M119 Final Exam

December 20, 1996

$\tolerance=3000 \input epsf \hsize=6truein \nopagenumbers \parindent=1.5truecm \rm \newcount\qno \newcount\ano \def\choi{\ignorespaces \ifnum\ano=1 A% \else\ifnum\ano=2 B% \else\ifnum\ano=3 C% \else\ifnum\ano=4 D% \else\ifnum\ano=5 E% \else F% \fi \fi \fi \fi \fi\ignorespaces} \def\qn {\ano=0\par\advance\qno by1\noindent\strut \hbox to\parindent{\the\qno.\hfil}\hangindent\parindent\ignorespaces} \def\an {\smallskip\advance\ano by1\noindent\strut \hbox to2\parindent{\hfil(\choi)\quad}\hangindent2\parindent\ignorespaces} \def\qnt#1 {\par\advance\qno by1\noindent\strut \hbox to\parindent{#1\hfil}\hangindent\parindent\ignorespaces} \def\qnn#1 {\medskip\noindent\strut \hbox to\parindent{\phantom{8}#1\hfil}\hangindent\parindent\ignorespaces} \def\boxfil{ \vrule \xleaders\vbox to1cm{\hrule\hbox{\ }\vfill\hrule}\hfill \vrule\ } \def\linefil{ \leaders\hrule\hfil\ } \def\answerbox#1 {{\noindent \vbox to1cm{\vfil\hbox{#1\quad}\vfil}\vrule \xleaders\vbox to1cm{\hrule\hbox{\ }\vfill\hrule}\hfill \vrule}} \def\picture #1 #2 #3 {\hbox to#2truein{\vbox to #3truein{% \special{anisoscale #1, #2in #3in}\vfil}\hfil}} %\font\bg=cmbx12 scaled 1000 %\font\hg=lcmssb8 scaled 2985 \font\bg=cmbx10 scaled \magstep2 \font\hg=cmr7 scaled 2985 \qno=0 %************************************************************************* \qn $\displaystyle\int\Big(x^{4/3}+e^{x/2}-{3\over x}+2\Big)\,dx$ is equal to which of the following? \an ${3\over 7}x^{7/3}-{1\over 2}e^{2x}+3\ln|x|-2x+C$ \an ${3\over 4}x^{4/3}+2e^{x/2}-3\ln|x|+2x+C$ \an ${3\over 7}x^{7/3}+2e^{x/2}-3\ln|x|+2x+C$ \an ${3\over 4}x^{4/3}+{1\over 2}e^{2x}+3\ln|x|-2x+C$ \an none of these$
$\qn $\displaystyle\int_2^5(2x^2+3)\,dx=$ \an $261/3$ \an $486/3$ \an $207/3$ \an $450/3$ \an none of these$
$\qn $\displaystyle\int e^{2x^2-1}\cdot 8x\,dx=$ \an $4e^{2x^2-1}+C$ \an $2e^{2x^2-1}+C$ \an ${1\over 4}e^{2x^2-1}+C$ \an ${1\over 2}e^{2x^2-1}+C$ \an none of these$
$\qn If $f(x)=10e^{4x^2}$, what is $f''(1/2)$? \an $270e$ \an $240e$ \an $300e$ \an $288e$ \an none of these$
$\qn The expression ${1\over 2}(\ln x+3\ln y)-4\ln x$ is equal to which of the following? \an $\displaystyle \ln{\sqrt{xy^3}\over z^4}$ \an $\displaystyle \ln{\sqrt x\over y^{3/2}z^4}$ \an $\displaystyle \ln{\sqrt x\, y^3\over z^{12}}$ \an $\displaystyle \ln{\sqrt x\, z^{12}\over y^3}$ \an none of these$
$\qn If the {\it derivative\/} $f'(x)=(x+2)(x-7)$, then $f(x)$ attains a relative minimum at: \an $x=2$ \an $x=7$ \an $x=-7$ \an $x=-2$ \an none of these$
$\qn The equation of the tangent line to $y=x+e^{x-2}$ at $x=2$ is: \an $y=1$ \an $y=2x-1$ \an $y=x-1$ \an $y=3x-1$ \an none of these$
$\qn Assuming that $\displaystyle\int \big(3g(x)+1\big)\,dx=2f(x)+C$, which of the following equations is correct? \an $\displaystyle g(x)={f'(x)-2\over 3}$ \an $\displaystyle g(x)={3f'(x)-1\over 2}$ \an $\displaystyle g(x)={f'(x)-3\over 2}$ \an $\displaystyle g(x)={2f'(x)-1\over 3}$ \an none of these$
$\qn $\displaystyle\int {x\over 3x^2+4}dx=$ \an $2\ln(3x^2+4)+C$ \an ${1\over 2}\ln(3x^2+4)+C$ \an $3\ln(3x^2+4)+C$ \an ${1\over 6}\ln(3x^2+4)+C$ \an none of these$
$\qn If $f(x)=1/x$ and $g(x)=(x+2)^2$, then $(f\circ g)(1)=$ \an $1/9$ \an $1/2$ \an $1/4$ \an $9$ \an none of these$
$\qn $\displaystyle \lim_{x\to -2}{x^2-4\over x^2+5x+6}=$ \an $-1$ \an $-4/3$ \an $-2$ \an $-4$ \an none of these$
$\qn If $\displaystyle f(x)=\Big({x\over x+1}\Big)^4$, then $f'(1)=$ \an $1/8$ \an $8$ \an $-8$ \an $-32$ \an none of these$
$\qn The {\it second derivative\/} of $f(x)$ is given as follows: $f''(x)=e^x(x-2)$. Where is $f(x)$ concave up? \an $(-\infty,2)$ \an $(-\infty,-2)$ \an $(2,\infty)$ \an $(-2,\infty)$ \an none of these$
$\qn The solutions to the equation $4e^{2\ln x}=6$ are: \an $x=\pm\sqrt{2/3}$ \an $x=\pm\sqrt{3/5}$ \an $x=\pm\sqrt{3/2}$ \an $x=\pm\sqrt{5/3}$ \an none of these$
$\qn If $f(x)=\big(\ln(x^2+x+1)\big)^3$, then $f'(x)=$ \an $3\big(\ln(x^2+x+1)\big)^2\cdot(2x+1)$ \an $\displaystyle {2x+1\over x^2+x+1}$ \an $\displaystyle {1\over (x^2+x+1)^3}$ \an $3\big(\ln(x^2+x+1)\big)^2\cdot\displaystyle {2x+1\over x^2+x+1}$ \an none of these$
$\qn How many years is required for an investment of \$500 to double if the interest is 8\% per year, compounded continuously? \an $25\ln 2$ \an $20\ln 2$ \an $12.5\ln 2$ \an $10\ln 2$ \an none of these$
$\qn The area of the region enclosed between the graphs of $y=x^2+2$ and $y=-2x^2+14$ is equal to \an $128/3$ \an $36$ \an $72$ \an $32$ \an none of these$
$\qn The growth rate of a certain bacteria culture is proportional to its size. If the bacteria population doubles every 10 mintues, how long (in minutes) will it take for the culture to grow to 8 times its original population? \an $36$ \an $30$ \an $20$ \an $24$ \an none of these$
$\qn The graph of $f(x)$ is given as follows:$

$\hangindent\parindent which of the following graphs resembles the graph of $f'(x)$ the most?$

$\an$

$\qn Estimate the area under the graph of the function $f(x)=2x^2$ from $x=1$ to $x=3$ by a Riemann sum with a partition of $4$ subintervals. Use right endpoints in constructing the rectangles. \an $43/2$ \an $261/4$ \an $87/2$ \an $129/4$ \an none of these$
$\qn The area of the region enclosed between the graph of $y=x^3+2$, the $x$-axis and the vertical lines $x=1$ and $x=3$ is equal to: \an $38/3$ \an $24$ \an $26$ \an $44/3$ \an none of these$
$\qn We wish to construct a rectangular box having a square base, but having no top. If the total area of the base and the four sides must be exactly 192 square inches, what is the largest possible volume for the box. \an $32$ \an $108$ \an $500$ \an $256$ \an none of these$
$\qn The graph of $f(x)$ is given below. Which of the following statements about $f(x)$ is correct?$
a graph is here

$\an $f'(-3)>0$ and $f''(2)<0$ \an $f'(-3)>0$ and $f''(2)>0$ \an $f'(-3)<0$ and $f''(2)>0$ \an $f'(-3)<0$ and $f''(2)<0$ \an none of these$
$\qn For what $x$-value is the slope of the tangent line to $y=\ln x$ equal to $\ln 4$? \an $x=e^4$ \an $x=\ln 4$ \an $x=1/\ln 4$ \an $x=\ln(1/4)$ \an none of these$
$\qn If $f(x)=e^{2x}$, which of the following is equal to $f'(0)$? \an $\displaystyle\lim_{h\to 0}{e^{2x+h}\over h}$ \an $\displaystyle\lim_{h\to 0}{e^{2h}-1\over h}$ \an $\displaystyle\lim_{h\to 0}{e^{2h}-e\over h}$ \an $\displaystyle\lim_{h\to 0}{e^{2x+h}-1\over h}$ \an none of these$