91Ó°ÊÓ

Rate of Change Determine whether there exist any values of \(x\) in the interval \([0,2 \pi)\) such that the rate of change of \(f(x)=\sec x\) and the rate of change of \(g(x)=\csc x\) are equal.

Satellites When satellites observe Earth, they can scan only part of Earth's surface. Some satellites have sensors that can measure the angle \(\theta\) shown in the figure. Let \(h\) represent the satellite's distance from Earth's surface, and let \(r\) represent Earth's radius. (a) Show that \(h=r(\csc \theta-1)\) (b) Find the rate at which \(h\) is changing with respect to \(\theta\) when \(\theta=30^{\circ} .\) (Assume \(r=3960\) miles.)

True or False? In Exercises \(87-92,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ g(x)=3 f(x), \text { then } g^{\prime}(x)=3 f^{\prime}(x) $$

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is continuous at a point, then it is differentiable at that point.

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 \(t=3\) .

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are $$\begin{array}{l}{P_{1}(x)=f^{\prime}(a)(x-a)+f(a) \text { and }} \\\ {P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)}\end{array}$$ In Exercises 123 and \(124,\) (a) find the specified linear and quadratic approximations of \(f,(b)\) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\) . $$ f(x)=\tan x ; \quad a=\frac{\pi}{4} $$

Differential Equations In Exercises \(125-128\) , verify that the function satisfies the differential equation. $$ \text{Function} \quad \text{Differential Equation} $$ $$ y=3 \cos x+\sin x \quad y^{\prime \prime}+y=0 $$

True or False? In Exercises \(129-134\) , determine whether the statement is true or false. If is false, explain why or give an example that shows it is false. The second derivative represents the rate of change of the first derivative.