Chapter 3

Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of \(15 \mathrm{m} / \mathrm{sec} .\) Because the acceleration of gravity at the planet's surface was \(g_{s} \mathrm{m} / \mathrm{sec}^{2}\), the explorers expected the ball bearing to reach a height of \(s=15 t-(1 / 2) g_{s} t^{2} \mathrm{m}\) \(t\) sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of \(g_{s} ?\)

Find the first and second derivatives. $$r=\frac{1}{3 s^{2}}-\frac{5}{2 s}$$

Find the indicated derivatives. $$\frac{d p}{d q}\( if \)p=q^{3 / 2}$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$f(x)=x^{2}+1 \quad \quad (2,5)$$

Find the values. $$\cot \left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$$

A cube's surface area increases at the rate of 72 in \(^{2} /\) sec. At what rate is the cube's volume changing when the edge length is \(x=3\) in?

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=\frac{x}{x+1}, \quad a=1.3$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\left(\frac{\sqrt{x}}{2}-1\right)^{-10}$$

Use implicit differentiation to find \(d y / d x\). $$x^{4}+\sin y=x^{3} y^{2}$$

Find \(d y / d x\). $$y=\frac{\cos x}{1+\sin x}$$

A 45 -caliber bullet shot straight up from the surface of the moon would reach a height of \(s=832 t-2.6 t^{2} \mathrm{ft}\) after \(t\) sec. On Earth, in the absence of air, its height would be \(s=832 t-16 t^{2} \mathrm{ft}\) after \(t\) sec. How long will the bullet be aloft in each case? How high will the bullet go?

Find the first and second derivatives. $$r=\frac{12}{\theta}-\frac{4}{\theta^{3}}+\frac{1}{\theta^{4}}$$

Find the indicated derivatives. $$\frac{d z}{d w}\( if \)z=\frac{1}{\sqrt{w^{2}-1}}$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$f(x)=x-2 x^{2}, \quad\quad(1,-1)$$

Volume The radius \(r\) and height \(h\) of a right circular cylinder are related to the cylinder's volume \(V\) by the formula \(V=\pi r^{2} h\). a. How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? b. How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? c. How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(t^{2}\right)$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\left(\frac{x^{2}}{8}+x-\frac{1}{x}\right)^{4}$$

Use implicit differentiation to find \(d y / d x\).. $$y \sin \left(\frac{1}{y}\right)=1-x y$$

Find \(d y / d x\). $$y=\frac{4}{\cos x}+\frac{1}{\tan x}$$

Had Galileo dropped a cannonball from the Tower of Pisa, 179 ft above the ground, the ball's height above the ground \(t\) sec into the fall would have been \(s=179-16 t^{2}\) a. What would have been the ball's velocity, speed, and acceleration at time \(t ?\) b. About how long would it have taken the ball to hit the ground? c. What would have been the ball's velocity at the moment of impact?

Find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$y=\left(3-x^{2}\right)\left(x^{3}-x+1\right)$$

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$f(x)=x+\frac{9}{x}, \quad x=-3$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$g(x)=\frac{x}{x-2}, \quad\quad(3,3)$$

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow-1^{+}} \cos ^{-1} x$$

The radius \(r\) and height \(h\) of a right circular cone are related to the cone's volume \(V\) by the equation \(V=(1 / 3) \pi r^{2} h\). a. How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? b. How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? c. How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. Show that the linearization of \(f(x)=(1+x)^{k}\) at \(x=0\) is \(L(x)=1+k x\)

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(t^{3 / 2}\right)+\sqrt{t}$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\sqrt{3 x^{2}-4 x+6}$$

Use implicit differentiation to find \(d y / d x\). $$x \cos (2 x+3 y)=y \sin x$$

Find \(d y / d x\). $$y=\frac{\cos x}{x}+\frac{x}{\cos x}$$

Find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$y=(2 x+3)\left(5 x^{2}-4 x\right)$$

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$k(x)=\frac{1}{2+x}, \quad x=2$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$g(x)=\frac{8}{x^{2}}, \quad\quad(2,2)$$

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \tan ^{-1} x$$

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. Show that the linearization of \(f(x)=(1+x)^{k}\) at \(x=0\) is \(L(x)=1+k x\)

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\sec (\tan x)$$

Use implicit differentiation to find \(d y / d x\). $$e^{2 x}=\sin (x+3 y)$$

Find \(d y / d x\). $$y=(\sec x+\tan x)(\sec x-\tan x)$$

Find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$y=\left(x^{2}+1\right)\left(x+5+\frac{1}{x}\right)$$

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$s=t^{3}-t^{2}, \quad t=-1$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$h(t)=t^{3}, \quad\quad(2,8)$$

The power \(P\) (watts) of an electric circuit is related to the circuit's resistance \(R\) (ohms) and current \(I\) (amperes) by the equation \(P=R I^{2}\). a. How are \(d P / d t, d R / d t\), and \(d I / d t\) related if none of \(P, R,\) and \(I\) are constant? b. How is \(d R / d t\) related to \(d I / d t\) if \(P\) is constant?

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow-\infty} \tan ^{-1} x$$

Use the linear approximation \((1+x)^{k} \approx 1+k x\) to find an approximation for the function \(f(x)\) for values of \(x\) near zero. a. \(f(x)=(1-x)^{6}\) b. \(f(x)=\frac{2}{1-x}\) c. \(f(x)=\frac{1}{\sqrt{1+x}}\) d. \(f(x)=\sqrt{2+x^{2}}\) e. \(f(x)=(4+3 x)^{1 / 3}\) \(f(x)=\sqrt[3]{\left(1-\frac{x}{2+x}\right)^{2}}\)

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\sin x)$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\cot \left(\pi-\frac{1}{x}\right)$$

Use implicit differentiation to find \(d y / d x\). $$e^{x^{2} y}=2 x+2 y$$

Find \(d y / d x\). $$y=x^{2} \cos x-2 x \sin x-2 \cos x$$

Find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$y=\left(1+x^{2}\right)\left(x^{3 / 4}-x^{-3}\right)$$

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable. $$y=\frac{x+3}{1-x}, x=-2$$