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\flushpar{\bf{Math 1251 Exam 1 April 17, 1997}} \qquad Your Name 
{$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$} 
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{\bf{Professor Stanton}}
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\flushpar{\it{Directions}}: 
{\eightpoint{This exam has 4 problems with the point values indicated. 
Do all four 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. An answer alone with no justification is worth 0 points.}}
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(15) 1. Find the equation of the tangent line to the curve $y=2x+1/x$ at the point (1,3).
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(50) 2. Find the following derivatives and limits.\newline
(a) If $y=(3x-2)^{45}$, then $\frac{dy}{dx}=?$\newline
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(b) If $f(x)=sin(4-1/\root{3}\of{x})$, then $f'(x)=?$\newline
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(c) If $f(x)=x\sqrt{1-x^2}$, then $f'(x)=?$
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(d) $$\lim_{t\to \infty} \frac{|t+2|+sin(t^2)}{5-4t}=$$
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(e) $$\lim_{x\to 3} \frac{x^2-3x}{x-3} =$$
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(15) 3. We motivated the limit definition of the derivative 
$$
f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}
\tag*
$$ 
by tangent lines. Draw an appropriate picture and explain 
in your own words how the limit definition (*) is related to a tangent line. 
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(20) 4. True or False. (You must give at least one reason for your answer in a complete sentence. An answer alone is worth zero.)\newline
(a) If $f(x)\ne 0$ and $f'(x)\ne 0$, then 
$\frac{d}{dx}(1/f(x))=1/f'(x)$. 
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(b) If $f(x)$ is differentiable for all real $x$, and with 
$|f(x)|\le 1$ for all real $x$, then
$|f'(x)|\le 1$ for all real $x$.
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