%
\def\frac#1#2{{{#1}\over{#2}}}
\def\textfrac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
\def\scriptfrac#1#2{{{\scriptstyle#1}\over{\scriptstyle#2}}}
\def\s{\hskip 3em}
\def\m{\hskip 1em}
\def\d{\displaystyle}

\let\xlim=\lim
\def\lim_#1{\xlim_{#1}\>}
\def\pt(#1,#2){\hbox{\mathsurround=0pt$\left(#1,#2\right)$}}
\def\nms#1{{\mathsurround=0pt{#1}}}

\global\pagenumberstyle=3

\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=1in
%\answersheetfootline{Each multiple choice problem is worth 5 points.          }
\pageannouncements={none} 
\input Perm30.3 
\title{Math 105, Test 3, Second draft}
\date{Novembar 24, 1992}
\lastversion=0

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\note 

{{\bf MULTIPLE CHOICE SECTION} (5 points each)}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
 
Simplify the following expression: $\displaystyle {e^{ \frac{\ln 5}{3}}}$

\correct $\root 3\of  5$
\wrong $\displaystyle{e^{\frac {5}{3}}}$
\wrong $5-e^3$
\wrong $125$
\wrong $\frac {5}{3}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 1.cm
Simplify the following expression: 
$\frac{18^{3x}}{9^{-2x}}$

\correct $3^{10x}2^{3x}$
\wrong $2\cdot 9^{5x}$
\wrong  $2^{3}9^{5x}$
\wrong  $\left(\frac {8}{9}\right)^{-{\frac{3}{2}}x}$
\wrong  $3\cdot\left(\frac{2}{3}\right)^{5x}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Solve the following equation: $2^{4x+3}=8^x$.

\correct $x=-3$ 
\wrong $x=0$
\wrong $x=3$
\wrong $ x=-1$
\wrong $x=-\frac{3}{7}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

$\frac{d^2}{dx^2}\Bigl[-(\ln x)^2\Bigr] $ is
\correct $2\frac{\ln x-1}{x^2}$
\wrong $\frac{2x\ln x}{x}$
\wrong $-\frac{\ln x}{2}$
\wrong $-\frac{\ln x}{x}$
\wrong $\frac{\ln x-x}{x^2}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
$\frac{d}{dx}\Bigl[ (5x+e^{x^3})^5\Bigr]$

\correct $(5x+e^{x^3})^4(25+15x^2e^{x^3})$
\wrong  $5(5x+e^{x^3})^4$
\wrong $ 5(5x+e^{x^3})^4(5+e^{x^3})$
\wrong $ 5(5x+e^{x^3})$
\wrong $5(5x+e^{x^3})^4(5+3e^{x^3}) $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

 The relative extreme point of the function
$f(x)=e^x-5x+5\ln 5$ is:

\correct $(\ln 5,5)$, relative minimum
\wrong $(\ln 5,5)$, relative  maximum
\wrong $(5, e^5-25+\ln 5)$, relative minimum
\wrong $(-\ln 5, \frac{1}{5}+10\ln 5)$, relative minimum
\wrong $(-\ln 5, \frac{1}{5}+10\ln 5)$ relative maximum



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

  Using logarithmic differentiation find  $\frac{d}{dx}\frac {(x+1)}{\sqrt
{2x^2+3}}$

\correct $(\frac{1}{x+1}-\frac{2x}{2x^2+3})\cdot\frac {(x+1)}{\sqrt
{2x^2+3}}$
 \wrong $(\frac{1}{x+1}+\frac{1}{2x^2+3})\cdot\frac
{(x+1)}{\sqrt {2x^2+3}}$
\wrong $(\frac{1}{x+1}-\ln \frac{1}{2x^2+3})\cdot\frac
{(x+1)}{\sqrt {2x^2+3}}$
\wrong $(\ln \frac{1}{x+1}-\frac{2x}{2x^2+3})\cdot\frac
{(x+1)}{\sqrt {2x^2+3}}$
\wrong $ \frac{1}{x+1}-\ln\frac{2x}{2x^2+3})$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Differentiate $(x+1)^x$

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Solve :
$(2x^2+4)3^{5x}-(10x-8)3^{5x}=0$

\correct $x=2, 3$
\wrong $x=0$
\wrong $x=3, 0$
\wrong $x=9, 5$
\wrong $ x=\frac{5+\sqrt 33}{2}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Determine all functions $y=f(x)$  such that $y'=-6y$  and  $f(0)=e^2$.

\correct $e^{-6x+2}$
\wrong $-6e^{2x}$
\wrong $e^{-6x}$
\wrong $e^{-6x-e^2}$
\wrong $-6e^{(e^2)}$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Determine the growth constant of a population that is growing at a rate
proportional to its size, where the population triples in size every 20 hours.

\correct $\frac {\ln 3}{20}$
\wrong $\frac {\ln 20}{3}$
\wrong $\frac {\ln 3}{5}$
\wrong  $\ln 20$
\wrong $\ln 3$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
The function $f(x)=2\ln(x+1)-(x+1)^2$ has its possible extreme for
$x\geq -1$ at the following point:

\correct $(0,-1)$
\wrong $(-1,0)$
\wrong $(1,2\ln 2-4)$
\wrong $(1,0) $
\wrong $(0,1)$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem
\afterproblemskip = 6cm

 Ten  grams of a certain radioactive material decays to 9 grams
in 15 years. After how many years will just 5 grams remain?
 
\correct $\frac {15 \ln ({1\over 2})}{\ln ({9\over 10})}$
\wrong $\frac {\ln 15}{2\ln 9}$
\wrong $\frac {2 \ln ({1\over 2})}{\ln ({3\over 5})}$
\wrong $\frac {15 \ln 5}{\ln ({10\over 9})}$
\wrong $75 $


\note

{{\bf PARTIAL CREDIT SECTION}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
 \afterproblemskip = 18cm
 ( {{\bf 20 points}})
Sketch the graph of the   following curve: $y=(1+x ) e^{-x}$ for $x\geq
-1$.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 6cm
 ({{\bf 15 points}})
 An investor initially invest \$10,000 in a speculative venture. Suppose
that the investment earns 10\% interest compounded continuously for 10
years and then 6\% interest compunded continuously for 10 years thereafter.
How much does \$10,000 grow to after 20 years? {\bf (Do not evalute
exponents.)}
\end