%\magnification 1200
\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=10pt

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

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

\input exam.perm6
\def\st{\perm1:abcde \perm2:edcba \perm3:abcde}
\title{Math 105{--}Test III}
\date{April 15, 1993}
\lastversion=3

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

The solution of the equation
{\bf $(2.3)^{1 - 2x} = (2.3)^{7 + x}$} is

\correct $x = -2$
\wrong $x = 1$
\wrong $x = -1$
\wrong $x = 2$
\wrong $x = 0$

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

The solution of the differential equation
{\bf $\frac{d y}{d x} = -3 y,\;\;\;y(0) = 3$} is

\correct $y = 3 e^{-3x}$
\wrong $y = 3 e^{3x}$
\wrong $y = \frac{2}{3} e^{-3x}$
\wrong $y = \frac{3}{2} e^{3x}$
\wrong $y = -3 e^{3x}$

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

The function
{\bf $y = \ln \left ( x^{2} + 1 \right ) $}
has one relative extreme point.
This point is

\correct $(0, 0)$, relative minimum
\wrong $(2, \ln 5)$, relative maximum
\wrong $(-1, \ln 2)$, relative minimum
\wrong $(1, \ln 2)$, relative maximum
\wrong $(\sqrt{2}, 3)$, relative minimum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

If $e^{\ln 5 + x} e^{x + 1} = 5$ then x = 

\correct $- \frac{1}{2}$
\wrong $1$
\wrong $2$
\wrong $\frac{1}{2}$
\wrong $-2$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The slope of the graph of $f(x) = \ln \left(1 + (x + 1)^{5} \right) $ 
at x = 0 is

\correct $\frac{5}{2}$
\wrong $\ln 2$
\wrong $\frac{5}{64}$
\wrong $\ln \frac{5}{64}$
\wrong $0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

Simplified form of the expression $e^{\ln x^{3} + \ln 2^{3} - \ln x}$ is

\correct $8 x^{2}$
\wrong $e^{\ln 2x}$
\wrong $x e^{\ln 8 x^{2}}$
\wrong $e^{8x^{3}}$
\wrong $8 x^{3}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

$e^{\ln (3 \ln e - 1)}$ =

\correct $2$
\wrong $e^{\ln 2}$
\wrong $e^{3\ln 2e}$
\wrong $0$
\wrong $e^{- \ln e^{2}}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

The derivative of $\ln (1 - x)^{8}$ is

\correct $- \frac{8}{1 - x}$
\wrong $- 8 \ln (1 - x)^{7}$
\wrong $\frac{1}{1 - x}$
\wrong $- \frac{1}{(1 - x)^{8}}$
\wrong $e^{(1 - x)^{8}}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

Find the derivative of $y = (x + 2)^{3} \ln (x + 2)$.

\correct $3(x + 2)^{2} \ln (x + 2) + (x + 2)^{2}$
\wrong $3(x + 2)^{2} + (x + 2)^{3} \ln (x + 2)$
\wrong $3(x + 2)^2+\frac1{x+2}$
\wrong $(x + 2)^{2} \ln (x + 2) - (x + 2)^{2}$
\wrong $3 \ln (x + 2)^{2} + (x + 2)^{2}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Solve the equation $\ln \sqrt{x+1} =2 $.

\correct $e^{4} - 1$
\wrong $e^2-2$
\wrong $ -1 $
\wrong $2e -1 $
\wrong doest not have a solution.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 

Find $x$ such that $10^x= e^{10x} $. 

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

$ \frac{d}{dx} \left[ \sqrt{\ln (2x) } + \ln 5] \right ] =$

\correct $\frac{1}{2x  \sqrt{\ln (2x) }}$
\wrong $\frac{1}{2x} + \frac{1}{5}$
\wrong $\frac{1}{2 x  \sqrt{\ln (2x) }} + \frac{1}{5}$
\wrong $\frac{1}{ x\sqrt{\ln (2x) }}$
\wrong $\frac{1}{ x\sqrt{\ln (2x) }} + \frac{1}{5}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

$ \frac{d}{dx} \left [ x e^{-x^{2}+1} \right ] = $

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

Iodine-131 is often used in medicine as a radioactive  tracer. Assume
that Iodine-131 has a half-life of 10 days. Find the decay  constant (measured in days).
(Assume that $\ln 2= 0.69$.)

\correct $\lambda= 0.069 $
\wrong $\lambda =0.69 $
\wrong $\lambda =6.9 $
\wrong $\lambda =-0.69 $
\wrong $\lambda =0.0239 $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Let $\ln 2=0.69, \; \ln 3=1.09 $. Find $\ln 12 $

\correct $2.47$
\wrong $1$
\wrong $3.89$
\wrong $1.78$
\wrong $0 $

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

\problem
Let $f(x)= \ln (xe^{-x}) $, then $f'(x)=$

\correct $\frac{1}{x}-1 $
\wrong $\frac{1}{xe^{-x}}$
\wrong $-\frac{1}{xe^{-x}}$
\wrong $\frac{1}{xe^{-x}}[e^{-x} +1] $
\wrong $\frac{1}{xe^{-x}}[xe^{-x} -1] $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem 
Using logarithmic differentiation, find the derivative of $f(x)= 4^xx^4$

\correct $4^xx^4\ln 4+4 ^{x+1}x^3$
\wrong $4^xx^4+4 ^{x+1}x^3$
\wrong $4^{x+1}x^4+4 ^xx^3$
\wrong $4^xx^4\ln 4+4 ^{x+1}x^3\ln x$
\wrong $4^xx^4+4 ^{x+1}x^3\ln x$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
Suppose a population of 100 fruit flies grows at an exponential rate. If the population reaches
300 in 10 days, what is the population  after 20 days?
\st
\wrong 600
\wrong 700
\wrong 800
\correct 900
\wrong 1000
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem

Assume that $ 2,000$ is deposited in a bank account with $10 \%$ compounded
continuously. How much money will be in the account after 10 years ? 
(Assume that $e = 2.718$.)

\correct $ 5,436.00$
\wrong $ 5, 000.00$
\wrong $ 4, 000.00$
\wrong $ 5, 400.00$
\wrong $ 4, 718.00$

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

\end

\problem %** 21 **%


\correct
\wrong 
\wrong 
\wrong 
\wrong 

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\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