%\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=1.5cm
\multiplechoiceskip=10pt

\pageannouncements={none}
%\marknumbers{1,20}
%\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.perm4
\def\st{\perm1:abcde \perm2:edcba}
\title{Math 105{--}Test I}
\date{Feb.\ 2, 1993}
\lastversion=3

\note In problems 1,2, and 3 find an equation of the straight line satisfying the given
conditions:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 1 **% 
With slope $-3$ and $y-$\!\!\!intercept \pt(0,-2).

\correct $y=-3x-2$
\wrong $3x-y=-2$
\wrong $3x-y=2$
\wrong $y=-3x+2$
\wrong $y=-3(x+2)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 2 **%
Through the point \pt(-1,3) and parallel to the $y-$\!\!\!axis.

\correct $x=-1$
\wrong $x=3$
\wrong $y=-1$
\wrong $y=3$
\wrong $y-3=-1(x+1)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 3 **%
Through the point \pt(1,2) and perpendicular to the straight line $x+y=1$.

\correct $x-y=-1$
\wrong $y=-x+1$
\wrong $x+y=2$
\wrong $y=2x+1$
\wrong $2x-y=1$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 4 **%
Let $f(x)=3x^2-2x+4$. Find an equation of the tangent line to the graph of $f(x)$ at the point
\pt(1,f(1)).

\correct $y-5=4(x-1)$
\wrong $y-4=5(x-1)$
\wrong $y-5=4(x+1)$
\wrong $y-4=5(x+1)$
\wrong $y+5=4(x-1)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 5 **%
The domain of $f(x)=\sqrt{2-x}$ is the set

\correct $(-\infty,2]$
\wrong \pt(-\infty,\infty)
\wrong \pt(-\infty,2)
\wrong \pt(2,\infty)
\wrong $[2,\infty)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 7 **%
If $f(x) = \sqrt{x+2}$ and $g(x) =x^4-4$ then $g\bigl(f(x)\bigr)$ is given by

\correct $x^2+4x$
\wrong$\sqrt{x-2}$
\wrong$x-2$
\wrong$(x^4-4)\sqrt{x+2}$
\wrong$\sqrt{x+2}-4$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 8 **%
The curves $y=x^2-2x$ and $y=-x+2$ meet at the point(s)

\correct\pt(-1,3) and \pt(2,0)
\wrong\pt(1,1) and \pt(-2,4)
\wrong\pt(-1,2)
\wrong\pt(-2,1)
\wrong there are no points of intersection

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 9 **%
The expression $4x^2+4x-8$ can be factored as

\correct$4(x-1)(x+2)$
\wrong$4(x+1)(x-2)$
\wrong$4(x+4)(x-2)$
\wrong$4(x-1)(x-2)$
\wrong$4(x+1)(x+2)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 11 **%
The expression $\bigl[(x^2+2x+1)^{-3}\bigr]^{1/6}$ is the same as

\correct$\frac 1{\sqrt{x^2+2x+1}}$
\wrong$x^2+2x+1$
\wrong$\frac1{2x+1}$
\wrong$x^2+x+1$
\wrong$(x+1)^{3/2}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 12 **%
A curve in the $xy-$\!\!\!plane is the graph of a function if and only if each vertical line
intersects the graph how many times?

\correct no more than once
\wrong once
\wrong any number of times
\wrong only a finite number of times
\wrong no more than twice, as long as these points are on the same vertical line

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 13 **%
The $x-$\!\!\!intercept of the line $3y+2x-6=0$ is

\correct\pt(3,0)
\wrong\pt(2,0)
\wrong\pt(0,3)
\wrong\pt(0,2)
\wrong the line has no $x$ intercept

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 14 **%
Find the point on the graph of $y=x^2$ where the tangent line is parallel to the line
$2x+y-1=0$.

\correct \pt(-1,1)
\wrong \pt(0,0)
\wrong \pt(1,1)
\wrong \pt(2,4)
\wrong \pt(3,9)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 15 **%
If $f(x)=\frac4{\root3\of{x^2}}$, then $f'(8)=$

\correct $-\frac1{12}$
\wrong $-\frac2{3}$
\wrong $\frac1{4}$
\wrong $\frac1{48}$
\wrong does not exist

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 16 **%
$\d \lim_{x\to1}\frac{x^2-x-6}{x-3}$ is equal to

\correct 3
\wrong 5
\wrong 0
\wrong 0.001
\wrong does not exist

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 17 **%
$\dsize \lim_{x\to0}\frac{x(x+1)}{x^2}$ is equal to

\st
\wrong 0
\wrong 2
\wrong $\frac12$
\wrong $-1$
\correct does not exist

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 18 **%
$\dsize \lim_{x\to1}\frac{\sqrt x-1}{x-1}$ is equal to

\st
\wrong 0
\wrong 2
\correct $\frac12$
\wrong $-1$
\wrong does not exist

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 19 **%
If $f(x)=\sqrt x-\frac1{x^3}$ then $f'(x)$ is equal to

\correct $\frac1{2\sqrt x}+\frac3{x^4}$
\wrong $\sqrt x-\frac1{x^4}$
\wrong 0
\wrong $\sqrt x-\frac1{x^3}$
\wrong $x^\frac32-\frac1{x^4}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 20 **%
If $f(x)=\frac1{x^2+1}$ then $f'(x)$ is equal to

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 21 **%
If $f(x)=\frac1{x-1}$ then $\dsize \lim_{x\to\infty}f(x)$ is equal to

\correct 0
\wrong 1
\wrong $\infty$
\wrong $-1$
\wrong does not exist

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 22 **%
If $f(x)=\frac1{x+1}$ then $\dsize \lim_{h\to 0}\frac{f(1+h)-f(1)}h$ is equal to

\st
\wrong $\frac12$
\wrong $-\frac1{2(1+h)}$
\wrong $-\frac1{(x+1)^2}$
\correct $-\frac14$
\wrong does not exist

\end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %** 14 **%
For  $f(x)=x^3$, the equation of the tangent line at $x=1$ is

\correct $y-3x=0$
\wrong  $y=3(x-1)$
\wrong $y-1=3x$
\wrong $y-3x+2=0$
\wrong $y=3x-1$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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