\magnification 1200
          \def\marknumbers#1{}
\marknumbers{1,20}
\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}%equivalent to \dsize in ams.tex
\def\pt(#1,#2){\hbox{\mathsurround=0pt$\left(#1,#2\right)$}}
\def\nms#1{{\mathsurround=0pt{#1}}}
\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=1.65cm
\multiplechoiceskip=5.5pt

%\pageannouncements={none}
\marknumbers{1,19}
%\twocolumn{16}
%\twocolumn{p.5}
%\def\section{{\bf Class Time:}}

\def\HonorCodeStatement{ 
\hbox{I have not violated the Honor Code in any way with regard to this
work.}\hskip -4in 
\lower .8cm \hbox{Signature: \vrule height .2pt depth 0pt width 3.74in} 
}
\answersheetfootline{\HonorCodeStatement}

\input exam.perm3
\def\st{\perm1:abcde \perm2:edcba \perm3:abcde}
\title{Math 105{--}Test III}
\date{April 12, 1994}
\lastversion=2

\pagenumberstyle=1
%%%%%%%%%1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The slope of the curve 
{\bf $x^3 + y^3 = 9xy$} at {\bf $(2,4)$} is

\correct $\frac{4}{5}$
\wrong $3$
\wrong $-4$
\wrong $-\frac{2}{5}$
\wrong $\frac{5}{3}$

%%%%%%%%%2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Let
{\bf $f(x) = \sqrt{x+1} \ln (x+1)$}. Then {\bf $f'(x) =$} 

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

%%%%%%%%%3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The solution of the differential equation
{\bf $y' + y = 0, y(0) = 1 $} is

\correct $y = e^{-x}$ 
\wrong $y = e^{-x} + e^{x}$ 
\wrong $y = e^{x}$ 
\wrong $y = 2 e^{-x}$ 
\wrong $y = 1$ 
%%%%%%%%4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

Find $x$ such that $\ln {10^x} = 10x$

\correct $x = 0$
\wrong $x = 1$
\wrong $x = \ln 10$
\wrong $x = e$
\wrong $x = 10$
%%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The solution of the equation $e^{x^2 - x} = e^2 $  is

\correct $x = 2$ or $x = -1$
\wrong $x = -2$
\wrong $x = 0$
\wrong $x = 1$ or $x = -2$
\wrong no solution
%%%%%%%%6%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The derivative of $y = (xe^{x} + 1)^{-1}$ is

\correct $-(xe^{x} + 1)^{-2} (e^{x} + xe^{x})$
\wrong $-(xe^{x} + 1)^{-2} + (e^{x} + xe^{x})$
\wrong $(xe^{x} + 1)^{-1} (e^{x} + xe^{x})$
\wrong $-(e^{x} + xe^{x}){-2}$
\wrong $-(xe^{x} + 1)^{-2}$
%%%%%%%%7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The function $y = (4x - 1)e^{3x - 2}$ has one relative extremum point. This
point is a 

\correct minimum at $x = -\frac{1}{12}$
\wrong minimum at $x = \frac{1}{4}$
\wrong maximum at $x = -\frac{1}{12}$
\wrong maximum at $x = \frac{1}{4}$
\wrong minimum at $x = \frac{2}{3}$
%%%%%%%%8%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

Solve the equation $\ln \big( \ln(x + 1) \big) = 1$ 

\correct $x = e^{e} - 1$
\wrong $x = e - 1$
\wrong $x = e^{e-1}$
\wrong $x = 0$
\wrong no solution
%%%%%%%%9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The equation of the tangent line to the graph of $y = \ln (x^{2} + 1)$ 
at $x=0$ is

\correct $y = 0$
\wrong $y = x + 1$
\wrong $x = -1$
\wrong $y = -1$
\wrong $y = 2x$
%%%%%%%%10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Use logarithmic differentiation to find the derivative of $f(x) = (x+1)^{(x+1)}$.

\correct $\big[\ln {(x+1)} +1\big](x+1)^{(x+1)}$
\wrong $(x+1)(x+1)^{x}$
\wrong $(x+1) \ln {(x+1)}$
\wrong $\frac{1}{x+1} \ln{(x+1)} $
\wrong $(x+1)\big[ \ln {(x+1)} + 1\big]$
%%%%%%%%11%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

From 1990 to 1993 the population of a town grew exponentially from 8000 to 
16000. How large will the population be in the year 1996?

\correct  $8000e^{2 \ln 2}$
\wrong $ 16000$
\wrong $\frac{e^{2 \ln 2}}{8000}$
\wrong $32000e^{\ln 2}$
\wrong $8000e^{ \ln 2}$
%%%%%%%%12%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

The half life of an radioactive element is 3 days. Its decay constant is 
(assume $ \ln 2 = 0.69$)

\correct $\lambda= 0.23$
\wrong $\lambda=0.069$
\wrong $\lambda=0.0.023$
\wrong $\lambda=0.69$
\wrong $\lambda=0.023$
%%%%%%%%13%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

$ \frac{d}{dx} [x^{2} \ln {\sqrt{x}}]= $

\correct $ x \ln {x} + \frac{x}{2}$ 
\wrong  $2x \ln {\sqrt{x}}$ 
\wrong  $2x \frac{1}{\sqrt x}$ 
\wrong  $2x \frac{1}{\ln {\sqrt{x}}}$ 
\wrong  $2x \ln {\frac{1}{2\sqrt x}}$ 
%%%%%%%%14%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

$ \frac{d}{dx} [(x+1)^{2} e^{x^3 - 2}] = $

\correct $2(x+1)e^{x^3 -2} + 3x^{2}(x+1)^{2}e^{x^3 -2} $
\wrong $2(x+1)e^{x^3 - 2} + (x+1)^{2}e^{x^3 -2} $
\wrong $2(x+1) + e^{3x^2} $
\wrong $2(x+1)e^{3x^2} $
\wrong $2(x+1) + 3x^2 e^{x^3 -2} $

%%%%%%%%15%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

If \$50 is deposited in an account with 10\% compounded continuously, how
much money will be in the account after 10 years? (assume $e$=2.7)

\correct $135$
\wrong $125$
\wrong $75$
\wrong $155$
\wrong $100 $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%16%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem
If $f(x) = \ln{[(x^2 +1)^{-3} (x^{3} -1)^{4}]}$, then $f'(x)$ is

\correct $-\frac {6x}{x^2 +1} + \frac{12x^2}{x^3 -1}$
\wrong $-3 \ln{(x^2 +1)^{-4}} + 4 \ln(x^3 -1)$
\wrong $-6x \ln{(x^2 + 1)^{-3}} + 12 x^2 \ln{(x^3 - 1)^{4}}$
\wrong $\frac{1}{(x^2 + 1)^3(x^3 - 1)^4}$
\wrong $\frac{6x}{x^2 +1} - 12 x^{2}(x^3 - 1) $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%17%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 
$ \frac{d}{dx} [x^{e + \ln{2}}] = $

\correct $(e + \ln 2 )x^{e + \ln {2} - 1}$
\wrong $(e + \ln 2 )x^{e + \ln {2}}$
\wrong $x^{e + \ln2}$
\wrong $\frac{1}{e + \ln 2} x^{e + \ln2}$
\wrong $ex^{e-1} + \ln {2} x^{\ln{2} -1}$

%%%%%%%%%18%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
$e^{2 \ln {x^3} - \ln{4} + \ln{x}} =$

\correct $\frac{x^7}{4}$
\wrong $\frac{x^4}{2}$
\wrong $2x^3 - 4 +x$
\wrong $\frac{2}{x^3} - \frac{1}{4} + \frac{1}{x}$
\wrong $-\frac{1}{2x^4}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%19%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

The derivative of $f(x) = \ln {(1 + e^{x})^{2}}$ is 

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

%%%%%%20%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

If $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 8$, then $y' =$

\correct $-\big( \frac{y}{x}\big) ^{\frac{1}{3}}$
\wrong   $-y^{-\frac{1}{3}} - x^{-\frac{1}{3}}$
\wrong   $\big( \frac{x}{y}\big) ^{\frac{1}{3}}$
\wrong   $y^{\frac{1}{3}} + x^{\frac{1}{3}}$
\wrong   $y^{\frac{1}{3}} - x^{\frac{1}{3}}$











%\note
%\vskip .2in
%{\bf Note!} 

%1). There were no problems on this exam ahout 
%implicit differentiation and/or related rates.
%There will surely be such problems on our exam on April 12, 1994.

%\vskip .2in
%2). There were only 19 problems on this exam.
%There will be 20 problems on our exam on April 12, 1994.

\end

\problem %** 21 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 22 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 23 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 24 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 25 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 26 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 27 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 28 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 29 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 30 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 


\end




