Chapter 2

find the second derivative of the function. $$ g(t)=32 t^{-2} $$

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

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. $$ f(x)=\frac{1}{x} ;[1,4] $$

Find the derivative of the function. $$ f(x)=4 x+1 $$

A spherical balloon is inflated with gas at a rate of 10 cubic feet per minute. How fast is the radius of the balloon changing at the instant the radius is (a) 1 foot and (b) 2 feet?

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{2}, u=4 x+7 $$

find the second derivative of the function. $$ f(x)=3\left(2-x^{2}\right)^{3} $$

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

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. $$ f(x)=\frac{1}{\sqrt{x}} ;[1,4] $$

Find the derivative of the function. $$ g(x)=3 x-1 $$

Volume The radius \(r\) of a right circular cone is increasing at a rate of 2 inches per minute. The height \(h\) of the cone is related to the radius by \(h=3 r\). Find the rates of change of the volume when (a) \(r=6\) inches and (b) \(r=24\) inches.

Find \(d y / d x\) \(\frac{x y-y^{2}}{y-x}=1\)

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{3}, u=3 x^{2}-2 $$

find the second derivative of the function. $$ f(x)=x \sqrt[3]{x} $$

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

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. $$ g(x)=x^{4}-x^{2}+2 ;[1,3] $$

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

A company that manufactures sport supplements calculates that its costs and revenue can be modeled by the equations \(C=125,000+0.75 x\) and \(R=250 x-\frac{1}{10} x^{2}\) where \(x\) is the number of units of sport supplements produced in 1 week. If production in one particular week is 1000 units and is increasing at a rate of 150 units per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

Find \(d y / d x\) \(\frac{x+y}{2 x-y}=1\)

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=\sqrt{u}, u=3-x^{2} $$

find the second derivative of the function. $$ y=\left(x^{3}-2 x\right)^{4} $$

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

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

Find the derivative of the function. $$ y=t^{2}-6 $$

cost, Revenue, and Profit A company that manufactures pet toys calculates that its costs and revenue can be modeled by the equations \(C=75,000+1.05 x\) and \(R=500 x-\frac{x^{2}}{25}\) where \(x\) is the number of toys produced in 1 week. If production in one particular week is 5000 toys and is increasing at a rate of 250 toys per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

Find \(d y / d x\) \(\frac{2 x+y}{x-5 y}=1\)

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=2 \sqrt{u}, u=5 x+9 $$

find the second derivative of the function. $$ y=4\left(x^{2}+5 x\right)^{3} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\ {f(x)=\frac{3 x}{x^{2}+1}} & {\left(-1,-\frac{3}{5}\right)} \end{array}$$

Find the derivative of the function. $$ f(t)=-3 t^{2}+2 t-4 $$

Volume All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{2}+y^{2}=16\) \(\quad\) \((0,4)\)

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{2 / 3}, u=5 x^{4}-2 x $$

find the second derivative of the function. $$ f(x)=\frac{x+1}{x-1} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\\{g(x)=\frac{2 x+1}{x-5}} & {(6,13)} \end{array}$$

Find the derivative of the function. $$ y=x^{3}-9 x^{2}+2 $$

Surface Area All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the surface area changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{2}-y^{2}=25\) \(\quad\) \((5,0)\)

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{-1}, u=x^{3}+2 x^{2} $$

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

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

Medicine The effectiveness \(E\) (on a scale from 0 to 1 ) of a pain-killing drug \(t\) hours after entering the bloodstream is given by $$ E=\frac{1}{27}\left(9 t+3 t^{2}-t^{3}\right), \quad 0 \leq t \leq 4.5 $$ Find the average rate of change of \(E\) on each indicated interval and compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ \begin{array}{llll}{\text { (a) }[0,1]} & {\text { (b) }[1,2]} & {\text { (c) }[2,3]} & {\text { (d) }[3,4]}\end{array} $$

Find the derivative of the function. $$ s(t)=t^{3}-2 t+4 $$

Moving Point A point is moving along the graph of \(y=x^{2}\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x .\) $$ \text { (a) } x=-3 \quad \text { (b) } x=0 \quad \text { (c) } x=1 \quad \text { (d) } x=3 $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(y+x y=4 \quad(-5,-1)\)

find the second derivative of the function. $$ y=x^{2}\left(x^{2}+4 x+8\right) $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6-2 x ;(2,2) $$

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

Chemistry: Wind Chill At \(0^{\circ}\) Celsius, the heat loss \(H\) (in kilocalories per square meter per hour) from a person's body can be modeled by $$ H=33(10 \sqrt{v}-v+10.45) $$ where \(v\) is the wind speed (in meters per second). (a) Find \(\frac{d H}{d v}\) and interpret its meaning in this situation. (b) Find the rates of change of \(H\) when \(v=2\) and when \(v=5\)

Find the derivative of the function. $$ y=2 x^{3}-x^{2}+3 x-1 $$