91Ó°ÊÓ

Differentiate each function $$ y=\sqrt{1+8 x} $$

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 3} \frac{x^{2}-8}{x-2} $$

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(F(x)=\frac{x^{3}+27}{x+3}\)

Find \(f^{\prime \prime}(x)\) $$ f(x)=\left(x^{3}+2 x\right)^{6} $$

Find an equation of the tangent line to the graph of \(f(x)=4-x^{2}\) at (a) (-1,3) (b) (0,4) (c) (5,-21) .

Differentiate each function $$ f(x)=-3 x(5 x+4)^{6} $$

Differentiate each function. \(G(x)=(5 x-4)^{2}\)

Find each derivative. $$ \frac{d}{d x}\left(5 x^{2}-7 x+3\right) $$

At the beginning of a trip, the odometer on a car reads \(30,680,\) and the car has a full tank of gas. At the end of the trip, the odometer reads \(31,077 .\) It takes 13.5 gal of gas to refill the tank. a) What is the average rate at which the car was traveling, in miles per gallon? b) What is the average rate of gas consumption in gallons per mile?

In \(t\) seconds, an object dropped from a certain height will fall \(s(t)\) feet, where $$ s(t)=16 t^{2} $$ a) Find \(s(5)-s(3)\). b) What is the average rate of change of distance with respect to time during the period from 3 to 5 sec? This is known as average velocity, or speed.