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The following are the verbal remarks that were made at the beginning of the exam:

For problem 3, find the vertical and horizontal asymptotes using limits and show all steps.  The answer alone does 
not give  full credit. For problem 6, you should write down the unit of the rate of the increase given that 
the unit of radius is cm.
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Exam 1 Oct. 6, 2005 \hfil SHOW ALL WORK \hfil
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Math 25 Calculus I \hfil Either circle your answers or place on answer line. \hfil


Find the following derivatives:

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[15]~  1.) ${d \over dx}[3x \cdot cos(x) \cdot  sin(2x)]$

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\centerline{Answer 1.) $\underline{\hskip 4in}$}
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[15]~  2.) ${d \over dx}[cos(\sqrt{e^{x^2 + 1}})]$

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\centerline{Answer 2.) $\underline{\hskip 4in}$}

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3.) Find the equations of all vertical and horizontal asymptotes for $f(x) = {-5(x^2 - 4)(2x - 9) \over (x-2)(x - 3)^2}$.
Show ALL steps.
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\centerline{[15]~ horizontal asymptotes) $\underline{\hskip 4in}$}

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\centerline{[15]~ vertical asymptotes) $\underline{\hskip 4in}$}

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[13] 4.) Find the derivative of $f(x) = {1 \over x}$ by using the definition
of derivative.


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\centerline{$f'(x) = \underline{\hskip 1in}$}

[12] 5.) Find the exact value of the following expression (SIMPLIFY your answer): 
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\centerline{$log_4 10 + 3log_4 2 - log_4 5 +  4^{log_4 3} + log_4 1 $ =  
$\underline{\hskip 1in}$}

\vfill \eject [7] 6.)  A spherical balloon is being inflated.  Find the rate of increase of the surface area ($S = 4\pi r^2$) with respect 
to the radius $r$ when $r$ is 10cm.  (note your answer should include units).  Find the average rate of increase of the surface area with 
respect to radius as $r$ increases from 10cm to 12cm.

rate of increast at $r = 10$cm =  $\underline{\hskip 1in}$

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average rate of increast as $r$ increase from 10cm to 12cm  =  $\underline{\hskip 1in}$
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[8] 7.)  Draw the graph of a function with the following properties:~~ \hfil \break
domain = $[-5, 7]$, range = $[-4, 6]$, \hfil \break
$f(-4) = 5$ $f'(x) = -2$ if  $-3 < x < -1$,  \hfil \break
$f$ is continuous, but not differentiable at 0,  \hfil \break
$f$ is not continuous at 2 \hfil \break
$f'(4) = 0$


\centerline{\includegraphics[width=36ex]{graph}}


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[4]  If  $f(x) = {1 \over x}$, find the slope of the secant line through the po
Find the slope of the tangent line at $x = 4$


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