Chapter 7

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

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(g(x)=\frac{x^{2}-4 x+4}{x^{2}-4}\)

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

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

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. $$ g(x)=3 x-1 $$

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(g(x)=\frac{x^{2}-9 x+20}{x^{2}-16}\)

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

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. $$ g(x)=x^{2}+5 x $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x^{2}-1}{x}\)

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

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{3 x}{x^{2}+4} \quad\left(-1,-\frac{3}{5}\right) $$

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. $$ y=t^{2}-6 $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{1}{x^{2}-4}\)

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 value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ g(x)=\frac{2 x+1}{x-5} $$

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

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x^{2}-1}{x+1}\)

Find the limit of (a) \(f(x)+g(x)\), (b) \(f(x) g(x)\), and (c) \(f(x) / g(x)\), as \(x\) approaches \(c\). $$ \begin{aligned} &\lim _{x \rightarrow c} f(x)=3 \\ &\lim _{x \rightarrow c} g(x)=9 \end{aligned} $$

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 value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{x+1}{x-1} $$

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

Find the limit of (a) \(f(x)+g(x)\), (b) \(f(x) g(x)\), and (c) \(f(x) / g(x)\), as \(x\) approaches \(c\). $$ \begin{aligned} &\lim _{x \rightarrow c} f(x)=\frac{3}{2} \\ &\lim _{x \rightarrow c} g(x)=\frac{1}{2} \end{aligned} $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{2}{1-x^{3}} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(t)=\frac{t^{2}-1}{t+4} $$

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. (a) \([0,1]\) (b) \([1,2]\) (c) \([2,3]\) (d) \([3,4]\)

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

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=x^{2}-2 x+1\)

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 limit of (a) \(\sqrt{f(x)}\), (b) \([3 f(x)]\), and (c) \([f(x)]^{2}\), as \(x\) approaches \(c\). $$ \lim _{x \rightarrow c} f(x)=16 $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ g(x)=\frac{4 x-5}{x^{2}-1} $$

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. $$ y=2 x^{3}-x^{2}+3 x-1 $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=3-2 x-x^{2}\)

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 x+4 ;(1,6) $$

Find the limit of (a) \(\sqrt{f(x)}\), (b) \([3 f(x)]\), and (c) \([f(x)]^{2}\), as \(x\) approaches \(c\). $$ \lim _{x \rightarrow c} f(x)=9 $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{8^{2}} $$

The height \(s\) (in feet) at time \(t\) (in seconds) of a silver dollar dropped from the top of the Washington Monument is given by \(s=-16 t^{2}+555\) (a) Find the average velocity on the interval \([2,3]\). (b) Find the instantaneous velocities when \(t=2\) and when \(t=3\) (c) How long will it take the dollar to hit the ground? (d) Find the velocity of the dollar when it hits the ground.

Find the derivative of the function. $$ y=4 t^{4 / 3} $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x}{x^{2}-1}\)

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{x^{2}} $$

A racecar travels northward on a straight, level track at a constant speed, traveling \(0.750\) kilometer in \(20.0\) seconds. The return trip over the same track is made in \(25.0\) seconds. (a) What is the average velocity of the car in meters per second for the first leg of the run? (b) What is the average velocity for the total trip?

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

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x-3}{x^{2}-9}\)

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

Find the marginal cost for producing \(x\) units. (The cost is measured in dollars.) $$ C=4500+1.47 x $$

Find the derivative of the function. $$ f(x)=4 \sqrt{x} $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x}{x^{2}+1}\)