91Ó°ÊÓ

Area The length of a rectangle is given by \(6 t+5\) and its height is \(\sqrt{t}\) , where \(t\) is time in seconds and the dimensions are in centimeters. Find the rate of change of the area with respect to time.

Volume The radius of a right circular cylinder is given by \(\sqrt{t+2}\) and its height is \(\frac{1}{2} \sqrt{t},\) where \(t\) is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.

Determining Differentiability In Exercises 89 and \(90,\) determine whether the function is differentiable at \(x=2\) . $$f(x)=\left\\{\begin{array}{ll}{\frac{1}{2} x+2,} & {x<2} \\ {\sqrt{2 x},} & {x \geq 2}\end{array}\right.$$

If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

Vertical Motion In Exercises 97 and \(98,\) use the position function \(s(t)=-4.9 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds?

A rock is dropped from the edge of a cliff that is 214 meters above water. (a) Determine the position and velocity functions for the rock. (b) Determine the average velocity on the interval \([2,5]\) . (c) Find the instantaneous velocities when \(t=2\) and \(t=5\) . (d) Find the time required for the rock to reach the surface of (e) Find the velocity of the rock at impact.

Depreciation The value \(V\) of a machine \(t\) years after it is purchased is inversely proportional to the square root of \(t+1 .\) The initial value of the machine is \(\$ 10,000 .\) (a) Write \(V\) as a function of \(t\) (b) Find the rate of depreciation when \(t=1\) (c) Find the rate of depreciation when \(t=3\) .

Conjecture Let \(f\) be a differentiable function of period \(p .\) (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x) .\) Is the function \(g^{\prime}(x)\) periodic? Verify your answer.