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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 26, 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 $L$ be the line through the points $(2,3)$ and $(4,6)$.  Find the
equation of the line that is perpendicular to $L$ and whose $y$-intercept is 7.

\correct $2x+3y=21$

\wrong $-2x+3y=21$

\wrong $2x-3y=21$

\wrong $3x-2y=14$

\wrong $2x+3y=13$

\note %2
\problem Find $\sin ({{-7\pi} \over 6})$.

\correct ${1 \over 2}$

\wrong $-{1 \over 2}$

\wrong ${{\sqrt{3}} \over 2}$

\wrong $-{{\sqrt{3}} \over 2}$

\wrong $-{{\sqrt{2}} \over 2}$

\note %3
\problem Find the domain of the function 
$$
f(x) = \sqrt{{x \over {x-1}}} 
$$
(Recall that $A \cup B$ means all $x$ which are either in $A$ or in $B$ or both.)

\correct $(-\infty,0] \cup (1,+\infty)$ 
\wrong $(-\infty,0) \cup (1,+\infty)$
\wrong $(0,1)$
\wrong $[0,1)$
\wrong all $x \neq 1$

\note %4
\problem The following equation is that of a circle.  Find its center and radius.
$$
x^2 + y^2 -6x+2y+6=0
$$
\correct center $(3,-1)$, radius 2
\wrong center $(-3,1)$, radius 2
\wrong center $(-3,1)$, radius 4
\wrong center $(3,-1)$, radius 4
\wrong center $(3,1)$, radius 2

\note %5 
\problem If a ball is thrown into the air with a velocity of 64 ft/sec, its height
in feet after $t$ seconds is given by $y = 64t-16t^2$.  Find the average velocity
of the ball for the first second of flight (i.e.\ from $t=0$ to $t=1$).

\correct 48 ft/sec
\wrong 32 ft/sec
\wrong 64 ft/sec
\wrong 16 ft/sec
\wrong 0 ft/sec

\note %6
\problem Let
$$
f(x) = \cases{x^2 +1 &if $x <1$; \cr
              6 &if $x=1$;\cr
              3x-1 &if $x>1$.}
$$
Find $\displaystyle \lim_{x\to 1} f(x)$, if it exists.

\correct 2
\wrong does not exist
\wrong 5
\wrong 6
\wrong 1

\note %7
\problem  Find $\displaystyle \lim_{x \to 1} {{x^2 -1} \over {x-1}}$, if it
exists.

\correct 2
\wrong 0
\wrong 1 
\wrong does not exist
\wrong $-2$

\note %8
\problem Find $\displaystyle \lim_{x \to 0^-} {{|x|} \over {x}}$, if it exists.

\correct $-1$
\wrong 1
\wrong 0
\wrong does not exist
\wrong $\pi$

\note %9
\problem Let $f(x)$ be a function.  Consider the following four limits:  

\bigskip

\centerline{I. $\displaystyle \lim_{h\to 0} {{f(a+h)-f(a)} \over h}$ \quad \quad
\quad \quad \quad II. $\displaystyle \lim_{x\to a} {{f(x)-f(a)} \over {x-a}}$}

\medskip

\centerline{
III. $f'(a)$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
IV. $\displaystyle \lim_{h\to 0} {{f(h)-f(a)} \over {a}}$
}

\bigskip

\noindent Which of
these limits represent(s) the slope of the tangent line to the graph
of
$y = f(x)$ at the point $(a,f(a))$?

\correct only I., II.\ and III.
\wrong they all do
\wrong only I.\ and III.
\wrong only III.
\wrong only I.\ and II.

\note %10
\problem
 Recall our notation that ``$|AB|$'' is the length of the
straight line segment joining $A$ to $B$, and ``arc $AB$'' is the length of the arc
joining
$A$ to $B$. In the following diagram, the radius of the circle is 1 (i.e.\ $|OA|
= |OB| = 1$) and the line segment $BD$ is tangent to the circle. 

\vskip 2.3in
\hskip 1.2in \triang

\bigskip
\noindent Which of the following is equal to $\tan \theta$?

\correct $|BD|$
\wrong arc $AB$
\wrong $|AB|$
\wrong $|AC|$
\wrong $|OC|$

\note %11
\problem Find $\displaystyle \lim_{x \to 0} {{\sin 3x} \over {\sin 4x}}$.

\correct $\displaystyle {3 \over 4}$
\wrong $\displaystyle {4 \over 3}$
\wrong  $\displaystyle {1}$
\wrong  does not exist
\wrong 0

\note %12
\problem Find $\displaystyle \lim_{x \to 0} {{x^2} \over {\sin 5x}}$.  (Hint:
$x^2 = x \cdot x$.)

\correct 0
\wrong $\displaystyle {1 \over 5}$
\wrong $\displaystyle {1}$
\wrong does not exist
\wrong $\displaystyle {2 \over 5}$

\note %13 
\problem 
Find the derivative of $\displaystyle f(x) = {{x^2 +1} \over {x^2 -1}}$

\correct $\displaystyle {{-4x} \over {(x^2 -1)^2}}$
\wrong $\displaystyle {{-4x^3} \over {(x^2 -1)^2}}$
\wrong $\displaystyle {{4x^3 -4x} \over {(x^2 -1)^2}}$
\wrong $\displaystyle {{4x} \over {(x^2 -1)^2}}$
\wrong $\displaystyle {{4x^3} \over {(x^2 -1)^2}}$

\note %14
\problem 
Find the derivative of $\displaystyle f(x) = {1 \over {\sqrt{x}}}$

\correct $\displaystyle{{-1} \over {2 \sqrt{x^3}}}$
\wrong $\displaystyle{{1} \over {2 \sqrt{x^3}}}$
\wrong $\displaystyle{{-2} \over { \sqrt{x^3}}}$
\wrong $\displaystyle{{2} \over { \sqrt{x^3}}}$
\wrong $\displaystyle{{2} \over { \sqrt{x}}}$

\note %15
\problem 
Consider the following equations and inequalities:

\centerline{
I. $\displaystyle {{\sin \theta} \over \theta} = 1$ \quad \quad \quad \quad
II. $\displaystyle \sin^2 \theta + \cos^2 \theta = 1$ \quad \quad \quad \quad
III. $-1 \leq \tan \theta \leq 1$ 
}

\noindent Which of them is/are true for {\it all} values of $\theta$?


\correct Only II.
\wrong I., II.\ and III.
\wrong II.\ and III.
\wrong I.\ and II.
\wrong I.\ and III.

\note %16
\problem Let
$$
f(x) = \cases{3 &if $x \leq -1$; \cr
              x &if $-1 < x < 1$;\cr
              {1 \over {\sqrt{x}}} &if $x\geq1$.
}
$$
Find all values of $x$ at which $f(x)$ is not continuous.

\correct $x=-1$
\wrong $x=-1$ and $x=1$
\wrong $x=-1$ and $x=0$
\wrong $x=-1$, $x=0$ and $x=1$
\wrong $x=1$

\note %17
\problem The following limit represents the derivative of some function $f(x)$ at
some number $a$:
$$
\lim_{h\to 0} {{\sqrt{4+h} -2} \over h}
$$
Find $f$ and $a$.

\correct $f(x) = \sqrt{x}$, $a = 4$
\wrong $f(x) = \sqrt{4+x}$, $a=4$
\wrong $f(x) = \sqrt{x}$, $a=2$
\wrong $f(x) = \sqrt{4+x}$, $a=2$
\wrong $f(x) = \sqrt{x}$, $a=0$

\note %18
\problem What is the distance between the points $(-1,-2)$ and $(3,4)$?

\correct $\sqrt{52}$
\wrong $\displaystyle {6 \over 4}$
\wrong $\displaystyle \sqrt{8}$
\wrong $52$
\wrong $\sqrt{10}$

\note %19
\problem 
Find the equation of the tangent line to the curve $y = x^3 -5$ at the point
$(2,3)$.

\correct $y-3 = 12(x-2)$
\wrong $y-3 = 11(x-2)$
\wrong $y-2 = 12(x-3)$
\wrong $y-2 = 11(x-3)$
\wrong $y-3 = 27(x-2)$

\appendix
%That's all folks!

 \end{document}
