Exam 1 Oct. 6, 2005 SHOW ALL WORK Math 25 Calculus I Either circle your answers or place on answer line. 3.) Find the equations of all vertical and horizontal asymptotes for f(x) = \frac{−5(x^{2}−4)(2x−9)}{(x−2)(x−3)^{2}}. Show ALL steps.

[15] horizontal asymptotes)

[15] vertical asymptotes)

Find the following derivatives:

[15] 1.) \frac{d}{dx}[3x · cos(x) · sin(2x)] Answer 1.)

[15] 2.) \frac{d}{dx}[cos(\sqrt{e^{x^{2}+1}})] Answer 2.)

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

f′(x) =

[12] 5.) Find the exact value of the following expression (SIMPLIFY your answer):

log_{4}10 + 3log_{4}2 − log_{4}5 + 4^{log_{4}3} + log_{4}1 =

[7] 6.) A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr^{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 = 10cm =

average rate of increast as r increase from 10cm to 12cm =

[8] 7.) Draw the graph of a function with the following properties: domain = [−5, 7], range = [−4, 6], f(−4) = 5 f′(x) = −2 if −3 < x < −1, f is continuous, but not differentiable at 0, f is not continuous at 2 f′(4) = 0