Chapter 2

Research and Development The table shows the amounts \(A\) (in billions of dollars per year) spent on R\&D in the United States from 1980 through 2004, where \(t\) is the year, with \(t=0\) corresponding to 1980 . Approximate the average rate of change of A during each period. $$ \begin{array}{ll}{\text { (a) } 1980-1985} & {\text { (b) } 1985-1990}&{\text { (c) } 1990-1995} \\ {\text { (d) } 1995-2000} & {\text { (e) } 1980-2004}&{\text { (f) } 1990-2004}\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline A & {63} & {72} & {81} & {90} & {102} & {115} & {120} \\\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {7} & {8} & {9} & {10} & {11} & {12} \\\ \hline A & {126} & {134} & {142} & {152} & {161} & {165} \\\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {13} & {14} & {15} & {16} & {17} & {18} \\ \hline A & {166} & {169} & {184} & {197} & {212} & {228} \\\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {19} & {20} & {21} & {22} & {23} & {24} \\ \hline A & {245} & {267} & {277} & {276} & {292} & {312} \\\ \hline\end{array} $$

use the given values to find \(d y / d t\) and \(d x / d t\). \(y=\sqrt{x}\) $$ \begin{array}{ll}{\text { (a) } \frac{d y}{d t}} & {x=4, \frac{d x}{d t}=3} \\\ {\text { (b) } \frac{d x}{d t}} & {x=25, \frac{d y}{d t}=2}\end{array} $$

Find \(d y / d x\) \(x y=4\)

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

find the second derivative of the function. $$ f(x)=9-2 x $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=x\left(x^{2}+3\right)} & {(2,14)} \end{array}$$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1) .\) $$ \text { (a) } y=x^{3 / 2} \quad \text { (b) } y=x^{3} $$

use the given values to find \(d y / d t\) and \(d x / d t\). $$ y=2\left(x^{2}-3 x\right) $$ $$ \begin{array}{ll}{\text { (a) } \frac{d y}{d t}} & {x=3, \frac{d x}{d t}=2} \\\ {\text { (b) } \frac{d x}{d t}} & {x=1, \frac{d y}{d t}=5}\end{array} $$

Find \(d y / d x\) \(3 x^{2}-y=8 x\)

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 second derivative of the function. $$ f(x)=4 x+15 $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {g(x)=(x-4)(x+2)} & {(4,0)} \end{array}$$

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. $$ f(t)=3 t+5 ;[1,2] $$

Find \(d y / d x\) \(y^{2}=1-x^{2}, 0 \leq x \leq 1\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=\left(4-x^{2}\right)^{-1}} \end{array}$$

find the second derivative of the function. $$ f(x)=x^{2}+7 x-4 $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=x^{2}\left(3 x^{3}-1\right)} & {(1,2)} \end{array}$$

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. $$ h(x)=2-x ;[0,2] $$

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

use the given values to find \(d y / d t\) and \(d x / d t\). $$ x^{2}+y^{2}=25 $$ $$ \begin{array}{ll}{\text { (a) } \frac{d y}{d t}} & {x=3, y=4, \frac{d x}{d t}=8} \\ {\text { (b) } \frac{d x}{d t}} & {x=4, y=3, \frac{d y}{d t}=-2}\end{array} $$

Find \(d y / d x\) \(4 x^{2} y-\frac{3}{y}=0\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=\left(x^{2}+1\right)^{4 / 3}} \end{array}$$

find the second derivative of the function. $$ f(x)=3 x^{2}+4 x $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=\left(x^{2}+1\right)(2 x+5)} & {(-1,6)} \end{array}$$

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. $$ h(x)=x^{2}-4 x+2 ;[-2,2] $$

Find the derivative of the function. $$ y=3 $$

Area The radius \(r\) of a circle is increasing at a rate of 3 inches per minute. Find the rates of change of the area when (a) \(r=6\) inches and (b) \(r=24\) inches.

Find \(d y / d x\) \(x^{2} y^{2}-2 x=3\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=\sqrt{5 x-2}} \end{array}$$

find the second derivative of the function. $$ g(t)=\frac{1}{3} t^{3}-4 t^{2}+2 t $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=\frac{1}{3}\left(2 x^{3}-4\right)} & {\left(0,-\frac{4}{3}\right)} \end{array}$$

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. $$ f(x)=x^{2}-6 x-1 ;[-1,3] $$

Find the derivative of the function. $$ f(x)=-2 $$

Volume The radius \(r\) of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the volume when (a) \(r=6\) inches and (b) \(r=24\) inches.

Find \(d y / d x\) \(x y^{2}+4 x y=10\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=\sqrt{1-x^{2}}} \end{array}$$

find the second derivative of the function. $$ f(x)=4\left(x^{2}-1\right)^{2} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=\frac{1}{7}\left(5-6 x^{2}\right)} & {\left(1,-\frac{1}{7}\right)} \end{array}$$

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. $$ f(x)=3 x^{4 / 3} ;[1,8] $$

Find the derivative of the function. $$ y=x^{4} $$

Area Let \(A\) be the area of a circle of radius \(r\) that is changing with respect to time. If \(d r / d t\) is constant, is \(d A / d t\) constant? Explain your reasoning.

Find \(d y / d x\) \(4 y^{2}-x y=2\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=(3 x+1)^{-1 / 2}} \end{array}$$

find the second derivative of the function. $$ f(t)=\frac{3}{4 t^{2}} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {g(x)=\left(x^{2}-4 x+3\right)(x-2)} & {(4,6)} \end{array}$$

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. $$ f(x)=x^{3 / 2} ;[1,4] $$

Find the derivative of the function. $$ h(x)=2 x^{5} $$

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.

Find \(d y / d x\) \(2 x y^{3}-x^{2} y=2\)

Identify the inside function, \(u=g(x),\) and the outside function, \(y=f(u) .\) $$\begin{array}{ll}{ { y=f(g(x))}} & {{ u=g(x)}} & { { y=f(u)}} \\\\{y=(x+2)^{-1 / 2}} \end{array}$$