



\def\section{{\bf Class Time:}}

\def\st{\perm1:abcde
\perm2:edcba}

\normalafterproblemskip=1cm

\twocolumn{p.5}

\title{Math 105{--}Final}
\date{December 16, 1991}


\gdef\junk#1#2#3@{%
\def\xxcc{#2}\def\xxaa{@}\ifx\xxcc\xxaa\relax%
\let\next=\relax\else%
\let\next=\junk%
\hbox to#2{$\hfil#1\hfil#3$}%
\fi%
\next}


\gdef\ss{\scriptstyle}

\def\zz{\quad$}

\problem
The $x${--}and $y${--}intercepts of the line \zz 5x+7y=140\zz are given by
\correct$(28,0)$ and $(0,20)$
\wrong$(5,0)$ and $(0,7)$
\wrong both intercepts are at $(0,0)$
\wrong $(0,5)$ and $(7,0)$
\wrong $(5,140)$ and $(140,7)$



\problem
The line \zz 2x+y=11\zz meets the parabola \zz y=x^2-4\zz in the point
\correct$(3,5)$ and $(-5,21)$
\wrong$(1,2)$
\wrong$(5,1)$
\wrong the two curves never meet
\wrong$(4,12)$ and $(-3,5)$

\problem
Which equation below is the equation of the line parallel
to \zz 2x+17y=131\zz which passes through the origin?
\correct$y=-\frac 2{17}x$
\wrong$y=-\frac 2{17}x +131$
\wrong$y=-\frac {17}2x$
\wrong$y=\frac {17}2x$
\wrong$x=-\frac 2{17}y + \frac{131}{17}$

\problem
Which equation below is the equation of a line
passing through the points
$(0,7)$ and $(1,0)$?
\correct$y=-7x+7$
\wrong$x+7y=7$
\wrong$y=-x+1$
\wrong$y=x+7$
\wrong$y=-6x+7$

\problem
Which equation below is the equation of the line
tangent to \zz y=\ln x\zz at the point $(1,0)$?
\correct$y=x-1$
\wrong $y=e^x$
\wrong$y=-x+1$
\wrong$y=e(x-1)$
\wrong$y=x$

\problem
Which number below is the limit
$$\lim_{t\to\infty}\frac{%
\junk{}{24pt}4t^3@%
-{24pt}7t^2@%
+{24pt}11t@%
-{18pt}4@@@@}
{\junk
{}{24pt}-t^4@%
+{24pt}3t^2@%
-{24pt}7t@%
+{18pt}7@@@@}\ ?$$
\correct$0$
\wrong$-\frac 14$
\wrong$-\frac34$
\wrong$4$
\wrong$-\frac 47$


\problem
Which number below is the limit
$$\lim_{x\to-1}\ \frac{%
x^2-1}{x^4-1}\ ?$$
\correct$\frac12$
\wrong$3$
\wrong$0$
\wrong$\frac13$
\wrong$2$

\problem
Which number below is the limit
$$\lim_{y\to-\infty}
e^{4y}\ ?$$
\correct$0$
\wrong$e$
\wrong$\root 4 \of e$
\wrong$-1$
\wrong$e^4-e^{1/4}$


\problem
Which number below is the derivative of $y$
with respect to $x$ at the point $(2,-1)$ on the
curve
$$y^3+x^2y+xy^2=-3\ .$$
\correct$1$
\wrong$7$
\wrong$\frac{10}{53}$
\wrong$3$
\wrong$-\frac 17$



\problem\mathsurround=4pt
At which one of the {\sl points\/} below is the function
$4x^3-3x$ decreasing?
\correct$(0,0)$
\wrong$(-2,-26)$
\wrong$(-1,-1)$
\wrong$(1,1)$
\wrong$(2,26)$


\problem\mathsurround=4pt
Which one of the points below is an inflection point for
the function
$x^3+ 12x^2 - 7x-19$?
\correct$(-4,137)$
\wrong$(2,23)$
\wrong$(0,-19)$
\wrong$(1,-11)$
\wrong There are no inflection points for this curve.

\problem\mathsurround=4pt
On which {\sl interval\/} below is the function
$\frac 1{x^2+1}$ increasing?
\wrong$(-1,\infty)$
\wrong$(-1,1)$
\wrong$(-2,0)$
\wrong$(-2,1)$
\correct$(-\infty,0)$

\problem\mathsurround=4pt %% 13 %%
What is the maximum {\sl value\/} assumed by the function
$x^3-3x^2-10$ on the interval $0\leq x \leq 4$?
\st
\wrong$-6$
\wrong$-3$
\wrong$0$
\wrong$3$
\correct$6$

\problem\mathsurround=4pt %% 14 %%
If $\frac{dx}{dt}=4$ at the point $(-1,1)$ on the curve $y=\ln(x+2) +1$, what
is $\frac{dy}{dt}$ at the same point?
\st
\wrong$8$
\correct $4$
\wrong$0$
\wrong$-4$
\wrong$-8$

\problem\mathsurround=4pt %% 15 %%
The derivative of
$(x^2+2x+1)(x^3+2)$ is
\correct$(2x+2)(x^3+2)+(x^2+2x+1)(3x^2)$
\wrong$5x^4+6x^3+3x+2+4$
\wrong$(x^2+2x+1)+(x^3+2)$
\wrong$(2x+1)(x^3+2)$
\wrong$x^2(x+1)^2$

\problem\mathsurround=4pt %% 16 %%
The derivative of
$f(x) = \frac{e^x}{\ln x}$ is
\correct$\frac{e^x(x\ln x-1)}{x(\ln x)^2}$
\wrong$\frac{e^x}{1/x}$
\wrong$\frac{\ln x + 1/x}{e^x}$
\wrong$\frac{e^x(x+\ln x)}{(\ln x)^2}$
\wrong$\frac{e^x}x$

\problem  %% 17 %%
The derivative of \zz \ln (x^2+x)\zz
is 

\correct$\frac{2x+1}{x(x+1)}$
\wrong$\frac 2x+ \frac 1x$
\wrong$\ln x +\ln (x+1)$
\wrong$2x+1$
\wrong$\frac1{(x^2+x)^2}$

\pagebreak

\problem  %% 18 %%
The function \zz e^x+e^{2-x}\zz
has a relative minimum with $x${--}coordinate 
\st
\correct$1$
\wrong$2$
\wrong$0$
\wrong$-2$
\wrong$-1$

\problem \mathsurround=4pt %% 19 %%
If $y=x^3+8x-5$ and $x=3e^t+2e$ find $\left .\frac{dy}{dt}\right|_{_{\ss t=1}}$
\correct$3\>e\>(75\>e^2+8)$
\wrong$11$
\wrong$3e$
\wrong$12e$
\wrong$0$



\problem\mathsurround=4pt %% 20 %%
\let\ss=\relax
Let $f(x)$ and $g(x)$ be differentiable functions such that
$$\matrix{\ss f(0)=1&;&\ss f(1)=2&;&\ss f(2)=3&;
&\ss f\>\!^{\prime}(0)=-3&; &\ss f\>\!^{\prime}(1)=-2&; &\ss f\>\!^{\prime}(2)=-1;\cr
\ss g(0)=1&; &\ss g(1)=2&; &\ss g(2)=3&;
&\ss g\>^{\prime}(0)=\hphantom{-}3&; &\ss g\>^{\prime}(1)=\hphantom{-}2&; &
\ss g\>^{\prime}(2)=\hphantom{-}1.\cr}$$ 
Which
number below is the value of the derivative of $\frac{f(x)}{g(x)}$ at $x=1$?
\st
\wrong$1$
\wrong$2$
\wrong$-6$
\correct$-2$
\wrong$-1$

\problem %% 21 %%
\afterproblemskip=2cm
Find the derivative of\qquad  $\frac{(x^2+1)^3\root 5 \of{e^x+2}}{x^3+3x^2+2x+1}$.


\wrong$\frac{(x^2+1)^3\root 5 \of{e^x+2}}{x^3+3x^2+2x+1}$
\wrong$(x^2+1)^3+\root 5 \of{e^x+2}-(x^3+3x^2+2x+1)$
\wrong$3(x^2+1)+\frac 15 (e^x+2)^{-45} - (3x^2+6x+2)$
\wrong$\ln\left(\frac{(x^2+1)^3\root 5 \of{e^x+2}}{x^3+3x^2+2x+1}\right)$
\correct $\frac{(x^2+1)^3\root 5 \of{e^x+2}}{x^3+3x^2+2x+1}
\left(6\>\frac{x}{x^2+1} + \frac 15\>\frac{e^x}{e^x+2} - 
\frac{3x^2+6x+2}{x^3+3x^2+2x+1}\right)$


\problem\mathsurround=4pt %% 22 %%
Which number below is the number given by using
differential approximation to compute the value
of $\ln(1.1)$?
\st
\wrong$0$
\correct$.1$
\wrong$.2$
\wrong$.3$
\wrong$\frac{.1}e$



\problem\mathsurround=4pt %% 23 %%
A firm estimates that its profits are given as a function of its sales
by the formula $P=.4s + \frac {.9}s\vrule width 0pt height 10pt depth 12pt$, 
where $P$ represents profits in millions of dollars 
and $s$ represents sales in millions of units.
How many million units should the company sell in order to maximize its profit?
\st
\wrong$1$
\correct$1.5$
\wrong$2$
\wrong$2.5$
\wrong$3$



\problem %% 24 %%
An object moving in a straight line travels $s(t)$ meters in $t$ minutes,
where \zz s(t)=16t^2-4t + 19$.
After how many minutes will its velocity be $12\> {\rm m./min.}$?
 %% 20 %%
\correct$\frac12$
\wrong$1$
\wrong$3$
\wrong$8$
\wrong$60$


\problem %% 25 %%
You have invested $\$20,000$ in a compound interest account in which the
interest is compounded continuously.
Six years later, your money has doubled.
What is the interest rate assuming that $\ln 2=.69$ ?
\correct $11.5\%$
\wrong $5\%$
\wrong $20.6\%$
\wrong$7.9\%$
\wrong$3.6\%$


\problem \mathsurround=4pt %% 26 %%
Find  $\int (x^4 + \root 3\of x + \frac 1x)\, dx$.

\correct$\frac {x^5}5 + \frac 34 x^{\frac{\ss 4}{\ss 3}} +\ln | x | + C$
\wrong$4x^3 +\frac 13 x^{-\frac{\ss2}{\ss3}} - \frac1{x^2} +C$
\wrong$4x^5 + \root 3\of{x^2} -\ln | x | +C$
\wrong$x^4 + \root 3\of x + \frac 1x$
\wrong$(x^4 + \root 3\of x + \frac 1x)\bigl(\frac{x^5}5 + 
\frac{\root 3\of x}3 + \frac 1{\ln x}\bigr)$


\problem \mathsurround=4pt %% 27 %%
If  $\frac{dy}{dx}= e^{2x} +\frac12$
and  $y(0) = 1$
then $y(1)=$

\correct$\frac{e^2}2+1$
\wrong$\frac{e^2-1}2$
\wrong$\frac{e^4}2-2$
\wrong$\ln 2 - 3$
\wrong$\ln (e^2-\frac 12)$


\problem\mathsurround=4pt %% 28 %%
A rocket is launched straight up into the air with velocity  
$-9.8\>t + \frac{20}{t^2+1}$ meters per second.
Which symbol below represents the distance traveled by the 
rocket in the first two seconds?

\correct $\int_0^2\ \left( -9.8\>t + \frac{20}{t^2+1}\right)\,dt$ meters
\wrong$\int_0^2\ \left(-9.8 - \frac{40t}{(t^2+1)^2}\right)\,dt$ meters
\wrong $\int_1^2\ \left(-9.8\>t + \frac{20}{t^2+1}\right)\,dt$ meters
\wrong$\int_1^2\ \left(-9.8 - \frac{40t}{(t^2+1)^2}\right)\,dt$ meters
\wrong$\int_0^2\ \left(-9.8\>t - \frac{40t}{(t^2+1)^2}\right)\,dt$ meters


\problem  %% 29 %%
Find the area under the curve  \zz y=5x^4+3x^2\zz
for  $x$ between  $-1$ and  $1$.
\st
\wrong$0$
\wrong$2$
\correct$4$ 
\wrong$6$
\wrong$8$

\problem  %% 30 %%
Calculate  \zz\int _1^4 \frac 3{x^2} \,dx$.

\correct  $\frac{9}{4}$
\wrong $3\ln 4$
\wrong $\frac34$
\wrong $-\frac13$
\wrong  $0$

\end



 