\def\whichexam{Final Exam}
\def\examdate{December 12, 2002}
\def\whichcourse{Math 125}
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\itemitem{$\bullet$} Be sure that you have all \numberpages\ pages of the test. 
\itemitem{$\bullet$} No calculators are to be used.
\itemitem{$\bullet$} The exam lasts for two hours.
\itemitem{$\bullet$} {\impno When told to begin, remove this answer sheet 
and keep it under the rest of your test.
When told to stop, hand in just this one page.}
\itemitem{$\bullet$}
The Honor Code is in effect for this examination, including keeping your answer
sheet under cover.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%
%%%%%%%     START OF TEST
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\Newp 1. 
Determine the following limit 
$$\lim_{x \rightarrow -1} \left (
\frac{x^2 +6x +5}{x^2+x} + \cos(x^2 \pi) \right ).$$
€5
(a)$-5$
(b)$-4$
(c)$2$
(d)$-3$
(e)$1$
\endp


\Newp 2. 
Calculate the following limit $$\lim_{x \rightarrow 3^-}
\frac{x^2\mid x-3 \mid}{x-3}.$$
€5
(a)$-9$
(b)$9$
(c)$-3$
(d)$3$
(e)$6$
\endp


\Newp 3. 
Find all points where the following function is discontinuous
$$f(x)= \cases{\frac{(x-1)(x+2)}{(x^2-1)x}&$x\neq 1$\cr\cr
\hskip 20pt\frac{3}{2}&$x = 1$\cr}\ .$$
€2
(a)$x=0,x=-1$
(b)$x=-1,x=1,x=0$
(c)$x=1,x=0$
(d)$x=-2,x=-1,x=1$
(e)$x=0,x=-2,x=1$
\endp

\Newp 4. 
Let $$ f(x)=\sin(x) \cos(2x) + \sin(2x) \cos(x).$$ Find
$f^\prime(\pi)=?$
€5
(a)$-3$
(b)$0$
(c)$2$
(d)$-1$
(e)$-2$
\endp

\Newp 5. 
Find the equation of the tangent line to the following curve at
$(1,2)$: $$f(x)=x^2-x+2.$$
€5
(a)$y=x+1$
(b)$y=x-1$
(c)$y=-x+3$
(d)$y=2x$
(e)$y=-x-1$
\endp

\Newp 6. 
Let $$f(x)=\frac{\sin x}{x}\ .$$ 
Find $f^{\prime\prime}(x) =\ ?$
€2
(a)$\frac{-x^2 \sin x - 2 x \cos x + 2 \sin x}{x^3}$
(b)$\frac{x^2 \sin x - 3 x \cos x + 2 \sin x}{x^3}$
(c)$\frac{-x^2 \sin x - 3 x \cos x + 3 \sin x}{x^3}$
(d)$\frac{-x^2 \sin x + 4 x \cos x + 5 \sin x}{x^3}$
(e)$\frac{x^2 \sin x + 4 x \cos x + 2 \sin x}{x^3}$
\endp

\Newp 7. 
If  $y = (x+1)^3 + \sin( 5x^3 + \sin x + \pi)$ 
what is $y^\prime(0)$?
€5
(a)$2$
(b)$3/2$
(c)$1/2$
(d)$4$
(e)$5/2$
\endp


\Newp 8. 
If  $\sin(x + y) = y^2 \cos x$, what is $y^{\prime}$ at the
point $(0,0)$?
€5
(a)$-1$
(b)$0$
(c)$-2$
(d)$4$
(e)$2$
\endp

\Newp 9. 
The mechanics at South Bend Automotive are reboring a
6-inch-deep cylinder to fit a new piston. 
The machine they are using increases the cylinder's radius 
one-thousandth of an inch every 3 minutes.
How rapidly is the cylinder volume increasing when its radius 
is 3.5 in.?
€3
(a)$\frac{14 \pi}{1000}$ $\mathrm{in.^3/min}$
(b)$\frac{7\pi}{3000}$ $\mathrm{in.^3/min}$
(c)$\frac{24.5 \pi}{1000}$ $\mathrm{in.^3/min}$
(d)$\frac{10.5 \pi}{3000}$ $\mathrm{in.^3/min}$
(e)$\frac{73.5 \pi}{1000}$ $\mathrm{in.^3/min}$
\endp


\Newp 10. 
Use differentials (or, equivalently, a linear approximation) to
estimate $f(3.2)$ if $f(3) = 72$ and $f^\prime(3) = 7$.
€5
(a)$73.4$
(b)$72.7$
(c)$70.6$
(d)$79$
(e)$65$
\endp

\Newp 11. 
Identify the $x$ values of the local maxima and minima of the
function $f(x) = (x+4)^5(x-3)^4$.
€1
(a)local minimum at $x=3$; local maximum at $x = -1/9$.
(b)local minima at $x=3$ and $x=-4$; local maximum at $x = -1/9$.
(c)local minimum at $x=3$; local maxima at $x = -4$ and $x = -1/9$.\hss
(d)local minimum at $x=-4$; local maximum at $x = 3$
(e)local minimum at $x=3$; local maximum at $x = -4$
\endp

\Newp 12. 
Consider the function $f(x) =\frac{x + 1}{x^2 + x + 1} $.
Which of the following statements is true? 
Note that $y=0$ is a horizontal asymptote and draw a rough sketch of the
graph after anaylyzing $f^\prime$.
€1
(a) $f$ has an absolute maximum at $x = 0$ and an absolute minimum at $x=-2$.
(b) $f$ has an absolute maximum at $x = 0$ and no absolute minimum.
(c) $f$ has no absolute maximum and an absolute minimum at $x=-2$.
(d) $f$ has no absolute maximum or minimum.
(e) $f$ has an absolute maximum at $x = 0$ and an absolute minimum
at $x=2$.
\endp

\Newp 13.  
The value $x = c$ given by the Mean Value Theorem applied to the
function $f(x)= x^3 + x - 1$ on the interval $[0,2]$ is
€5
(a) $x = \frac{2\sqrt{3}}{3}$
(b) $x = 1$
(c) $x = \sqrt{\frac{7}{6}\ }$
(d) $x = 2$
(e) $x = \frac{\sqrt{3}}{3}$
\endp


\Newp 14.  
If $y=2x^2-x^4$, which one of the following is true?
€1
(a)There are two local maxima and two inflection points.
(b)There are two local minima and two inflection points.
(c)There is just one inflection point.
(d)There is one inflection point, one local minima and one local maxima.
(e)The line $y=4x-4$ is a slant asymptote.
\endp

\Newp 15.  
Find $$\lim_{x\rightarrow -\infty}{\sqrt{4x^4+7x+2}\over
{\root 3 \of {27x^6+17}}}$$
€5
(a)$2\over 3$
(b)$-{2\over 3}$
(c)$\infty$
(d)$-\infty$
(e)$\frac{\sqrt 2}{\root 3 \of {17}}$
\endp

\Newp 16.  
A page of a book is to have a total area of 150 square inches, with 1 inch
margins at the top and sides, and a 2 inch margin at the bottom.  Find the 
dimensions in inches of the page which will have the largest print area.
€5
(a)$15 \times 10$
(b)$13 \times 11\smfrac{7}{13}$
(c)$\frac{30}{\sqrt{3\,}} \times 5\sqrt{3\,}$
(d)$3\sqrt{7\,} \times \frac{50}{\sqrt{7\,}}$
(e)$30 \times 5$
\endp

\Newp 17.  
Newton's method is to be used to find a root of the equation
$$x^3-x-1=0\ .$$
\hskip 60pt{If $x_1=1$, find $x_2$.}
€5
(a)$1.50$
(b)$3$
(c)$0.95$
(d)$1.35$
(e)$1.75$
\endp

\Newp 18.  
Express the limit below as a definite integral.
Calculate that integral.
$$ \lim_{n\rightarrow \infty} \sum_{i=1}^{n}
{\pi\over{4n}}\sec^2\Bigl({i\pi\over{4n}}\Bigr)$$
€5
(a)$1$
(b)$-\pi$
(c)$-1$
(d)$0$
(e)$\pi$
\endp

\Newp 19.  
Find $f^\prime (2)$ if $f(x)=\displaystyle\int_{2}^{x} \sqrt{1+t^3}\ dt $.
€5
(a)$3$
(b)$0$
(c)$4$
(d)$1$
(e)$9$
\endp


\Newp 20.  
A car racing on a straight road crosses the starting line with a
velocity of $88$ ft/sec.  
From this point on it accelerates at $\frac{60}{\sqrt{t+1}}$
ft/sec/sec.  How fast in ft/sec will the car be going in $3$ seconds?
€5
(a)$208$
(b)$292$
(c)$152$
(d)$244$
(e)$328$
\endp

\Newp 21. 
 Find $$\int_3^6{\pi\over {x^2}}\cos({\pi\over x})\ dx$$
€5
(a)$\frac{\sqrt 3 -1}{2}$
(b)$\frac{\sqrt 3 +1}{2}$
(c)$\frac{1}{2}$
(d)$\frac{1}{\sqrt 2}$
(e)$0$
\endp

\Newp 22.
A solid in space intersects the $x y$ plane in the region between $x=0$ and 
$x=\pi$; bounded above by the graph of $y = \sin x$ and 
below by the graph of $y = -2\sin x$.
Each cross section perpendicular to the $x${--}axis is a rectangle whose short 
side is the line in the $x y$ plane and whose long side is $2$ times the length 
of the short side.
Which integral below gives the volume?
€3
(a)$18\Int_0^\pi \sin^2 x\ dx$
(b)$-\Int_0^\pi \sin^2 x\ dx$
(c)$6\Int_0^\pi \sin x\ dx$
(d)$6\Int_0^\pi \sin^2 x\ dx$
(e)$18\Int_0^\pi \sin x\ dx$
\endp

\Newp 23.
The plane region bounded below by the graph of $y = \sqrt x$ 
and above by the graph of $y = \root 3 \of{x}$ is rotated
about the line $x=5$.
Which integral below gives the volume?
€2
(a)$2\pi\Int_0^1\bigl(5-x\bigr)\cdot\bigl(\root 3\of{x}-\sqrt{x}\ \bigr)\> dx$
(b)$\pi\Int_0^1\bigl(5-\sqrt x\ \bigr)^2 - \bigl( 5 - \root 3\of{x}\ \bigr)^2\ dx$
(c)$2\pi\Int_0^1\bigl(x - 5 \bigr)\cdot\bigl(\root 3\of{x}-\sqrt{x}\ \bigr)\> dx$
(d)$2\pi\Int_0^1\bigl(5-x\bigr)\cdot\bigl(\sqrt{x} - \root 3\of{x}\ \bigr)\> dx$
(e)$\pi\Int_0^1 \bigl( 5 - \root 3\of{x}\ \bigr)^2 - \bigl(5-\sqrt x\ \bigr)^2 \ dx$
\endp

\Newp 24.
Consider the plane region
bounded by the graphs of $y = \sqrt x$, 
$y=0$, $x=0$ and $x=4$.
Rotate this region about the line $y=-1$ and calcualte the volume.
€5
(a)$\frac{56\pi}{3}$
(b)$\frac{112\pi}{3}$
(c)$\frac{28\pi}{3}$
(d)$28\pi$
(e)$\frac{61\pi}{3}$
\endp

\Newp 25.
The function $f(x)=\root 3 \of{x}$ is continuous on the interval $[1,8]$.
Which number below is its average value on this interval?
€5
(a)$\frac{45}{28}$
(b)$\frac{45}{8}$
(c)$\frac{5}{2}$
(d)$\frac{5}{3}$
(e)$\frac{7}{5}$
\endp
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