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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



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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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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[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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