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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%
%%%%%%%     START OF TEST
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\Newp 1.
If $f(x) = (x^2+3x)(6x^5 - 2 x^{15})$ compute $f^\prime(1)$.
�5
(a)$20$
(b)$30$
(c)$40$
(d)$36$
(e)$10$
\endp

\Newp 2.
Compute the left handed limit $\Lim_{u \to 1^{-}}\ \frac{u^2+1}{u^2-1}$
�4
(a)$-\infty$
(b)$\infty$
(c)$1$
(d)$0$
(e)Does not exist and is not $\infty$ or $-\infty$.\hss
\endp


\Newp 3.
Compute the right handed limit $\Lim_{y\to -\smfrac{\pi}{2}^{+}}\ \sec y$.
�4
(a)$\infty$
(b)$-\infty$
(c)$1$
(d)$0$
(e)Does not exist and is not $\infty$ or $-\infty$.\hss
\endp

\Newp 4.
If $f(x) = \frac{\sqrt{x} + 1}{\root 3\of{x} }$, then $f^\prime(x) =\ ?$
\par\noindent{\bf Hint:} Write $f$ as a sum of two powers of $x$.
�3
(a)$\smfrac{1}{6} x^{-\smfrac{5}{6}} - \smfrac{1}{3} x^{-\smfrac{4}{3}}$
(b)$\smfrac{1}{6} x^{-\smfrac{5}{6}} + \smfrac{1}{3} x^{\smfrac{4}{3}}$
(c)$-\smfrac{1}{6} x^{-\smfrac{5}{6}} + 3 x^{\smfrac{4}{3}}$
(d)$\smfrac{7}{6} x^{\smfrac{7}{6}} + \smfrac{3}{2} x^{\smfrac{4}{3}}$
(e)$6 x^{-\smfrac{5}{6}} - 2 x^{-\smfrac{3}{2}}$
\endp

\Newp5.
If $f(u) = \sin^2(u^2)$, compute $f^\prime(u)$
�3
(a) $4u\bigl(\sin(u^2)\bigr)\bigl(\cos(u^2)\bigr)$
(b) $2u\bigl(\sin^3(u^2)\bigr)$
(c) $4u\bigl(\cos^2(u^2)\bigr)$
(d) $\smfrac{2}{3}\bigl(\sin^3(u^2)\bigr)$
(e) $2u \sin^2(u^2)$
\endp

\Newp 6. 
If $y = x \sin y$, find $\frac{d y}{d x}$
�2
(a) $\frac{\sin y}{1 - x\cos y}$
(b) $\frac{\cos y}{1 - x y}$
(c) $\sin y$
(d) $\sin y + x \cos y$
(e) $x \cos y$
\endp

\Newp 7. 
If we define $f(x) = \Int^3_{x^2} \frac{1}{1+ w^2}\ dw$, find $f^\prime(x)$.
�5
(a) $\frac{-2x}{1+x^4}$
(b) $\frac{-1}{1+x^4}$
(c) $\frac{1}{10} - \frac{1}{1+x^4}$
(d) $\frac{2x}{1+x^2}$
(e) $\frac{-2x}{(1+x^2)^2}$
\endp

\Newp 8. 
Consider the function $f(x) = x^4 + x^3$ defined on the real line.
Which of the following is true.
�1
(a) $f$ has a local minimum at $-\frac{3}{4}$
(b) $f$ has a local minimum at both $0$ and $-\frac{3}{4}$
(c) $f$ has a local minimum at \hide both\endhide $0$ and a local maximum $-\frac{3}{4}$
(d) $f$ has no local minima
(e) $f$ has a local minimum at $0$
\endp

\Newp 9.
Find the value of $x$ for which the function $f$ below assumes
its absolute maximum value on the interval $(0,\infty)$.
$$f(x) = 24 - 2x -\frac{8}{x}$$
�5
(a) $2$
(b) $28$
(c) $3$
(d) $5$
(e) $10$
\endp

\Newp 10.
On which of the following intervals is the function 
$f(x) = -\frac{x^2}{4} - \cos(x)$ concave upwards?
�5
(a) $\bigl(-\smfrac{\pi}{3},\smfrac{\pi}{3}\bigr)$
(b) $\bigl(\smfrac{\pi}{6},\pi\bigr)$
(c) $\bigl(-\smfrac{\pi}{2},\smfrac{\pi}{3}\bigr)$
(d) $\bigl(-\smfrac{\pi}{3},\smfrac{5\pi}{6}\bigr)$
(e) $\bigl(0,\pi\bigr)$
\endp

\Newp 11. 
Find an equation of the tangent line to the curve $x^2 + y^2 = 4$
at the point $(1, \sqrt{3 })$.
�2
(a) $y = -\frac{1}{\sqrt{3}}\bigl( x -1 \bigr) + \sqrt{3}$
(b) $y = -\frac{1}{\sqrt{3}}\bigl( x -\sqrt{3} \bigr) + 1$
(c) $y = \frac{1}{\sqrt{2}}\bigl( x -1 \bigr) + \frac{3}{\sqrt{5}}$
(d) $y = \frac{2}{\sqrt{3}}\bigl( x -1 \bigr) + \sqrt{3}$
(e) $y = \frac{2}{\sqrt{3}}\bigl( x - \sqrt{3} \bigr) + 1$
\endp

\Newp 12. 
Find the second derivative of the function
$y = \sin(x^2+1) + \cos(x+1)$.
�1
(a) $2 \cos(x^2+1) - 4x^2\sin(x^2+1) - \cos(x+1)$
(b) $2x \cos(x^2+1) - 4x\sin(x^2+1) - \cos(x+1)$
(c) $2x\cos(x^2+1) + 4x^2\sin(x^2+1) - \sin(x+1)$
(d) $2 \cos(x^2+1) - 4x^2\sin(x^2+1) + \cos(x+1)$
(e) $2 \cos(x^2+1) + 4x\sin(x^2+1) + \cos(x+1)$
\endp

\Newp 13. 
You have run an experiment which has yielded the following measurements.
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Which number below is the Riemann sum for $\Int^3_1\ f(t)\ dt$ using
{\bf right{--}hand endpoints} and the given data?
\par\noindent{\bf Comment:} Answers (a) and (d) are equal.
Answer (a) is what you get if you just plug into the 
right{--}hand{--}endpoint{--}Riemann sum formula; 
answer (d) is what you get by plugging into the 
left{--}hand{--}endpoint{--}Riemann sum formula.
Since the value of $f$ the same at the beginning and the end,
these two answers are equal.
�1
(a) $\smfrac{1}{2}\bigl( 1.1 + 0.9 + 0.7 + 0.8 \bigr)$
(b) $\smfrac{1}{2}\bigl( (1.5\times 1.1) + (2.0\times 0.9) + 
(2.5\times 0.7) + (3.0\times 0.8) \bigr)$
(c) $\smfrac{2}{5}\bigl( 0.8 + 1.1 + 0.9 + 0.7 + 0.8 \bigr)$
(d) $\smfrac{1}{2}\bigl( 0.8 + 1.1 + 0.9 + 0.7 \bigr)$
(e) $\smfrac{2}{5}\bigl( (1.0\times 0.8) + (1.5\times 1.1) + (2.0\times 0.9) + 
(2.5\times 0.7) + (3.0\times 0.8) \bigr)$
\endp

\Newp14. 
Let $f$ be a continuous function and suppose $\Int^3_0 f(x)\ dx = 5$.
Find $\Int^9_0 \frac{f\bigl(\sqrt{x }\bigr)}{\sqrt{x}}\ dx$.
�5
(a)$10$
(b)$5/3$
(c)$3/2$
(d)$5/2$
(e)$0$
\endp

\Newp 15.
The vertical asymptotes and horizontal asymptotes of the graph of the
function
$$ f(x) = \frac{\sqrt{x^6+1 }}{x(x^2-1)}$$
are:
�2
(a)$x=0$, $x=1$, $x=-1$; $y=1$, $y=-1$
(b)$x=1$, $x=-1$; $y=1$
(c)$x=1$, $x=-1$; $y=0$, $y=1$, $y=-1$
(d)$x=0$, $x=1$, $x=-1$; $y=1$
(e)$x=0$, $x=1$, $x=-1$; $y=0$, $y=1$, $y=-1$
\endp


\Newp 16. 
Examine the function $$f(x) = x^4 - 4x^3 + 4x^2 +1 $$
for regions of increase, decrease, absolute maxima (if any) and
absolute minima (if any).
\par\noindent{\bf Comment:}This problem caused more trouble than any other.
The critical points are the roots of $f^\prime(x)=4x^3-12x^2+8x=4x(x-1)(x-2)=0$: hence
$x=0$, $x=1$ and $x=2$. The function $f$ is increasing on the intervals $(0,1)$ and $(2,\infty)$
and decreasing on $(-\infty,0)$ and $(1,2)$.
Since $\displaystyle\mathop{\lim}_{x\to\pm\infty} f(x) =\infty$, $f$ has no absolute maxima.
There is a local maximum at $x=1$, with $f(1)=2$ and two local minima at $x=0$ and $x=2$
where $f(x)=1$. It follows that $1$ is an absolute minimum.
Hence (c) is the correct answer.
�1
(a) Increasing on $(0,1)$; absolute minimum value of the function is $1$ 
(b) Increasing on $(1,2)$; absolute minimum value of the function is $1$
(c) Increasing on $(0,1)$; absolute minimum value of the function is $2$ 
(d) Decreasing on $(1,2)$; absolute minimum value of the function is $2$
(e) Decreasing on $(0,1)$; absolute minimum value of the function is $1$
\endp

\Newp 17. 
Suppose two motorboats leave from the same point at the same time.
If one travels north at 3 miles per hour and the other travels east at 4 miles
per hour, how fast will the distance between them be changing after $3$ hours?
�5
(a)$5$ mph
(b)$6$ mph
(c)$4$ mph
(d)$7$ mph
(e)$3$ mph
\endp

\Newp 18. 
\def\cm{\ {\rm cm}}%
A rectangular box with a square base and open top with volume $4000 \cm^3$
is to be constructed.
Find the minimum area of material necessary in its construction.
�3
(a) $1200 \cm^2$
(b) $800 \cm^2$
(c) $1400 \cm^2$
(d) $1600 \cm^2$
(e) $600 \cm^2$
\endp

\Newp 19. 
A train is traveling at a constant speed of $120$ mph ( $176\ {\rm ft}/{\rm sec}$) 
when a signal triggers its emergency brakes.
The brakes give a constant deceleration of $4\ {\rm ft}/{\rm sec}^2$.
How far will the train travel before coming to rest?
�5
(a) $3872$ ft.
(b) $1760$ ft.
(c) $5280$ ft.
(d) $2420$ ft.
(e) $1936$ ft.
\endp

\Newp 20. 
Find the area of the region in the right half{--}plane bounded by the curves
$y = x$ and $y = x^2 - 2$.
�5
(a)$10/3$
(b)$8/5$
(c)$5/3$
(d)$6$
(e)$19/3$
\endp

\Newp 21. 
Find the volume of the solid obtained by rotating the region bounded by
$y=x$ and $y = \sqrt{x}$ about the line $x = 2$.
�5
(a)$8\pi/15$
(b)$2\pi$
(c)$4\pi/3$
(d)$4/15$
(e)$2\pi/15$
\endp

\Newp 22. 
The base of a solid is the region in the $xy${--}plane bounded above by 
$y = \tan x$, below by the $x${--}axis between $x=0$ and $x = \smfrac{\pi}{4}$.
Slices perpendicular to the $x${--}axis are triangles of height $3$.
Which integral below is the volume of this solid?
�3
(a)$\smfrac{3}{2}\Int_0^{\smfrac{\pi}{4}}\ \tan x\ dx$
(b)$\smfrac{3}{2}\Int_0^{\smfrac{\pi}{4}}\ \tan^2 x\ dx$
(c)$\Int_0^{\smfrac{\pi}{4}}\ (\pi - \tan x)\ dx$
(d)$\pi\Int_0^{\smfrac{\pi}{4}}\ \tan^2 x\ dx$
(e)$\Int_0^{\smfrac{\pi}{4}}\ \tan^2 x\ dx$
\endp


\Newp23. 
Which integral below computes the volume of the solid of revolution
obtained by rotating the region above the $x${--}axis, below the curve
$y = \sin x$ and between $x=0$ and $x = \pi$, about the line $x = 2\pi$.
Use the shell method.
�2
(a)$2\pi \Int_0^\pi (2\pi - x) \sin x\ dx$
(b)$\pi \Int_0^{2\pi} (\pi^2 - \sin^2 x)\ dx$
(c)$2\pi \Int_0^\pi (\pi^2 - \sin^2 x)\ dx$
(d)$2\pi \Int_0^\pi x \sin x\ dx$
(e)$2\pi \Int_0^\pi (\pi - \sin x) dx$
\endp

\Newp24. 
Compute the volume of the solid obtained by rotating the region in the
right half{--}plane bounded between $y = x^2 - 1$ and $y = 1 - x^2$
about the line $x = 1$.
�5
(a)$\frac{5\pi}{3}$
(b)$\frac{8\pi}{7}$
(c)$\frac{10\pi^2}{7}$
(d)$\frac{11\pi}{6}$
(e)$\frac{10\pi}{7}$
\endp


\Newp 25.  
Find the average value of the function $y = \cos\bigl(x/2\bigr)$ 
on the interval $[0,\pi]$.
�5
(a)$\frac{2}{\pi}$
(b)$-\frac{1}{\pi}$
(c)$0$
(d)$\frac{\pi}{2}$
(e)$\frac{2}{3\pi}$
\endp
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