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\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl November 1, 1994}\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}

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\note %1
\problem
Find $\displaystyle {d \over {dx}} \left ( {{\sin x +1} \over {\cos x + 1}}
\right )$ \ \ \ (Hint: remember that $\sin^2 x + \cos^2 x = 1$.)
\correct
$\displaystyle {{1 + \sin x + \cos x} \over {(\cos x + 1)^2}}$
\wrong
$\displaystyle \sec^2 (x+1)$
\wrong
$\displaystyle \sec^2 x$
\wrong
$\displaystyle {{1} \over {(\cos x + 1)^2}}$
\wrong
$0$

\note %2
\problem
Find the slope of the tangent line to the
curve $\displaystyle y = \cos 3x$ at the point $\displaystyle \left ({\pi \over
4}, - {{\sqrt{2}} \over 2}  \right )$.
\correct
$\displaystyle - {{3 \sqrt{2}} \over 2}$
\wrong
$\displaystyle {{3 \sqrt{2}} \over 2}$
\wrong
$-3$
\wrong
3
\wrong 
$\displaystyle - {1 \over 2}$

\note %3
\problem
Find $f'(x)$ if $f(x) = (1 + 2x)^3$.
\correct 
$f'(x) = 6(1+2x)^2$
\wrong
$f'(x) = 12(1+2x)^2$
\wrong
$f'(x) = 24(1+2x)^2$
\wrong
$f'(x) = 3(1+2x)^2$
\wrong
$f'(x) = (1+2x)^2$

\note %4
\problem
Find $\displaystyle {{dy} \over {dx}}$ by implicit differentiation:
$$
x^2 + xy^2 + y^3 = 1
$$
\correct
$\displaystyle {{dy} \over {dx}} = {{-2x-y^2} \over {2xy+3y^2}}$
\wrong
$\displaystyle {{dy} \over {dx}} = {{1-2x-y^2} \over {2xy+3y^2}}$
\wrong
$\displaystyle {{dy} \over {dx}} = {{-2x-y^2-2xy} \over {3y^2}}$
\wrong 
$\displaystyle {{dy} \over {dx}} = {{-2x-2xy} \over {3y^2}}$
\wrong
$\displaystyle {{dy} \over {dx}} = {{1-2x-2xy} \over {3y^2}}$

\note %5
\problem
Find the slope of the tangent line to the following curve at the given point:
$$
x^2 - y^2 = 3, \ \ (2,1)
$$
\correct
2
\wrong
$\displaystyle {1 \over 2}$
\wrong
$\displaystyle -{1 \over 2}$
\wrong
$\displaystyle -2$
\wrong
$\displaystyle {3 \over 2}$

\note %6
\problem
Find the second derivative of the function $f(x) = \tan x$.
\correct
$2 \sec^2 x \tan x$
\wrong 
$2\sec x$
\wrong
$-2 \sec x \tan x$
\wrong 
$-2\sec x$
\wrong
$2 \sec x \tan x$

\note %7
\problem
A particle moves along a straight line, with equation of motion given by $s = t^3
- 12t$, with $t \geq 0$.  Find the acceleration at the instant when the velocity is
0.
\correct
12
\wrong
$-12$
\wrong
24
\wrong
$\sqrt{12}$
\wrong
0

\note %8
\problem
A square with side $x$ and area $A$ is growing with respect to time $t$ in
such a way that $x$ grows at a rate of 3 feet per second.  Find
$\displaystyle {{dA} \over {dt}}$ at the moment when $x = 4$.
\correct
24
\wrong
9
\wrong
$\displaystyle {{16} \over 9}$
\wrong
16
\wrong
12

\note %9
\problem
If $xy = 1$ and $\displaystyle {{dx} \over {dt}} = 4$, find $\displaystyle {{dy}
\over {dt}}$ when $x = 2$.
\correct
$-1$
\wrong
$\displaystyle - {1 \over 2}$
\wrong
$\displaystyle {1 \over 4}$
\wrong
$-4$
\wrong
$\displaystyle  {1 \over 2}$

\note %10
\problem
Find (all) the critical numbers of the function $f(x) = | x-1|$.
\correct
$x=1$
\wrong
$x = -1$
\wrong
$x = 0$
\wrong
$x = 1$ and $x = -1$
\wrong
The function has no critical numbers

\note %11
\problem
Find the absolute maximum value (i.e.\ the $y$-coordinate) of the function $f(x)
= - x^2 + 4x + 1$ on the interval $0 \leq x \leq 3$.
\correct
5
\wrong
2
\wrong
4
\wrong
10
\wrong
3

\note %12
\problem
Find (all of) the critical numbers of the function $f(x) = \sin 2x$ on the interval
$0 < x < \pi$.
\correct
$\displaystyle x = {{\pi} \over 4}, {{3\pi} \over 4}$
\wrong
$\displaystyle x = {{\pi} \over 2}$
\wrong
$\displaystyle x = {{\pi} \over 4}$
\wrong
$\displaystyle x = {{\pi} \over 2}$
\wrong
The function has no critical numbers on the given interval

\note %13
\problem
Joe and Robin start from the same point, at the same time.  Joe walks east at a
rate of 4 mph and Robin walks north at a rate of 3 mph.  How quickly is the
distance between them changing after one hour?
\correct
5 mph
\wrong
10 mph
\wrong
7 mph
\wrong
12 mph
\wrong
15 mph

\note %14
\problem
If $g(t) = t^4 - 4t^2 + 2$, find $g^{(3)} (1)$.  (Remember that $f^{(n)}(x)$ is
the $n$-th derivative of $f(x)$.)
\correct
24
\wrong
0
\wrong
4
\wrong
$-4$
\wrong
$-1$

\note %15
\problem
Consider the curve $x^2 + 2y^2 = 8$.  It is a fact that
$\displaystyle {{dy} \over {dx}} = - {1 \over 2} \cdot{x \over y}$.  Using this
fact, find all the points where the tangent line is horizontal.
\correct
$(0,2)$ and $(0,-2)$
\wrong
$(0,2)$
\wrong
$(\sqrt{8},0)$ and $(-\sqrt{8},0)$
\wrong
$(0,0)$
\wrong
$(\sqrt{8},0)$

\note %16
\problem
Find a second degree polynomial $P(x)$ such that $P(0) = 1$, $P'(0) = 3$ and
$P''(0) = 4$.  (In other words, if $P(x) = ax^2 + bx + c$, this question is
asking you to figure out what $a$, $b$ and $c$ have to be.)
\correct
$P(x) = 2x^2 + 3x + 1$
\wrong
$P(x) = 4x^2 + 3x + 1$
\wrong
$P(x) = x^2 + 3x + 2$
\wrong 
$P(x) = x^2 + 3x + 4$
\wrong
$P(x) = x^2 + 2x + 3$

\note %17
\problem
If $\displaystyle f(x) = \tan {1 \over x}$, find $f'(x)$.
\correct
$\displaystyle -\left ( {1 \over {x^2}} \right ) \left ( \sec^2 {1 \over x}
\right )$
\wrong
$\displaystyle \left ( \tan x \right ) \left ( - {1 \over {x^2}} \right ) + \left
( {1 \over x} \right ) \left ( \sec^2 x \right )$
\wrong
$\displaystyle \left ( {1 \over {x^2}} \right ) \left ( \csc^2 {1 \over x}
\right )$
\wrong
$\displaystyle \left ( {1 \over {x^2}} \right ) \left ( \sec^2 {1 \over x}
\right )$
\wrong
$\displaystyle -\left ( {1 \over {x^2}} \right ) \left ( \csc^2 {1 \over x}
\right )$

\note %18
\problem
Let $\displaystyle f(x) = 8x^{1 \over 2} - {2 \over 3} x^{3 \over 2} + 1$.  It is
a fact that
$\displaystyle f'(x) = {4 \over \sqrt{x}} -
\sqrt{x}$.  Find (all of) the critical numbers of $f(x)$.
\correct
$x = 0$ and $x = 4$
\wrong
$x = 4$
\wrong
$x = 0$ and $x = 2$
\wrong
$x = 0$, $x = 2$ and $x = -2$
\wrong
$x = 0$ and $x = 16$

\note %19
\problem
Let $f(x) = x \sin x$.  Find $f'(\pi)$.
\correct
$-\pi$
\wrong
0
\wrong
$\pi$
\wrong
$-1$
\wrong
1


\appendix
%That's all folks!

 \end{document}
