91Ó°ÊÓ

Suppose that an object moves along a coordinate line so that its directed distance from the origin after \(t\) seconds is \(\sqrt{2 t+1}\) feet. (a) Find its instantaneous velocity at \(t=\alpha, \alpha>0\). (b) When will it reach a velocity of \(\frac{1}{2}\) foot per second? (see Example \(5 .)\)

Find \(f^{\prime \prime}(2)\). $$ f(s)=s\left(1-s^{2}\right)^{3} $$

Find \(D_{x} y .\) \(y=\cos \left(\frac{3 x^{2}}{x+2}\right)\)

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). \(F(x)=\frac{6}{x^{2}+1}\)

In Problems \(5-29\), find the indicated derivative by using the rules that we have developed. $$ D_{\theta}^{2}\left(\sin \theta+\cos ^{3} \theta\right) $$

A light in a lighthouse 1 kilometer offshore from a straight shoreline is rotating at 2 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point \(\frac{1}{2}\) kilometer from the point opposite the lighthouse?

Find \(D_{x} y\) using the rules of this section. $$y=\pi x^{7}-2 x^{5}-5 x^{-2}$$

Find \(D_{x} y .\) \(y=x^{2} \cos x\)

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). \(F(x)=\frac{x-1}{x+1}\)

An aircraft spotter observes a plane flying at a constant altitude of 4000 feet toward a point directly above her head. She notes that when the angle of elevation is \(\frac{1}{2}\) radian it is increasing at a rate of \(\frac{1}{10}\) radian per second. What is the speed of the airplane?