (shaded in the picture below).

\vskip 2in
\hskip .6in \e-to-x

\correct $\displaystyle \int_{-1}^1 \pi \left [ (3-x)^2 - (3-e^x )^2 \right ] \
dx$
\wrong $\displaystyle \int_{-1}^1 \pi \left [ 3 - (e^x - x)^2 \right ]\ dx$
\wrong $\displaystyle \int_{-1}^1 \pi \left [ 3- e^{2x} - x^2 \right ] \ dx$
\wrong $\displaystyle \int_{-1}^1 \pi \left [ (3 - e^{2x} ) + (3 - x^2) \right ]
\ dx$
\wrong $\displaystyle \int_{-1}^1 \pi \left [ 3 - (e^x + x)^2 \right ]\ dx$

\note %7
\problem Set up, but do not evaluate, an integral for the volume of the
solid obtained by rotating about the $y$-axis the region between the $y$-axis and
the curve $y = x^2$, from $y = 0$ to $y=9$ (shaded, below).

\vskip 1.5in
\hskip 1.7in \region

\correct $\displaystyle \int_0^9 \pi y \ dy$
\wrong $\displaystyle \int_0^9 {1 \over 2} \pi y^2 \ dy$
\wrong $\displaystyle \int_0^3 \pi x^4 \ dx$
\wrong $\displaystyle \int_0^3 \pi \left [ 81 - x^4 \right ] \ dx$
\wrong $\displaystyle \int_0^9 \pi \left [ 9 - \sqrt{y}\ \right ] \ dy$

\note %8
\problem Solve for $x$: $\displaystyle \log_3 (\log_2 (\log_5 x)) = 1$

\correct $\displaystyle 5^8$
\wrong $\displaystyle 5^9$
\wrong $\displaystyle 3^{32}$
\wrong $\displaystyle 3^{25}$
\wrong $\displaystyle 2^{125}$

\note %9
\problem Let $\displaystyle f(x) = {1 \over 2} e^{2x} + e^{-x}$.  Find the
interval(s) on which $f(x)$ is increasing.

\correct $[0,\infty)$
\wrong $[-\infty, 0]$
\wrong $(-\infty,\infty)$
\wrong $\displaystyle \left [ {1 \over 3}, \infty \right )$
\wrong $\displaystyle \left [{1 \over {\sqrt{2}}}, \infty \right )$

\note %10
\problem Find the degree 3 Taylor polynomial for the function $\displaystyle f(x)
= \sqrt{1+x}$.  (Note that $\displaystyle \sqrt{1+x} = (1+x)^{1/2}$.)

\correct $\displaystyle 1 + {1 \over 2}x - {1 \over 8}x^2 + {1 \over {16}}x^3$
\wrong $\displaystyle 1 + {1 \over 2}x - {1 \over 4}x^2 + {3 \over {8}}x^3$
\wrong $\displaystyle 1 + {1 \over 2}x - {1 \over 2}x^2 + {9 \over {4}}x^3$
\wrong $\displaystyle 1 + {1 \over 2}x + {1 \over 4}x^2 + {1 \over {8}}x^3$
\wrong $\displaystyle 1 - {1 \over 2}x + {1 \over 4}x^2 - {1 \over {8}}x^3$

\note %11
\problem Find the $n$th Taylor polynomial for $\displaystyle g(x) = e^{-2x}$. 
(You can either compute it directly, or else use the fact that the $n$th Taylor
polynomial for $\displaystyle f(x) = e^x$ is $\displaystyle 1 + x + {{x^2} \over
{2!}} + \cdots + {{x^n} \over {n!}} = \sum_{k=0}^n {{x^k} \over {k!}}$.)

\correct $\displaystyle \sum_{k=0}^n (-1)^k 2^k {{x^k} \over {k!}}$
\wrong $\displaystyle \sum_{k=0}^n 2^k {{x^k} \over {k!}}$
\wrong $\displaystyle \sum_{k=0}^n (-1)^k {{x^k} \over {k!}}$
\wrong $\displaystyle \sum_{k=0}^n 2^k {{x^{-k}} \over {k!}}$
\wrong $\displaystyle \sum_{k=0}^n {{x^{-2k}} \over {k!}}$

\note %12
\problem Giuseppe's Pizza has 8 different toppings that they can put on their
pizzas.  A customer ordering their Deluxe pizza can choose any 3 of the
toppings.  How many different Deluxe pizzas can be made?  (For example, a pizza
consisting of pepperoni, sausage and green pepper is different from one
consisting of pepperoni, sausage and mushrooms.)

\correct 56
\wrong 28
\wrong 24
\wrong 3
\wrong 48

\note %13
\problem A fair die has 6 sides, of which two are painted white and four are
painted black.  Jack is told to roll the die a total of five times, each time
recording whether it comes up white or black.  He is told that if he rolls at
least four whites, he will win \$1,000,000.  What is the probability that he will
win \$1,000,000?

\correct $\displaystyle {{11} \over {243}}$
\wrong $\displaystyle {{10} \over {243}}$
\wrong $\displaystyle {{3} \over {243}}$
\wrong $\displaystyle {{112} \over {243}}$
\wrong $\displaystyle {{6} \over {243}}$

\note %14
\problem Find $\displaystyle \lim_{x \rightarrow -\infty} x^2 e^x$.

\correct 0
\wrong $\infty$
\wrong $-\infty$
\wrong 2
\wrong $-2$

\note %15
\problem Find $\displaystyle \lim_{x \rightarrow 1} {{x^8 -1} \over {x^5 -1}}$,
if it exists.

\correct $\displaystyle {8 \over 5}$
\wrong 1
\wrong 0
\wrong $\infty$
\wrong limit does not exist

\note %16
\problem Evaluate $\displaystyle \int {{x^4 +1} \over {x^2 -1}} \ dx$.

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

\note %17
\problem It is a fact, which you don't have to verify, that 
$$
{{4x^2} \over {x^4 - 2x^2 +1}} =
{1 \over {x-1}} + {1 \over {(x-1)^2}} - {1 \over {x+1}} + {1 \over {(x+1)^2}}
$$
(Note that the third term is a minus.)  Using this fact, evaluate $\displaystyle
\int {{4x^2} \over {x^4 - 2x^2 +1}}\ dx$.

\correct $\displaystyle \ln \left | {{x-1} \over {x+1}} \right | - {1 \over
{x-1}} - {1 \over {x+1}} +C$
\wrong $\displaystyle \ln \left | {{x-1} \over {x+1}} \right | + {1 \over
{x-1}} + {1 \over {x+1}} +C$
\wrong $\displaystyle \ln \left | (x-1)(x+1) \right | + {1 \over
{x-1}} + {1 \over {x+1}} +C$
\wrong $\displaystyle \ln \left | (x-1)(x+1) \right | - {2 \over
{(x-1)^3}} - {2 \over {(x+1)^3}} +C$
\wrong $\displaystyle \ln \left | {{x-1} \over {x+1}} \right | + {2 \over
{(x-1)^3}} + {2 \over {(x+1)^3}} +C$

\note %18
\problem Evaluate $\displaystyle \int_0^{\pi/2} x \cos (2x) \ dx$

\correct $\displaystyle -{1 \over 2}$
\wrong 0
\wrong $\displaystyle {1 \over 2}$
\wrong 1
\wrong $\displaystyle {3 \over 2}$

\note %19
\problem Find the average value of the function $f(x) = x^2$ on the interval
$[0,2]$.

\correct $\displaystyle {4 \over 3}$
\wrong $\displaystyle {8 \over 3}$
\wrong $\displaystyle 2$
\wrong  1
\wrong $\displaystyle {3 \over 2}$

\note %20
\problem If you were to use a trigonometric substitution to evaluate the integral
$\displaystyle \int {{x^2} \over {\sqrt{5+x^2}}} \ dx$, what substitution should
you use?

\correct $x = \sqrt{5} \tan \theta$
\wrong $x = 5 \tan \theta$
\wrong $x = \sqrt{5} \sin \theta$
\wrong $x = 5 \sin \theta$
\wrong $x = 5 \sec \theta$

\note %21
\problem Find the equation of the tangent line to the curve $y = \ln (x^2 +1)$ at
the point $(1, \ln 2 )$.

\correct $y = x - 1+ \ln 2$
\wrong $y = 2x+\ln 2$
\wrong $y = x-1$
\wrong $y = 2x-2 + \ln 2$
\wrong $y = x - \ln 2$

\note %22
\problem A certain sum of money is deposited in an account which earns an annual
interest rate of 10\%, compounded continuously.  No further deposit is made into
the account.  How long does it take (in years) before the total amount of
money in the account is twice the original amount?

\correct $10 \cdot \ln 2$ (approx.\ 6.9)
\wrong $10$
\wrong 5
\wrong $2 \cdot \ln 10$ (approx.\ 4.6)
\wrong It depends on how much was originally deposited.

\note %23
\problem The initial amount of a certain radioactive substance is 100g.  After 3
minutes, 50g remains.  How much remains after 4 minutes?

\correct $\displaystyle 100 \left ( {1 \over 2} \right )^{4/3}$
\wrong $\displaystyle 100 e^{4/3}$
\wrong $\displaystyle 100 \left ( {1 \over e} \right )^{4/3}$
\wrong $\displaystyle 100 \left ( {1 \over 2} \right )^{4}$
\wrong $\displaystyle 100 \left ( {1 \over 3} \right )^{4}$

\note %24
\problem Evaluate $\displaystyle \int_{\pi/2}^{\pi} {{\sin x} \over {2+\cos x}}\
dx$.

\correct $\ln 2$
\wrong 1
\wrong 0
\wrong $\ln 3$
\wrong $\ln 3 - \ln 2$

\note %25
\problem Suppose that a certain spring requires 2 J of work to stretch from its
natural length of 0.30 m to a length of 0.40 m.  What is the constant in Hooke's
Law for this spring?  (I.e.\ Hooke's Law says that the force is proportional to
the displacement, 
$f(x) = kx$.  I'm asking what $k$ is in this case.)

\correct 400
\wrong $\displaystyle {{400} \over 7}$
\wrong 25
\wrong 4
\wrong $\displaystyle {4 \over 3}$

\note %26
\problem A tank has the shape of an inverted circular cone with height 10 m and
radius 5 m.  It is filled with water to a height of 7 m.  Which of the
following integrals represents the work required to empty the tank by pumping all
the water to the top of the tank.  (The density of water is 1000 kg/m$^3$ and the
acceleration due to gravity is 9.8 m/sec$^2$.  Notice that the answers don't all
have the same limits of integration, but they all assume that $x$ is measured
from the {\it top} of the tank, as indicated in the diagram.)

\vskip 1.7in
\hskip 1.2in \cone

\correct $\displaystyle \int_3^{10} 9800 \pi \left ( {{10-x} \over 2} \right )^2
x \ dx$
\wrong $\displaystyle \int_3^{10} 9800 \pi \left ( {{x} \over 2} \right )^2
(10-x) \ dx$
\wrong $\displaystyle \int_0^{7} 9800 \pi \left ( {{x} \over 2} \right )^2
x \ dx$
\wrong $\displaystyle \int_0^{3} 9800 \pi \left ( {{10-x} \over 2} \right )^2
x \ dx$
\wrong $\displaystyle \int_0^{3} 9800 \pi \left ( {{10-x} \over 2} \right )^2
(10-x) \ dx$

\note %27
\problem Find $\displaystyle \lim_{x \rightarrow 1^-} e^{2/(x-1)}$

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

\note %28
\problem Find $\displaystyle \int \cos^3 4x \ dx$

\correct $\displaystyle {1 \over 4} \sin 4x - {1 \over {12}} \sin^3 4x +C$
\wrong $\displaystyle {1 \over {16}} \cos^4 4x +C$
\wrong $\displaystyle -{1 \over 4} \sin 4x + {1 \over {12}} \sin^3 4x +C$
\wrong $\displaystyle \sin 4x - {1 \over 3} \sin^3 4x +C$
\wrong $\displaystyle {1 \over 4} \cos^4 4x +C$

\note %29
\problem Find the slope of the tangent line to the curve $\displaystyle y =
e^{-x^2}$ at the point $\displaystyle (1, {1 \over e})$.

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

\appendix
I enjoyed teaching this course.  Have a great summer vacation, and good luck next
year!!

\end
