91Ó°ÊÓ

In Exercises 5-8, show that the slopes of the graphs of \(f\) and \(f^{-1}\) are reciprocals at the indicated points. $$\begin{array}{ll}\text { Function } & \text { Point } \\\\\hline f(x)=x^{3} & \left(\frac{1}{2}, \frac{1}{8}\right)\\\f^{-1}(x)=\sqrt[3]{x} & \left(\frac{1}{8}, \frac{1}{2}\right)\end{array}$$

In Exercises 11-14, find \(d y / d x\) at the indicated point for the equation. $$ x=y^{3}-7 y^{2}+2,(-4,1) $$

In Exercises 15-28, find the derivative of the function. $$ y=x \arctan 2 x-\frac{1}{4} \ln \left(1+4 x^{2}\right) $$

Find an equation of the tangent line to the graph of \(g(x)=\arctan x\) when \(x=1\)

\mathrm{\\{} T a n g e n t ~ L i n e s ~ \(\quad\) The graph of \(f(x)=-\sin x\) has infinitely many tangent lines that pass through the origin. Use Newton's Method to approximate the slope of the tangent line having the greatest slope to three decimal places.

Find the derivative of the transcendental function. $$ f(\theta)=(\theta+1) \cos \theta $$

Famous Curves In Exercises 35-38, find the slope of the tangent line to the graph at the indicated point. Witch of Agnesi: \(\left(x^{2}+4\right) y=8\) Point: (2,1)

A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad. Let \(\theta\) be the angle of elevation of the shuttle and let \(s\) be the distance between the camera and the shuttle (as shown in the figure). Write \(\theta\) as a function of \(s\) for the period of time when the shuttle is moving vertically. Differentiate the result to find \(d \theta / d t\) in terms of \(s\) and \(d s / d t\).

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).

A population of 500 bacteria is introduced into a culture and grows in number according to the equation $$ P(t)=500\left(1+\frac{4 t}{50+t^{2}}\right )$$ where \(t\) is measured in hours. Find the rate at which the population is growing when \(t=2\).