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

\preface
\def\a{{\bf a}}
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\def\0{{\bf 0}}
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\def\problemskip{\vskip0.5in}

\title
\noindent
{\bf Math 119}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final exam}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl May 8, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}


\def\answerskip{\vskip0.25in}


\note %1
\problem
Find $\ \lim_{h\to0}\>{{\cos ({{\pi}\over 2} + h)}\over h}$
\correct $-1$
\wrong $0$
\wrong $\pi$
\wrong $-\pi$
\wrong $-2$


\note %2
\problem
Find all the horizontal and vertical asymptotes of
$y={{x+1}\over{{\sqrt {x-1}}}}$.
\correct
No horizontal asymptote; $x=1$
\wrong
$y=1$; $x=1$
\wrong
$y=-1$; $x=-1$
\wrong
$y=0$; $x=-1$
\wrong
$y=-1$; no vertical asymptote


\note %3
\problem
A Norman window is constructed from a rectangular sheet of glass 
surmounted by a semicircular sheet of glass. The light that enters
through a window is propotional to the area of the window. What is the
dimension of the semicircle's {\bf diameter} of the Norman window 
having a perimeter of 30 feet which 
admits the most light?
\correct
${60\over {4+\pi}}$
\wrong
${30\over {4+\pi}}$
\wrong
${15\over {2+\pi}}$
\wrong
${20}$
\wrong
$15$


\note %4
\problem
Let $F(x)=\int_0^x {\sqrt {{u-1}\over{u+1}}}\, du$. Calculate
$F'(1)$.
\correct $0$
\wrong $1$
\wrong $-1$
\wrong $1\over 2$
\wrong $-{1\over 2}$



\note %5
\problem
Evaluate the integral $\int_{-1}^1 (1-|x|)\, dx$
\correct $1$
\wrong $-1$
\wrong ${1\over 2}$
\wrong $-{1\over 2}$
\wrong $0$


\note %6
\problem
Where is the graph of the function
$$f(x) \ = \ x^3 - 6x^2 + 12x - 5$$
concave up, and where is it concave down?
\correct concave up on $(2,\infty)$, concave down on $(-\infty,2)$
\wrong concave up on $(-\infty,0)$, concave down on $(0,\infty)$
\wrong concave up on $(-\infty,0)$ and $(2,\infty)$, concave down on $(0,2)$
\wrong concave up on $(0,2)$, concave down on $(-\infty,0)$ and $(2,\infty)$
\wrong concave up on $(-\infty,2)$, concave down on $(2,\infty)$


\note %7
\problem
 $$f(x) = \cases{1 &if $x \le 0$ \cr
                  \cos x &if $0 < x < {\pi/2}$  \cr
                 -1 &if $x \ge {\pi/2}$}$$
Where is $f$ continuous?
\correct Everywhere but at $x = \pi/2$
\wrong Everywhere but at $x = 0$
\wrong Everywhere but at $x=0$ and $x=\pi/2$
\wrong Everywhere
\wrong Nowhere


\note %8
\problem
If $f(x) = \sqrt{2x + 3}$, which of the following limits conforms to
the definition of $f'(1)$?
\correct $\lim_{h\to0}\,{{\sqrt{5+2h} - \sqrt5} \over h}$
\wrong $\lim_{h\to0}\,{{\sqrt{2h+3} - \sqrt5}\over h}$
\wrong $\lim_{h\to0}\,{{\sqrt{5+h} - \sqrt5}\over h}$
\wrong $\lim_{h\to0}\,{{\sqrt3 + \sqrt{h} - \sqrt5}\over h}$
\wrong $\lim_{h\to0}\,{{\sqrt{2h} + \sqrt3 - \sqrt5}\over h}$



\note %9
\problem
Suppose $f$ and $g$ are functions satisfying the following conditions:
\bigskip
\hskip 1in $f(0) = 1$  \hfil  $f(1) = 0$  \hfil  $g(0) = 3$  \hfil  $g(1) = -2$ 
             \hskip 1in
\bigskip
\hskip 1in $f'(0) = 2/3$  \hfil  $f'(1) = - 1/2$  \hfil  $g'(0) = -5$ 
                    \hfil  $g'(1) = 1/3$ \hfil
\bigskip
If $h(x) = g\big(f(x)\big)$, what is $h'(1)$?
\correct $5/2$
\wrong $-{1/6}$
\wrong $2/9$
\wrong $-{10/3}$
\wrong 0


\note %10
\problem
Find the slope of the tangent to the graph of the equation
$$x^2 + y^2 +xy =1$$
at $(-1, 0)$.
\correct -2
\wrong 1/2
\wrong 1
\wrong 2
\wrong does not exist


\note %11
\problem
Suppose a tank holds 5000 gallons of water, and it takes 40 minutes
to drain a full tank. According to Toricelli's law, the volume $V$ of water
remaining in the tank at the end of $t$ minutes is given by the formula
$$V \ =\ 5000\Big(\,1 - {t\over{40}}\,\Big)^2 \hbox{\quad for\quad }
     0 \le t \le 40$$
How fast is the tank {\bf draining} at the end of 20 minutes? (All answers are in
gallons per minute.)
\correct 125
\wrong 5000
\wrong 1250
\wrong 250
\wrong 500



\note %12
\problem
Find the absolute maximum and minimum values of the function
$$f(x) = {x\over4} + {9\over x}$$
on the interval $[1,8]$
\correct maximum value $= 37/4$,  minimum value $= 3$
\wrong maximum value $= 37/4$,  minimum value $= 25/8$
\wrong maximum value $= 12$, minimum value $= 25/8$
\wrong maximum value $= 39/4$, minimum value $= 13/4$
\wrong maximum value $= 12$, minimum value$= 3$


\note %13
\problem
If $\ f(0) = 4 \hbox{\ and\ } f'(0) = 2\,,$ find the slope of the
graph
$$y = \sqrt{x + f(x)}$$
at the point $(0,2)$
\correct $3/4$
\wrong $1/4$
\wrong $\sqrt3/2$
\wrong $1/\sqrt2$
\wrong $\sqrt2$


\note %14
\problem
Letting $ f(x) ={{x-1}\over {x+1}}$, find ${d\over{dx}} \int_1^{3x} f$
\correct ${{3(3x-1)}\over{3x+1}}$
\wrong ${{3x-1}\over{3x+1}}$
\wrong ${{x-1}\over{x+1}}-1$
\wrong $-{{x-1}\over{x+1}}$
\wrong ${{3(3x-1)}\over{3x+1}}-1$

\note %15
\problem
For what $x$ is the line tangent to the graph
$$y = x^2 + x + 1$$
parallel to the line through the points $(0,3)$ and $(-1,1)$.
\correct $1/2$
\wrong $-3/2$
\wrong $-2$
\wrong $3$
\wrong $1$

\note %16
\problem The volume of a sphere of radius $r$ is given by the formula 
$$V = {4\over3}\,\pi r^3$$
When helium is pumped into a spherical balloon, both the
radius and the volume change with respect to time. Suppose that at a certain
moment the volume is increasing at the rate of $8\pi$ cu. ft. per minute, and
the radius is increasing at the rate of 1/2 ft per minute. What is the volume of
the balloon at that moment?
\correct ${{32\pi}\over3}$
\wrong ${{\pi}\over4}$
\wrong $16\pi$
\wrong ${\pi\over{16}}$
\wrong ${{4\pi}\over{3\sqrt2}}$


\note %17
\problem
Find $\ \lim_{x\to4}\,{{x-4}\over{\sqrt{x}-2}}\,,\ $ if it exists
\correct 4
\wrong 2
\wrong 1
\wrong 0
\wrong does not exist


\note %18
Suppose $f(x)$ is defined and has a derivative for all $x$ in some 
interval $a<x<b$, and for some point $c$ in the interval $f'(c)=0$
and $f''(c)=0$. Which of the following statements is true? (Only one
of them is true.)
\correct
The function can not have either a local maximum or minimum at $x=c$
\wrong The function must have a local minimum at $x=c$
\wrong The function must have a local maximum at $x=c$
\wrong The function must have an inflection point at $x=c$
\wrong No conclusion can be drawn about whether there is an inflection
point or local extremum without further information



\note %19
\problem A box with a square base and a top is to be built for a cost of \$60.
The material for the base costs \$3 per sq. ft.; the material for the top and
sides costs \$2 per sq. ft.
What is the maximum volume of such a box? (Answers are in cubic feet.)
\correct 10
\wrong 24
\wrong 16
\wrong 20
\wrong there is no maximum

\note %20
\problem
How many real roots does the cubic equation 
$x^3+1$ have?
\correct 1
\wrong none
\wrong 2
\wrong 3
\wrong 4


\note %21
\problem Find all the critical points of the function
$$f(x) = \sin x + \cos x$$
in the interval $(0,2\pi)$.
\correct $\pi/4$ and $5\pi/4$
\wrong $\pi$
\wrong $3\pi/4$ and $7\pi/4$
\wrong $\pi/2$, $\pi$ and $3\pi/2$
\wrong $\pi/2$ and $3\pi/2$

\note %22
\problem
Which statement is false for the curve $f(x)={x\over {1+x}}$?
\correct its graph is symmetric around the origin
\wrong its critical number is $-1$
\wrong it does not have an inflection point
\wrong it is continuous everywhere except at $x=-1$
\wrong it is concave up on $(-1, \infty )$

\note %23
\problem
Suppose $f$ is a differentiable function such that $f(g(x))=x$ and
$f'(x)=1+[f(x)]^2$. What is $g'(x)$?
\correct
${1\over {1+x^2}}$
\wrong
${1\over {1+x}}$
\wrong
$-{1\over {1+x^2}}$
\wrong
${1\over x}$
\wrong
$-{1\over x}$


\note %24
\problem
Find $f(x)$ if $f'(x)=\sec ^2 x$ and $f(0)=0$.
\correct
$\tan x$
\wrong 
$\tan x +1$
\wrong
$\tan x -1$
\wrong
$\sec x$
\wrong 
$\sec x -1$


\note %25
\problem
Which of the following is equivalent to
$$\lim_{n\rightarrow \infty} \sum_{i=1}^{n} {3\over n}
\sqrt{1+\bigl( {{3i}\over n}\bigr) ^2}$$?
\correct
$\int_0^3 \sqrt{1+ x^2}\, dx$
\wrong
$\int_0^1 \sqrt{1+ x^2}\, dx$
\wrong
$\int_0^3 \sqrt{x^2}\, dx$
\wrong
$\int_0^3 \sqrt{1+ 3x^2}\, dx$
\wrong
$\int_0^1 \sqrt{1+ 3x^2}\, dx$


\note %26
\problem
Which of the following statements is false?
\correct 
If $f$ and $g$ are integrable on $[a,b]$, then
$\int_a^b [f(x)g(x)]\, dx=\Bigl( \int_a^b f(x)\, dx\Bigr)
                          \Bigl( \int_a^b g(x)\, dx\Bigr) $
\wrong If $f$ is  differentiable at $a$, then $f$ is 
continuous at $a$
\wrong 
All continuous fuctions have antiderivatives
\wrong If $p$ is a polynomial and $b$ is a real number, then 
$\lim _{x\rightarrow b} p(x)=p(b)$
\wrong If $f'(a)$ exists, then 
$\lim _{x\rightarrow a} f(x)=f(a)$

\note %27
\problem
 Find $\int \cos ^4 x\, \sin x\, dx$
\correct $-{1\over 5} \cos ^5 x + C$
\wrong ${1\over 5} \cos ^5 x + C$
\wrong $-{1\over 4} \cos ^4 x + C$
\wrong ${1\over 5} \sin ^5 x + C$
\wrong $-{1\over 4} \sin ^5 x + C$


\note %28
\problem
Which of the following statements is true?
\correct
If $f$ and $g$ are integrable on $[a,b]$, then
$\int_a^b [f(x)+g(x)]\, dx= \int_a^b f(x)\, dx +\int_a^b g(x)\, dx$
\wrong
$D(\sec x)=D(\tan ^2 x)$
\wrong
If $f$ and $g$ are differentiable and $f(x)\ge g(x)$ for $a\le x\le b$,
then $f'(x)\ge g'(x)$ for $a<x<b$.
\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 %29
\problem
Which of the following  is true?
\correct
if $f(-x)=-f(x)$, then $f$ is symmetric around the origin
\wrong
if $f$ has an absolute minimum value at $c$, then $f'(c)=0$
\wrong
if $f''(c)=0$, then $(c,f(c))$ is an inflection point of the
curve $y=f(x)$
\wrong
if $f'(x)=g'(x)$ for $0<x<1$, then $f(x)=g(x)$ for $0<x<1$.
\wrong
$\sum_{i=1}^{n} a_i b_i 
=\big( \sum_{i=1}^{n} a_i\big) \big(\sum_{i=1}^{n} b_i\big) $

\note %30
\problem There is a unique number that falls between the lower and upper sums
of the function
$$f(x) = x^2 -2\, ,\quad 1 \le x \le 4$$
for every partition of the interval $[1, 4]$. What is that number?
\correct $15$
\wrong $21$
\wrong $-15$
\wrong $1$
\wrong $0$

\note %31
\problem
Which of the following sums is the Riemann sum for the function
$$f(x) = {1\over{1+x}}\,, \quad 0 \le x \le 2$$
if the interval $[0,2]$ is partitioned into four equal segments and $c_i$ is
the midpoint of the~$i^{\rm th}$~interval?
\correct ${2\over5} + {2\over7} + {2\over9} + {2\over{11}}$
\wrong ${1\over5} + {1\over6} + {1\over7} + {1\over8}$
\wrong ${1\over4} + {1\over5} + {1\over6} + {1\over7}$
\wrong ${1\over2} + {1\over4} + {1\over6} + {1\over8}$
\wrong ${2\over5} + {2\over6} + {2\over7} + {2\over8}$

\note %32
\problem
State the Fundamental Theorem of Calculus (two parts).
\vskip 3cm


\appendix
That's all folks!

\end