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\flushpar{\bf{Math 1251 Exam 2 May 23, 1997}} \qquad Your Name 
{$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$} 
\newline
{\bf{Professor Stanton}}
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\flushpar{\it{Directions}}: 
{\eightpoint{This exam has 5 problems with the point values indicated. 
Do all five problems. You may use your notes, books, and calculators. 
If you use a table (or calculator) to find your answer, write down the word ``table'' (or ``calculator'') next to your answer. If your answer is correct, you will receive full credit for that problem, otherwise you will be 
awarded 0 points. This means if you copy a minus sign incorrectly you will be given 0 points.}}
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(15) 1. Suppose that $y$ is given implicitly as a function of $x$ by 
$\displaystyle{y^7+e^{x} y=e^2+\frac{x}{2}}$.
Find the equation of the best linear approximation to $y$ 
near the point $(2,1)$.
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(30) 2. Find the following derivatives and limits.\newline
(a) If $\displaystyle{y=ln(x^2+arctan(4x))}$, 
then $\displaystyle{\frac{dy}{dx}=?}$\newline
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(b) 
$\displaystyle{
\lim_{x\to 0} \frac{sin(xe^{-50})}{tan(2x)}}
$
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(c) $\displaystyle{\lim_{x\to\infty}
\frac{x^{47}-ln(x)^2-e^x}{e^{-4x}-x^{66}-e^{x}}}$
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(20) 3. Find the location of all local max-mins, 
and find the absolute maximum of the function 
$\displaystyle{f(x)=\frac{x^4}{4}-x^2}$ on the interval $[-1,3]$.
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(15) 4. Suppose that a particle moves along the hyperbola $xy=1$. 
If $\displaystyle \frac{dx}{dt}=3$ when the particle passes (2,1/2), 
what is $\displaystyle\frac{dy}{dt}$ at this same time?
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(20) 5. True or False. (If true give a proof, if false give a counterexample.
 An answer alone is worth zero.)\newline
(a)  If $f'(x)=e^{-x^2}$ for all real $x$, then $f(x)$ 
 has no local maxima.
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(b)  If $f'(0)=0$, then there must be a negative number $c$ and a positive number $d$ such that $\displaystyle{f(c)=f(d)}$.
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