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\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl September 29, 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}

\def\problemskip{\vskip1in}


\note %1
\problem
Find $\displaystyle {d \over {dx}} \left ( {{x^5 + 1} \over {x^5 - 1}} \right )$
\correct
$\displaystyle {{-10 x^4} \over {(x^5 - 1)^2}}$
\wrong
$\displaystyle {{10 x^4} \over {(x^5 - 1)^2}}$
\wrong
$1$
\wrong
$\displaystyle {{10 x^9} \over {(x^5 - 1)^2}}$
\wrong
$0$

\note %2
\problem
Let 
$
f(x) = 6 \sqrt{x} - x + 1.
$
Find $f'(9)$.
\correct $f'(9) = 0$
\wrong $f'(9)$ is undefined
\wrong $f'(9) = 1$
\wrong $f'(9) = 3$
\wrong $f'(9) = 10$

\note %3
\problem
Consider the curve $y = x^3 -3x$.  Find the $x$-coordinate(s) of the point(s)
where the tangent line to this curve is parallel to the $x$-axis.
\correct $x= 1$ and $x=-1$
\wrong  only $x=1$
\wrong the tangent line is never parallel to the $x$-axis
\wrong $x=3$ and $x=-3$
\wrong $x=0$, $x=\sqrt{3}$ and $x=-\sqrt{3}$

\note %4
\problem  
Let $f(x) = \sqrt{-x}$.  Find the natural domain of $f(x)$.
\correct $(- \infty , 0]$
\wrong $(- \infty ,0)$
\wrong $[0,\infty)$
\wrong $(0,\infty)$
\wrong $f(x)$ is not defined for any value of $x$

\note %5
\problem
Find $\displaystyle \lim_{x \rightarrow 1} {{x-1} \over {\sqrt{4x-3} -1}}$
\correct $\displaystyle {1 \over 2}$
\wrong undefined
\wrong $2$
\wrong $\displaystyle {1 \over 4}$
\wrong $0$

\note %6
\problem
Which of the following is equal to $\displaystyle \sin (x - {\pi \over 6})$ for all
$x$?
\correct $\displaystyle {{\sqrt{3}} \over 2} \sin x - {1 \over 2} \cos x$
\wrong $\displaystyle {{\sqrt{3}} \over 2} \sin x + {1 \over 2} \cos x$
\wrong $\displaystyle {1 \over 2}  \sin x - {{\sqrt{3}} \over 2} \cos x$
\wrong $\displaystyle -{{\sqrt{3}} \over 2} \sin x + {1 \over 2} \cos x$
\wrong $\displaystyle {1 \over 2} \sin x + {{\sqrt{3}} \over 2} \cos x$

\note %7
\problem
Find $\displaystyle \lim_{x \rightarrow 1} {{x^2 - 1} \over {x-1}}$
\correct 2
\wrong 0
\wrong $-1$
\wrong 1
\wrong $-2$

\note %8
\problem
Let $f(x)$ be a function with $f(1) = 3$ and $f'(1) = 4$.  Let $y = x^2 f(x)$. 
Find  $\displaystyle  \left. {{dy} \over {dx}} \right |_{x=1}$
\correct 10
\wrong 7
\wrong 12
\wrong 8
\wrong 6

\note %9
\problem Which of the following limits
represents the slope of the graph of $\displaystyle y = {{x-1} \over {x+1}}$ at
$x=1$?
\correct $\displaystyle \lim_{h \rightarrow 0} {1 \over h} \left [ {h \over {2+h}}
- 0 \right ]$
\wrong $\displaystyle \lim_{h \rightarrow 0} 0$
\wrong $\displaystyle \lim_{h \rightarrow 0} {1 \over h} \left [ {{2+h} \over h}
- 0 \right ]$
\wrong $\displaystyle \lim_{h \rightarrow 0} {1 \over h} \left [ {{h-1} \over
{h+1}} - 0 \right ]$
\wrong $\displaystyle \lim_{h \rightarrow 0} {1 \over h} \left [ {{1-h} \over h}
- 0 \right ] $

\note %10
\problem
Suppose $f(x)$ and $g(x)$ are two functions with
$$
\lim_{x \rightarrow 0} f(x) = 2 \quad \hbox{and} \quad \lim_{x \rightarrow 0}
g(x) = 3.
$$
Find $\displaystyle \lim_{x \rightarrow 0} {{f(x) + 4} \over {f(x)g(x)}}$
\correct 1
\wrong $\displaystyle {1 \over 6}$
\wrong $\displaystyle {2 \over 3}$
\wrong $\displaystyle {3 \over 2}$
\wrong the limit does not necessarily exist

\note %11
\problem
Find $\displaystyle \lim_{\theta \rightarrow 0} {{ \theta} \over {
\sin 3 \theta}}$, if it exists.
\correct $\displaystyle {1 \over 3}$
\wrong 3
\wrong 1
\wrong 0
\wrong does not exist

\note %12
\problem
Find $\displaystyle \lim_{\theta \rightarrow \pi} { {\theta} \over {\sin 3 \theta}
}$, if it exists.
\correct does not exist
\wrong ${1 \over 3}$
\wrong $ \pi$
\wrong $- \pi$
\wrong 0

\note %13
\problem
If $\tan \theta = {3 \over 4}$ and $0 \leq \theta \leq {\pi \over 2}$, find $\cos
2 \theta$.
\correct $\displaystyle {7 \over {25}}$
\wrong $\displaystyle {3 \over 5}$
\wrong $\displaystyle {16 \over 25}$
\wrong $\displaystyle {12 \over 25}$
\wrong $\displaystyle {{3 \pi} \over 8}$

\note %14
\problem
Find the center of the circle determined by the equation
$$
x^2 + y^2 + 6x -4y - 1 = 0
$$
\correct $(-3 , 2)$
\wrong $(3,2)$
\wrong $(3,-2)$
\wrong $(-3, -2)$
\wrong $(0,0)$

\note %15
\problem
Let
$$
f(x) = \cases{x + 1, & if $x < 1$; \cr
              3,       & if $x = 1$; \cr
              3x-1,     & if $x > 1$. \cr}
$$
Which of the following statements is/are true?
\vskip .2cm
I. $f(x)$ is continuous at $x=1$.
\vskip .2cm
II. $\displaystyle \lim_{h \rightarrow 0} f(1+h)$ exists
\vskip .2cm
III. $\displaystyle \lim_{h \rightarrow 0} {{f(1+h) - f(1)} \over h}$ exists.
\correct II.\ but not I.\ or III.
\wrong all three are true
\wrong none of the three is true
\wrong I.\ and II.\ but not III.
\wrong I.\ but not II.\ or III.

\note %16
\problem
An object is thrown
upward from the top of a platform.  Its height (in feet) at the end of $t$ seconds
is given by the function
$$
f(t) = -16t^2 + 20t + 10.
$$
Find the average velocity of the object for the time interval starting at the
instant that it is thrown and ending at time $t=2$.
\correct $-12$ feet/second
\wrong $-7$ feet/second
\wrong $-14$ feet/second
\wrong $-24$ feet/second
\wrong $-44$ feet/second

\note %17
\problem
An object is thrown
upward from the top of a platform.  Its height (in feet) at the end of $t$ seconds
is given by the function
$$
f(t) = -16t^2 + 20t + 10.
$$
Find the instantaneous velocity of the object at time $t=2$.
\correct $-44$ feet/second
\wrong $-32$ feet/second
\wrong $-10$ feet/second
\wrong $-14$ feet/second
\wrong $-64$ feet/second

\note %18
\problem
Let $y = x^2 + 1$.  Find the equation of the tangent line to this curve at the
point $(2,5)$.
\correct $y = 4x - 3$
\wrong $y = 2x + 1$
\wrong $y = 5x - 10$
\wrong $y = 5x - 5$
\wrong $y = 4x + 5$

\note %19
\problem
Let $f(x) = x^{3 \over 2} (x-2)$.  Find $f'(x)$.
\correct $\displaystyle x^{3 \over 2} + {3 \over 2} x^{1 \over 2} (x-2)$
\wrong $\displaystyle x^{3 \over 2} - {3 \over 2} x^{1 \over 2} (x-2)$
\wrong $\displaystyle -2 x^{3 \over 2} + {3 \over 2} x^{1 \over 2} (x-2)$
\wrong $\displaystyle -2 x^{3 \over 2} - {3 \over 2} x^{1 \over 2} (x-2)$
\wrong $\displaystyle {3 \over 2} x^{1 \over 2}$


\appendix
%That's all folks!

\end



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