\input ignore.tex %This line causes the mce macros to be ignored.

\preface
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\def\problemskip{\vskip0.5in}

\title
\noindent
{\bf Math 119}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl March 17, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}


\def\answerskip{\vskip0.25in}


\note %1
\problem
Let $f(x)=\sqrt x$. For what value of $x$ is the tangent to 
this curve parallel to the line $-x+8y=10$?
\correct
$16$
\wrong
$4$
\wrong
$2$
\wrong
$-2$
\wrong
$-4$



\note %2
\problem
A ladder $8\, m$ long leans against a wall $6\, m$ high. 
If the lower
end of the ladder is pulled away from the wall at a rate of
$2\, m/sec $, how fast $(radian/sec)$ is the angle between the top of the ladder
and the wall changing when the angle is $60 ^{\circ} ={\pi\over 3}$ radians?
\correct
${1\over 2}$
\wrong
$2$
\wrong
$\pi$
\wrong
${1\over 2}\pi$
\wrong
$1$


\note %3
\problem
If $y=\sin (x+y)$, find ${dy\over dx}$.
\correct
${{\cos (x+y)}\over {1-\cos (x+y)}}$
\wrong
${{- \cos (x+y)}\over {1-\cos (x+y)}}$
\wrong
${{\cos (x+y)}\over {1+\cos (x+y)}}$
\wrong
${{\cos (x+y)}\over {1-\sin (x+y)}}$
\wrong
${{-\sin (x+y)}\over {1-\cos (x+y)}}$


\note %4
\problem
Find ${d\over dx} \tan ^2 \sqrt x$.
\correct
${{\tan {\sqrt x} \, \sec ^2 {\sqrt x}}\over \sqrt x}$
\wrong
${{\tan {\sqrt x} \, \sec ^2  x}\over \sqrt x}$
\wrong
${{-\tan {\sqrt x} \, \sec ^2 {\sqrt x}}\over x}$
\wrong
${{-\sec {\sqrt x} \, \sec ^2  x}\over \sqrt x}$
\wrong
${{\sec {\sqrt x} \, \tan ^2 {\sqrt x}}\over x}$


\note %5
\problem
Find ${{dy}\over{dx}}$ if ${1\over x}+{1\over y}=1$.
\correct
$-{{y^2}\over {x^2}}$
\wrong
${{y^2}\over {x^2}}$
\wrong
${{y}\over {x}}$
\wrong
$-{{y}\over {x}}$
\wrong
${{1}\over {x^2}}-{{1}\over {y^2}}$


\note %6
\problem
Find $\lim_{x\rightarrow 2} {{\sin (x-2)}\over 
{x^2 -2x}}$.
\correct
$1\over 2$
\wrong
$1$
\wrong
$-1$
\wrong
$0$
\wrong
does not exist.


\note %7
\problem
 Suppose $f$ has a derivative at $c$ and $f'(c) < 0$. Which of the
following statements is true?
\correct $f$ can not have either a local maximum or local minimum at $c$.
\wrong $f$ must have a local maximum at $c$.
\wrong $f$ must have a local minimum at $c$.
\wrong $c$ must be a critical point of $f$.


\note %8
\problem
Find the equation of the line tangent to the curve $xy+x^2 y^2=2$
at $(1,1)$
\correct $y=-x+2$
\wrong $y=-x-2$
\wrong $y=x+1$
\wrong $y=2$
\wrong $y=x+2$


\note %9
\problem
Suppose
$$f(x) = \cases{\sin x &if $x \le 0$  \cr
                  \tan x &if $0 < x < \pi/2$  \cr
                    \cos x &if $x \ge \pi/2$  \cr}$$
Where is $f$ continuous?
\correct Everywhere but $x=\pi/2$
\wrong Everywhere but $x=0$
\wrong Everywhere but $x=0$ and $x=\pi/2$
\wrong Everywhere
\wrong Nowhere


\note %10
\problem
If a ball is thrown upward with an initial velocity of $40\, ft/sec$ from
the top of a 30 foot building, its height after $t$ seconds is given by the
formula
$$h = -16t^2 + 40t + 30\,.$$
How many seconds after it is thrown does it stop moving upwards and start
moving downwards?
\correct 5/4
\wrong 3/5
\wrong 6/5
\wrong 3/2
\wrong 7/8


\note %11
\problem
Which of the following statements is true for a function
$f(x)=x^3$?
\correct
it has a critical point at $x=0$ but neither a local maximum
nor local minimum there
\wrong
it has no critical point 
\wrong
it has a critical point, local maximum
and local minimumat $x=0$
\wrong
it has a local maximum
and local minimum at $x=0$ but not a critical point there
\wrong
it has a critical point at $x=0$ and a local maximum at $x=0$
but not a  local minimum there



\note %12
\problem
Which statement is true?
\correct
If $f' (r)$ exists, then $\lim_{x\rightarrow r}f(x)=f(r)$
\wrong
$D(\sec x)=D(\tan ^2 x)$
\wrong
If $f' (x) > 0$ for all $x$ in $(a,b)$, then $f$ is decreasing
on $[a,b]$.
\wrong
${{d^2 y}\over dx^2}={\bigl( {{dy }\over{dx }} \bigr) }^2$
\wrong
If $f$ has an absolute minimum value at $c$, then $f'(c)=0$

\note %13
\problem
Suppose that $h(x)=f(x)g(x)$ and $F(x)=(f\circ g)(x)$ where
$f(1)=2, f'(1)=-1, f'(0)=3, g(1)=0$ and $g'(1)=1$.
Find $h'(1)$ and $F'(1)$.
\correct
$2$ and $3$
\wrong
$-1$ and $3$
\wrong
$-1$ and $0$
\wrong
$2$ and $0$
\wrong
$0$ and $-2$

\note %14
\problem
Suppose $f$ is a differentiable function such that $f(g(x))=x$
and $f'(x)=2+ f(x)$. What is $g''(x)$?
\correct
${-1\over {(x+2)^2}}$
\wrong 
${-1\over {(x+2)}}$
\wrong 
${1\over {(x+2)^2}}$
\wrong 
${1\over {(x+2)}}$
\wrong
$-x$

\note %15
\problem {\bf Partial Credit}
\hskip .5cm
Find the local, absolute maximum and absolute minimum values
of $f(x)=x- {\sqrt 2}\sin x$ on $[0, \pi ]$.
\vskip 2cm

\note %16
\problem {\bf Partial Credit}
\hskip .5cm
Determine whether the following statement is true or false and
state the reason of your determination.
\vskip .2cm
{\bf If $f'(c)=0$, then $f$ has a local maximum or minimum at $c$.}
\vskip 2cm

\appendix
That's all folks!

\end
