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\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl October 29, 1996}\hfill
Time of MWF class:\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  
Let 
$$
f(x) = \cases{x^2 + 1, &if $x < 0$;\cr
              3, &if $x = 0$;\cr
              2x+3, &if $x>0$.}
$$
Which of the following is true:
\correct $f(x)$ is continuous from the right but not from the left, at $x=0$.
\wrong $f(x)$ is continuous from the left but not from the right, at $x=0$.
\wrong $f(x)$ is continuous at $x=0$.
\wrong $f(x)$ is neither continuous from the left nor from the right, at $x=0$.
\wrong $f(x)$ is undefined at $x=0$.

\note %2
\problem
Convert $\displaystyle {{13\pi} \over 6}$ from radians to degrees.

\correct $390^\circ$
\wrong $420^\circ$
\wrong $330^\circ$
\wrong $300^\circ$
\wrong $360^\circ$

\note %3
\problem Find all values of $x$ in the interval $[0,2\pi]$ that satisfy the
equation 
$$
2 \sin x = \tan x
$$

\correct $\displaystyle x = 0, {{\pi} \over 3}, \pi, {{5\pi} \over 3}, 2\pi$
\wrong $\displaystyle x = 0, {{\pi} \over 6}, \pi, {{11\pi} \over 6}, 2\pi$
\wrong $\displaystyle x = 0, {{2\pi} \over 3}, \pi, {{4\pi} \over 3}, 2\pi$
\wrong $\displaystyle x = 0, {{5\pi} \over 6}, \pi, {{7\pi} \over 6}, 2\pi$
\wrong $\displaystyle x = 0, {{\pi} \over 2}, \pi, {{3\pi} \over 2}, 2\pi$

\note %4
\problem
Find $\displaystyle \lim_{x\rightarrow 0} {{\sin^2 5x} \over {x^2}}$, if it
exists.

\correct 25
\wrong 5
\wrong 1
\wrong it does not exist
\wrong 0

\note %5
\problem 
Find $\displaystyle {{dy} \over {dx}}$ if $\displaystyle y = {{\sin x} \over x}$

\correct $\displaystyle {{x \cos x - \sin x} \over {x^2}}$
\wrong 1
\wrong $\cos x$
\wrong $\displaystyle {{x\sin x - \cos x} \over {x^2}}$
\wrong $\displaystyle {{\sin x - \cos x} \over {x^2}}$

\note %6
\problem
If $\displaystyle f(x) = \sin^2 (x^2 +1)$, find $f'(x)$.

\correct $4x \sin (x^2 +1) \cos (x^2 +1)$
\wrong $4x \sin (x^2 +1)$
\wrong $4x  \cos (x^2 +1)$
\wrong $2 \sin (x^2 +1)$
\wrong $2 \sin (x^2 +1) \cos (x^2 +1)$

\note %7
\problem
If $f(x) = \tan^2 x$, it can be shown that $\displaystyle f'(x) = 2 {{\sin x}
\over {\cos^3 x}}$ (you don't have to verify this).  Find the equation of the
tangent line to the graph of $y = \tan^2 x$ at $\displaystyle x = {{\pi} \over
3}$.

\correct $\displaystyle y - 3 = 8 \sqrt{3} \left (x- {{\pi} \over 3} \right )$
\wrong $\displaystyle y - {1 \over 3} = 8 \sqrt{3} \left (x- {{\pi} \over 3} \right
)$
\wrong $\displaystyle y - 3 =  {{\sqrt{3}} \over 8} \left (x- {{\pi} \over 3}
\right )$
\wrong $\displaystyle y - {1 \over 3} = {{\sqrt{3}} \over 8} \left (x- {{\pi}
\over 3} \right )$
\wrong the graph has no tangent line at $\displaystyle x = {{\pi} \over 3}$

\note %8
\problem
Suppose that $F(x) = f(g(x))$ and suppose that you have the following information:
$$
\matrix{
f(2) = 7 & f(4) = 6 & g(2) = 4 & g(4) = 6 \cr \cr
f'(2) = -3 & f'(4) = 5 & g'(2) = 3 & g'(4) = -2 \cr}
$$
Find $F'(2)$.

\correct 15
\wrong $-10$
\wrong $-9$
\wrong $-12$
\wrong 21

\note %9
\problem
Use implicit differentiation to find $\displaystyle {{dy} \over {dx}}$ if 
$\displaystyle x^2 + xy + y^3 = 3$.

\correct $\displaystyle {{-2x-y} \over {x+3y^2}}$
\wrong $\displaystyle {{-x^2 - x - y} \over {3y^2}}$
\wrong $\displaystyle {{-2x + 3y^2} \over x}$
\wrong $\displaystyle {{-2x} \over {x + 3y^2}}$
\wrong $\displaystyle {{-2x} \over {3y^2}}$

\note %10
\problem
For what value of $c$ is the following function continuous?
$$
f(x) = \cases{cx^2 + 3, &if $x \leq 1$;\cr
              2x-3, &if $x>1$.}
$$

\correct $c = -4$
\wrong $c=2$
\wrong $c = 3$
\wrong $c = -3$
\wrong $c = 0$

\note %11
\problem
Find a second-degree polynomial $P(x)$ such that $P(1) = 2$, $P'(1) = 5$, and
$P''(1) = 8$.  (Recall that a second degree polynomial is one of the form $ax^2 +
bx + c$ for some constants $a$, $b$ and $c$.)  Once you've found this polynomial,
find $P(0)$.

\correct $1$
\wrong $0$
\wrong $2$
\wrong $8$
\wrong $5$

\note %12
\problem
Let $f(x) = \sqrt{3x+1}$.  Find $f''(x)$.

\correct $\displaystyle -{{9} \over 4} \cdot {1 \over {(3x+1)^{3/2}}}$
\wrong $\displaystyle -{{1} \over 4} \cdot {1 \over {(3x+1)^{3/2}}}$
\wrong $\displaystyle -{3 \over {(3x+1)^{2}}}$
\wrong $\displaystyle -{{9} \over 4} \cdot {1 \over {\sqrt{3x+1}}}$
\wrong $\displaystyle -{{1} \over 4} \cdot {1 \over {\sqrt{3x+1}}}$

\note %13
\problem Car A is moving east at 40 mph, while Car B is moving south at 30 mph. 
At noon, Car A is 3 miles east of point P, while Car B is 4 miles north of point
P.  (See diagram.)  Which of the following is true about the distance between Car
A and Car B at noon?

\vskip 1.5 in
\hskip 1.6in  \cars
\vskip .2in

\correct the distance is not changing
\wrong the distance is increasing by 5 mph
\wrong the distance is decreasing by 5 mph
\wrong the distance is increasing by 10 mph
\wrong the distance is decreasing by 10 mph

\note %14
\problem
The equation of motion for a certain particle moving along a straight line is
given by $s = 2t^3 - 4t^2 + 3t -5$ (where $s$ is in meters and $t$ is in
seconds).  Find the acceleration after 1 second.

\correct 4 m/$\hbox{sec}^2$
\wrong $-8$ m/$\hbox{sec}^2$
\wrong $8$ m/$\hbox{sec}^2$
\wrong $0$ m/$\hbox{sec}^2$
\wrong $6$ m/$\hbox{sec}^2$

\note %15
\problem
A balloon is constructed in the shape of a perfect cube, for a parade.  It is
being inflated at a rate of 36 $\hbox{ft}^3$/min, always maintaining the shape of
a cube.  How quickly is the side of the balloon growing at the instant when the
side is 2 ft.\ long?

\vskip 1.1in
\hskip 2in \cube

\correct 3 ft/min
\wrong 1 ft/min
\wrong 2 ft/min
\wrong 4 ft/min
\wrong 5 ft/min

\note %16
\problem
Find $\displaystyle \lim_{t \rightarrow 0} {{\cos t - 1} \over {\sin 2t}}$, if it
exists.

\correct 0
\wrong 1
\wrong the limit does not exist
\wrong $\displaystyle {1 \over 2}$
\wrong 2

\note %17
\problem
Find the slope of the tangent line to the curve $y^2 - 6x^2 = 1$ at the point
$(2,5)$.

\correct $\displaystyle {{12} \over 5}$
\wrong $\displaystyle {{2} \over 5}$
\wrong $\displaystyle {{24} \over 5}$
\wrong $\displaystyle 0$
\wrong $3$

\note %18
\problem
If a ball is thrown vertically upward from the top of a 48-foot tower with a
velocity of 32 ft/sec, its height after $t$ seconds is $s = -16 t^2 + 32 t +
48$.  What is its velocity as it hits the ground?

\correct $-64$ ft/sec
\wrong $-32$ ft/sec
\wrong $-16$ ft/sec
\wrong $-48$ ft/sec
\wrong $-80$ ft/sec

\note %19
\problem
Of the five expressions listed below, which one is equal to $\sec x - \cos x$?

\correct $\tan x \sin x$
\wrong $\tan^2 x$
\wrong $\cos^2 x$
\wrong $\sin^2 x$
\wrong 0

\note %20
\problem
Find the third derivative, $f^{(3)} (x)$, of $f(x) = \sin 2x$.

\correct $-8 \cos 2x$
\wrong $8 \cos 2x$
\wrong $- \cos 2x$
\wrong $\cos 2x$
\wrong $-2 \cos 2x$

\appendix
%That's all folks!

 \end{document}
