91Ó°ÊÓ

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=(6 x-5)^{4}} \end{array}$$

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=\left(x^{2}-2 x+3\right)^{3}} \end{array}$$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1) .\) $$ \text { (a) } y=x^{-1 / 2} \quad \text { (b) } y=x^{-2} $$

Volume Let \(V\) be the volume of a sphere of radius \(r\) that is changing with respect to time. If \(d r / d t\) is constant, is \(d V / d t\) constant? Explain your reasoning.

A spherical balloon is inflated with gas at a rate of 10 cubic feet per minute. How fast is the radius of the balloon changing at the instant the radius is (a) 1 foot and (b) 2 feet?

Volume The radius \(r\) of a right circular cone is increasing at a rate of 2 inches per minute. The height \(h\) of the cone is related to the radius by \(h=3 r\). Find the rates of change of the volume when (a) \(r=6\) inches and (b) \(r=24\) inches.

A company that manufactures sport supplements calculates that its costs and revenue can be modeled by the equations \(C=125,000+0.75 x\) and \(R=250 x-\frac{1}{10} x^{2}\) where \(x\) is the number of units of sport supplements produced in 1 week. If production in one particular week is 1000 units and is increasing at a rate of 150 units per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ g(x)=x^{3}-1 ;[-1,1] $$

Velocity The height \(s(\text { in feet) at time } t\) (in seconds) of a silver dollar dropped from the top of the Washington Monument is given by $$ s=-16 t^{2}+555 $$ (a) Find the average velocity on the interval \([2,3] .\) (b) Find the instantaneous velocities when \(t=2\) and when \(t=3 .\) (c) How long will it take the dollar to hit the ground? (d) Find the velocity of the dollar when it hits the ground.

Physics: Velocity A racecar travels northward on a straight, level track at a constant speed, traveling 0.750 kilometer in 20.0 seconds. The return trip over the same track is made in 25.0 seconds. (a) What is the average velocity of the car in meters per second for the first leg of the run? (b) What is the average velocity for the total trip?