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\NoBlackBoxes
\nologo
\flushpar{\bf{Math 1251 Exam 1 January 27, 1995}} \qquad Your Name 
{$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$} 
\newline
{\bf{Professor Stanton}}
\vskip 3pt
\flushpar{\it{Directions}}: 
{\eightpoint{You may use your notes, books, and calculators.
This exam has 8 problems with the point values indicated.
Circle your answer to the multiple choice questions
.
If you use a table (or calculator) to find your answer,
write down the word ``table'' (or ``calculator'') next to your answer.
If your answer is correct, you will receive full credit for that
problem, otherwise you will be 
awarded 0 points. This means if you copy a minus sign incorrectly
you will be given 0 points. An answer alone with no justification is worth 0 points.}}
\vskip 3pt
\flushpar
(10) 1. If $f(x)=(5-2/x)^{100}$, then $f'(x)=$\newline
(a) $100(5-2/x)^{99}$\flushpar
(b) $200(5-2/x)^{99}/x$\flushpar
(c) $400(5-2/x)^{99}/x^2$\flushpar
(d) $1000(5-2/x)^{99}$\flushpar
(e) none of the above.
\vskip5pt\flushpar
(10) 2. If $y=tan(x)/x^2$, then $\frac{dy}{dx}$=\newline
(a) $sec(x)/x^2$\flushpar
(b) $sec^2(x)/x^2+sin(x)/(2x)$\flushpar
(c) $tan(x)/(2x)$\flushpar
(d) $-2 tan(x)/x^3 +sec^2(x)/x^2$\flushpar
(e) none of the above.
\vskip 5pt
\flushpar
(10) 3. If $f(x)= \sqrt{1+cos(x^2)}$, then $f'(x)=$\newline
(a) $(1+cos(x^2))^{-1/2}/2$\flushpar
(b) $\sqrt{|sin(x^2)|}$\flushpar
(c) $\frac{-xsin(x^2)}{\sqrt{1+cos(x^2)}}$\flushpar
(d) $-sin(x)$\flushpar
(e) none of the above.
\vskip 5pt
\flushpar
(10) 4. $$
\lim_{x\to 0}\frac{xtan(x)(x^6+12)(cos(sin(x))}
{sin(2x)sin(3x)(x^{12}+14x^5+3x+1)}=
$$
\newline
(a) $0$\flushpar
(b) $12$\flushpar
(c) $3$\flushpar
(d) $2$\flushpar
(e) does not exist.
\vfill
\pagebreak
\flushpar
(12) 5. Find the equation of the tangent line to the graph of $y=x\sqrt{x}+x$ 
at $(4,12)$.\flushpar
\vskip100pt\flushpar
(12) 6. The position function of a particle is given by $s=4t^3-12t$.
\flushpar
(a) Find the average velocity during $1\le t\le3$.\flushpar
(b) Find the instantaneous velocity at $t=3$.\vskip 120 pt\flushpar
(8) 7. Give an example of a function $f(x)$, which is continuous at 
all real numbers $x$, but is not differentiable at integer values 
of $x$.\vskip100pt\flushpar
(8) 8. True or False. (You must give at least one reason
for your answer in a complete sentence.
An answer alone is worth zero.)\newline
If $f(x)$ is a continuous function for all
real $x$ which satisfies $0\le f(x)\le1$, then 
$\lim_{x\to\infty} f(x)$ must exist.
 
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\magnification=\magstep1
\input amstex
\documentstyle{amsppt}
\document
\NoBlackBoxes
\nologo
\flushpar{\bf{Math 1251 Exam 2 February 23, 1995}} \qquad Your Name 
{$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$} 
{\bf{Professor Stanton}}
\vskip 3pt
\flushpar{\it{Directions}}: 
{\eightpoint{You may use your notes, books, and calculators.
This exam has 8 problems with the point values indicated.
Circle your answer to the multiple choice questions.
If you use a table (or calculator) to find your answer,
write down the word ``table'' (or ``calculator'') next to your answer.
If your answer is correct, you will receive full credit for that problem,
otherwise you will be 
awarded 0 points. This means if you copy a minus sign
incorrectly you will be given 0 points.
An answer alone with no justification is worth 0 points.}}
\vskip 3pt
\flushpar
(10) 1. If $f(x)=e^{-x^2 ln(x)}$, then $f'(x)=$\newline
(a) $e^{-x^2ln(x)}(-2xln(x))$\flushpar
(b) $e^{-2xln(x)}$\flushpar
(c) $e^{-x^2ln(x)}(-x)$\flushpar
(d) $x^{-x^2}$\flushpar
(e) none of the above.
\vskip5pt\flushpar
(10) 2. If $x\thinspace tan^{-1}(y)=y^2$, then $\frac{dy}{dx}$=\newline
(a) $x/(1+y^2)+tan^{-1}(y)$\flushpar
(b) $(2y-tan^{-1}(y))(1+y^2)/x$\flushpar
(c) $tan^{-1}(y)/(2y-\frac{x}{1+y^2})$\flushpar
(d) $x/(2y(1+y^2))$\flushpar
(e) none of the above.
\vskip 5pt
\flushpar
(10) 3. If $f(x)=x\thinspace e^{2x}$ then $f''(1)=$\newline
(a) $8e^2$\flushpar
(b) $e^2$\flushpar
(c) $0$\flushpar
(d) $3e^2$\flushpar
(e) none of the above.
\vskip 5pt
\flushpar
(10) 4. 
$$
\lim_{x\to \infty}\frac{tan^{-1}(x)(2e^{2x}+ln(x))}
{\sqrt{x}+e^{2x}}=
$$
\newline
(a) $0$\flushpar
(b) $2$\flushpar
(c) $1$\flushpar
(d) $\pi$\flushpar
(e) does not exist.
\vskip 5pt
\flushpar
(10) 5. The best linear approximation to $y=\sqrt{x}$ at $x=4$ is\newline
(a) $y=x$\flushpar
(b) $y-2=x-4$\flushpar
(c) $y-4=x-2$\flushpar
(d) $y-2=(x-4)/4$\flushpar
(e) none of the above.
\vskip5pt\flushpar
\vfill
\pagebreak
\flushpar
(10) 6. A population of bacteria is exponentially decaying. In one hour 
the population decreases from 2000 to 1500.  What is the population after 
2.5 additional hours?\flushpar
\vskip150pt\flushpar
(10) 7. A gas is trapped in a balloon at a constant temperature. 
It is known that the volume, $V$, and the pressure $P$ 
of such a gas are related by
$PV=c$, where $c$ is a fixed constant. The volume of the balloon 
is increasing at the constant rate of 4 ml/sec. Find the rate of change of 
the pressure with respect to time when
the pressure is 2 and the volume is 5. 
\vskip 160 pt\flushpar
(10) 8. Give an explicit function $f(x)$ such that
$$\lim_{x\to\infty}\frac{f(x)}{x}=0 \quad{\text{and}}\quad 
\lim_{x\to\infty}\frac{f(x)}{ln(x)}=\infty.
$$
 
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