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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 October 31, 1995}\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}

\def\problemskip{\vskip1in}

\note %1
\problem  
let $f(x) = \sin x + \cos x$. Find all values of $x$ between $0$ and $2\pi$,
inclusive, where the tangent line to the graph of $f(x)$ is horizontal.
\correct $\displaystyle x = {{\pi} \over 4}, {{5\pi} \over 4}$
\wrong $\displaystyle x = {{3\pi} \over 4}, {{7\pi} \over 4}$
\wrong $\displaystyle x = {{\pi} \over 4}, {{3\pi} \over 4}, {{5\pi} \over 4},
{{7\pi} \over 4}$
\wrong $x = 0, \pi, 2\pi$
\wrong $\displaystyle x = {{\pi} \over 2}, {{3\pi} \over 2}$

\note %2
\problem
Let $\displaystyle f(x) = \tan (x^2 +1)$.  Find $f'(x)$.
\correct $\displaystyle 2x \sec^2 (x^2 +1)$
\wrong $\displaystyle \sec^2 (x^2 +1)$
\wrong $\displaystyle \sec^2 (2x)$
\wrong $\displaystyle \tan (2x) \cdot \sec^2 (x^2 +1)$
\wrong $\displaystyle \sec^2 (x^2 +1) + \tan (2x)$

\note %3
\problem
If $xy + 1 = y^2$, use implicit differentiation to find $\displaystyle {{dy} \over
{dx}}$.
\correct $\displaystyle {{-y} \over {x-2y}}$
\wrong $\displaystyle {{x+y} \over {2y}}$
\wrong 0
\wrong $\displaystyle {{y} \over {x}}$
\wrong $\displaystyle {{-y-1} \over {x-2y}}$

\note %4
\problem Let $f(x) = \sin x$.  Find the one hundredth derivative $f^{(100)}(x)$.
\correct $\displaystyle \sin x$
\wrong $\displaystyle \cos x$
\wrong $\displaystyle -\sin x$
\wrong $\displaystyle -\cos x$
\wrong $\displaystyle (100!)\cdot \sin x$

\note %5
\problem Let $\displaystyle f(x) = x^{10} + x^5 + 3$.  Find $f^{(10)} (x)$.
\correct $10!$
\wrong $10! + 5!$
\wrong $10! + 5! + 3!$
\wrong 0
\wrong $10! + 5! + 3$

\note %6
\problem If a ball is thrown vertically upward with a velocity of 16 ft/sec,
then its height after $t$ seconds is $s = 16t - 16t^2$.  What is the maximum
height reached by the ball?
\correct 4 ft.
\wrong 0 ft.
\wrong 16 ft.
\wrong 8 ft.
\wrong 32 ft.

\note %7
\problem The equation of motion for a certain particle is $s = t^4 - t^2 + 1$. 
What is the equation for the acceleration of the particle?
\correct $12t^2 -2$
\wrong $12t^2$
\wrong $4t^3 - 2t$
\wrong $24t$
\wrong $4t^3 - 2t + 1$

\note %8
\problem Dave is standing at the edge of a canal directly across from point $A$. 
The canal is 8 feet across.  Mary is walking along the canal, away from point $A$,
at a rate of 5 ft/sec.  When she is 6 ft away from point $A$, at what rate is the
distance between Dave and Mary increasing?

\vskip 1.2in
\hskip 1.4in \canal

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

\note %9
\problem A spherical snowball is melting in such a way that its volume is {\it
decreasing} at a rate of 4 $\hbox{cm}^3$/min.  Recall that the formula for the
volume of a sphere is 
$$V = {{4} \over 3} \pi r^3
$$
where $V$ is the volume and $r$ is the radius of the sphere.
Find $\displaystyle {{dr} \over {dt}}$ at the moment when $r = 5$~cm. (Hint: 
don't forget that the snowball is getting smaller.)
\correct $\displaystyle {{-1} \over {25\pi}}$
\wrong $\displaystyle {{1} \over {25\pi}}$
\wrong $\displaystyle {{3} \over {125\pi}}$
\wrong $\displaystyle {{-3} \over {125\pi}}$
\wrong $\displaystyle {{-1} \over {125\pi}}$

\note %10
\problem Let $f(x) = x^3 - 12x$.  Find the absolute maximum value of $f(x)$ on
the interval $[0,3]$.  (You are looking for a $y$-value, not an $x$-value.)
\correct 0
\wrong $-9$
\wrong $-16$
\wrong 16
\wrong 2

\note %11
\problem Let 
$$
f(x) = {{x} \over {x^2 +1}}
$$
Find (all) the critical numbers of $f(x)$, if there are any.
\correct $x = -1, 1$
\wrong $x = 1$
\wrong $x = -1$
\wrong there are no critical numbers
\wrong $x=0$

\note %12 
\problem Let $\displaystyle f(x) = 5x^{2/3} + x^{5/3}$.  It happens to be true
that 
$$
f'(x) = {{10+5x} \over {3x^{1/3}}}
$$
Find the value(s) of $x$, if any, where $f(x)$ has a local minimum
\correct $x = 0$
\wrong $x = -2, 0$.
\wrong $x = -2$
\wrong $x = 2$
\wrong $f(x)$ has no local minimum

\note %13
\problem Let $f(x) = x^4 - 6x^2$.  Find the interval(s) where $f(x)$ is concave
up.
\correct $(-\infty,-1) \cup (1,\infty)$
\wrong $(-1,1)$
\wrong $(-\infty, 1)$
\wrong $f(x)$ is concave up everywhere
\wrong $f(x)$ is not concave up anywhere

\note %14
\problem Consider the curve $x = y^2$.  Find the slope of the tangent line at the
point $(4,2)$.
\correct $\displaystyle {1 \over 4}$
\wrong $4$
\wrong $\displaystyle {1 \over 8}$
\wrong 8
\wrong $\displaystyle {1 \over 2}$

\note %15
\problem Let 
$$
f(x) = {1 \over 4} x^4 - {1 \over 3} x^3 + 5.
$$  
Find {\it all} the local extrema of $f(x)$.
\correct local minimum at $x=1$
\wrong local maximum at $x=0$, local minimum at $x=1$
\wrong local minimum at $x=0$, local maximum at $x=1$
\wrong there are no local extrema
\wrong local minimum at $x=0$, local minimum at $x=1$

\note %16
\problem Let $\displaystyle f(x) = {1 \over 2} x^2 + \sin x$.  Find the
inflection points, if any, of $f(x)$ in the range $0 \leq x \leq 2\pi$.
\correct there are no inflection points
\wrong $\displaystyle x = {{\pi} \over 2}, {{3\pi} \over 2}$
\wrong $\displaystyle x = {{\pi} \over 2}$
\wrong $\displaystyle x = {{\pi} \over 2}, \pi, {{3\pi} \over 2}$
\wrong $\displaystyle x = {{3\pi} \over 2}$

\note %17
\problem Let $f(x) = {\root 3 \of {x^4 - x}}$.  Find $f'(x)$.
\correct $\displaystyle {{4x^3 - 1} \over {3(x^4 - x)^{2/3}}}$
\wrong $\displaystyle {1 \over {3(x^4 - x)^{2/3}}}$
\wrong $\displaystyle {{3(4x^3 - 1)} \over {(x^4 - x)^{2/3}}}$
\wrong $\displaystyle {{(x^4 - x)^{2/3}} \over {4x^3 -1}}$
\wrong $\displaystyle {{4(x^3 - 1)} \over {3(x^4 - x)^{2/3}}}$

\note %18
\problem Consider the curve $4x^2 + y^2 = 4$.  It is a fact that
$$ {{dy} \over {dx}} = -4 \cdot{x \over y}$$  Using this fact, find all
the points on the curve where the tangent line is horizontal.
\correct $(0,2)$ and $(0,-2)$
\wrong $(1,0)$ and $(-1,0)$
\wrong $(0,2)$
\wrong $(1,0)$
\wrong $(0,0)$

\note %19
\problem An inverted conical tank has radius 4 meters and height 10 meters.  It 
is being filled with water at a rate of 8 $\hbox{m}^3$/min.  How quickly is the
water rising when the tank is half full (i.e.\ when the height of the water
reaches 5 meters)?  Recall that the formula for the volume of a cone of radius
$r$ and height $h$ is $\displaystyle V = {1 \over 3} \pi r^2 h$.

\vskip 1.2in
\hskip 1.5in  \cone

\correct $\displaystyle {2 \over \pi}$
\wrong $\displaystyle {4 \over \pi}$
\wrong $\displaystyle {\pi \over 3}$
\wrong $\displaystyle {\pi \over 10}$
\wrong $\displaystyle {6 \pi}$

\appendix
%That's all folks!

 \end{document}
