\input ignore.tex %This line causes the mce macros to be ignored.

\preface
\magnification=1200

\def\a{{\bf a}}
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\def\c{{\bf c}}
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\def\0{{\bf 0}}
\def\F{{\bf F}}
\def\N{{\bf N}}
\def\T{{\bf T}}

\def\problemskip{\vskip0.5in}

\title
\noindent
{\bf Math 119}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl February 10, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}


\def\answerskip{\vskip0.25in}


\note %1
\problem
Find the equation of the line parallel to the line
 $2x+5y=6$ which passes through the point $(-1,1)$.
\correct
$y=-{2 \over 5}x + {3\over 5}$
\wrong
$y={3\over 5}x + -{2 \over 5}$
\wrong
$y=-x+1$
\wrong
$y=-{3\over 5}x + -{2\over 5}$
\wrong
$y={2\over 5}x + {2\over 5}$



\note %2
\problem
Find the domain of the function
\hskip 3pt  $y=\sqrt {x^2-3x+2}$.
\correct
$\{x : x\le 1 \quad {\rm or}\quad x\ge 2 \}$
\wrong
$\{x : x\le -2 \quad {\rm or}\quad x\ge -1 \}$
\wrong
$\{x : 1\le x\le 2 \}$
\wrong
$\{x : -2 < x < -1 \}$
\wrong
$\{x : x< 1 \quad {\rm or}\quad x> 2 \}$


\note %3
\problem
Which {\it equation} does {\bf not} determine a {\it function}?
\correct
$x^2+y^2=4$
\wrong
${x\over y}=2, \quad {\rm for}\quad y\ne 0$ 
\wrong
$2x+3y=6$
\wrong
$y=\sqrt{x^3-1}, \quad {\rm for}\quad x\ge 0$
\wrong
$5y=\tan 3x$ 


\note %4
\problem
On what intervals is the following function $f$ continuous?
$$f(x)=\sqrt{x+2} + {{x+1}\over {x-1}}+ |x-2|$$
\correct
$[-2, 1) \quad {\rm and} \quad (1, \infty )$
\wrong
$[-2, \infty )$
\wrong
$(-2, 2)$
\wrong
$(-2, -1) \quad {\rm and} \quad (1, \infty )$
\wrong
$(-\infty , 2]$


\note %5
\problem
If $f(x)=x^2$, which one is the possible function $g$ such that
$(f\circ g)(x)=x^2-10x+25$?
\correct
$g(x)=x-5$
\wrong
$g(x)=x+5$
\wrong
$g(x)=x^2 +5$
\wrong
$g(x)=x^2 -5$
\wrong
$g(x)=-x-5$


\note %6
\problem
Which one has the well-defined limit?
\correct
$\lim_{x\rightarrow 1} {{x-1}\over {\sqrt{x} -1}}$
\wrong
$\lim_{x\rightarrow 4}\sqrt{x^2-25}$
\wrong
$\lim_{x\rightarrow -1}\sqrt{x+1}$
\wrong
$\lim_{x\rightarrow 0}{1\over{x^4}}$
\wrong
All of them are well-defined.


\note %7
\problem
For what value of $\alpha$ is the function
$$
f(x)=\cases{{{x^3 -1}\over {x-1}} &if $x\ne 1$;\cr
                        \alpha   &if $x=1$ \cr}
$$        
continuous at $x=1$?
\correct
$3$
\wrong
$0$
\wrong
$1$
\wrong
$-1$
\wrong
It cannot be continous for any $\alpha$.


\note %8
\problem
If the distance of a particle travels is given by 
$s(t)=t^3+t^2+6$ kilometers after $t$ hours, how fast is it
traveling (in km/hr) after 2 hours?
\correct
$16$
\wrong
$5$
\wrong
$7$
\wrong
$17$
\wrong
$18$

\note %9
\problem
Let
$$f(x) = \cases{0 &if $x<0$;\cr
             x &if $0\le x\le 1$;\cr
             x^3 &if $x>1$\cr}$$
Which of the following statements is true about the function  $f$?  
\correct It is continuous at every  $x$, and it has 
a derivative for all  $x$  except  $x = 0$ and $1$
\wrong It is continuous at all  $x$  except  $x = 0$ and $1$
\wrong It is continuous at every  $x$, and it has 
a derivative for all  $x$  except  $x = 0$
\wrong It is continuous at all  $x$  except  $x = 0$ and $1$, and 
it has a derivative for all  $x$  except  $x = 1$
\wrong It is continuous at every  $x$  except  $x = 1$, and it 
has a derivative for all  $x$  except  $x = 1$


\note %10
\problem
Find $\lim _{x\rightarrow 3^{-}} {{|x-3|}\over{-x+3}}$
\correct
$1$
\wrong
$-1$
\wrong
$0$
\wrong
$2$
\wrong
It does not exist.

\note %11
\problem
Find the graph of $y=\cos { x\over 2}$.
\correct
\hskip 5cm
\wrong
\hskip 5cm
\wrong
\hskip 5cm
\wrong
\hskip 5cm
\wrong
\hskip 5cm

\note %12
\problem
If $\lim _{x\rightarrow 1} f(x)g(x)=-1$ and $\lim _{x\rightarrow 1}
(f(x)+g(x))=0$, find all possible $\lim _{x\rightarrow 1} f(x)$. (Assume that $\lim _{x\rightarrow 1} f(x)$
and $\lim _{x\rightarrow 1} g(x)$ exist.)
\correct
$\pm 1$
\wrong
$1$
\wrong
$-1$
\wrong
$0$
\wrong
$2$

\note %13
\problem
Find the equation of the tangent line to 
$y=x\sqrt{x} +1$ at $(1,2)$.
\correct
$y={3\over 2}x+ {1\over 2}$
\wrong
$y=-{3\over 2}x+ {1\over 2}$
\wrong
$y={2\over 3}x$
\wrong
$y={2\over 3}x+ {1\over 2}$
\wrong
$y=-{2\over 3}x$

\note %14
\problem
Which statement is false?
\correct If $f$ and $g$ are differentiable, then
${d\over dx}[f(x)g(x)]=f'(x)g'(x)$
\wrong If $f$ is  differentiable at $a$, then $f$ is 
continuous at $a$
\wrong 
$\lim _{x\rightarrow a} {{f(x)-f(a)}\over {x-a}}
=\lim _{h\rightarrow 0} {{f(a+h)-f(a)}\over h}$
\wrong If $p$ is a polyomial, then 
$\lim _{x\rightarrow b} p(x)=p(b)$
\wrong If $f'(a)$ exists, then 
$\lim _{x\rightarrow a} f(x)=f(a)$


\note %15
\problem {\bf (Partial Credit)}\quad
Let $f(x)={|x|\over x}$ and $g(x)=x^2$. Find $f\circ g$ and
$g\circ f$, and determine the domain of each.

\note %16
\problem {\bf (Partial Credit)}\quad
Explain why $\lim _{x\rightarrow 0} {1\over x^3}$ does not exist.

\appendix
That's all folks!

\end
