\magnification=\magstep1 \input amstex \documentstyle{amsppt} \document \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. \vfill \enddocument \end \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 $$\lim_{x\to\infty}\frac{f(x)}{x}=0 \quad{\text{and}}\quad \lim_{x\to\infty}\frac{f(x)}{ln(x)}=\infty.$$ \vfill \enddocument \end