Chapter 9

Suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=\frac{x f(x)}{x+g(x)}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{2}{x^{2}}-\frac{3}{x^{1 / 3}}\)

At a temperature of \(20^{\circ} \mathrm{C}\), the volume \(V\) (in liters) of \(1.33 \mathrm{~g}\) of \(\mathrm{O}_{2}\) is related to its pressure \(p\) (in atmospheres) by the formula \(V=1 / p\) a. What is the average rate of change of \(V\) with respect to \(p\) as \(p\) increases from \(p=2\) to \(p=3 ?\) b. What is the rate of change of \(V\) with respect to \(p\) when \(p=2 ?\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{-}} \frac{1+x}{1-x}\)

Find the indicated limit. \(\lim _{x \rightarrow 2} \frac{2 x+1}{x+2}\)

Find the derivative of each function. \(f(x)=\left(\frac{x+1}{x-1}\right)^{5}\)

Suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=\frac{f(x) g(x)}{f(x)-g(x)}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{3}{x^{3}}+\frac{4}{\sqrt{x}}+1\)

The total cost \(C(x)\) (in dollars) incurred by Aloha Company in manufacturing \(x\) surfboards a day is given by $$ C(x)=-10 x^{2}+300 x+130 \quad(0 \leq x \leq 15) $$ a. Find \(C^{\prime}(x)\). b. What is the rate of change of the total cost when the level of production is ten surfboards a day?

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{+}} \frac{1+x}{1-x}\)

Find the indicated limit. \(\lim _{x \rightarrow 1} \frac{x^{3}+1}{2 x^{3}+2}\)

Find the derivative of each function. \(s(t)=\left(\frac{t}{2 t+1}\right)^{3 / 2}\)

In Exercises 35-38, find the derivative of each of the given functions and evaluate \(f^{\prime}(x)\) at the given value of \(x\). \(f(x)=(2 x-1)\left(x^{2}+3\right) ; x=1\)

Find the derivative of the function \(f\) by using the rules of differentiation. Let \(f(x)=2 x^{3}-4 x\). Find: a. \(f^{\prime}(-2)\) b. \(f^{\prime}(0)\) c. \(f^{\prime}(2)\)

The quarterly profit (in thousands of dollars) of Cunningham Realty is given by $$ P(x)=-\frac{1}{3} x^{2}+7 x+30 \quad(0 \leq x \leq 50) $$ where \(x\) (in thousands of dollars) is the amount of money Cunningham spends on advertising per quarter. a. Find \(P^{\prime}(x)\). b. What is the rate of change of Cunningham's quarterly profit if the amount it spends on advertising is \(\$ 10,000 /\) quarter \((x=10)\) and \(\$ 30,000 /\) quarter \((x=30) ?\)

Find the indicated limit. \(\lim _{x \rightarrow 2} \sqrt{x+2}\)

Find the derivative of each function. \(g(s)=\left(s^{2}+\frac{1}{s}\right)^{3 / 2}\)

Find the derivative of each of the given functions and evaluate \(f^{\prime}(x)\) at the given value of \(x\). \(f(x)=\frac{2 x+1}{2 x-1} ; x=2\)

Find the derivative of the function \(f\) by using the rules of differentiation. Let \(f(x)=4 x^{5 / 4}+2 x^{3 / 2}+x\). Find: a. \(f^{\prime}(0)\) b. \(f^{\prime}(16)\)

The demand function for Sportsman \(5 \times 7\) tents is given by $$ p=f(x)=-0.1 x^{2}-x+40 $$ where \(p\) is measured in dollars and \(x\) is measured in units of a thousand. a. Find the average rate of change in the unit price of a tent if the quantity demanded is between 5000 and 5050 tents; between 5000 and 5010 tents. b. What is the rate of change of the unit price if the quantity demanded is \(5000 ?\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow-3^{+}} \frac{\sqrt{x+3}}{x^{2}+1}\)

Find the indicated limit. \(\lim _{x \rightarrow-2} \sqrt[3]{5 x+2}\)

Find the derivative of the function. \(g(u)=\sqrt{\frac{u+1}{3 u+2}}\)

Find the derivative of each of the given functions and evaluate \(f^{\prime}(x)\) at the given value of \(x\). \(f(x)=\frac{x}{x^{4}-2 x^{2}-1} ; x=-1\)

In Exercises 37-40, find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative. \(\lim _{h \rightarrow 0} \frac{(1+h)^{3}-1}{h}\)

The gross domestic product (GDP) of a certain country is projected to be $$ N(t)=t^{2}+2 t+50 \quad(0 \leq t \leq 5) $$ billion dollars \(t\) yr from now. What will be the rate of change of the country's GDP 2 yr and 4 yr from now?

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow 0^{-}} f(x)\), where $$ f(x)=\left\\{\begin{array}{ll} 2 x & \text { if } x<0 \\ x^{2} & \text { if } x \geq 0 \end{array}\right. $$

Find the indicated limit. \(\lim _{x \rightarrow-3} \sqrt{2 x^{4}+x^{2}}\)

Find the derivative of the function. \(g(x)=\sqrt{\frac{2 x+1}{2 x-1}}\)

Find the derivative of each of the given functions and evaluate \(f^{\prime}(x)\) at the given value of \(x\). \(f(x)=(\sqrt{x}+2 x)\left(x^{3 / 2}-x\right) ; x=4\)

Find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative. \(\lim _{x \rightarrow 1} \frac{x^{5}-1}{x-1}\) Hint: Let \(h=x-1\).

Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time \(t\) (in minutes) is described by the function $$ P=f(t)=3 t^{2}+2 t+1 $$ Find the rate of population growth at \(t=10 \mathrm{~min}\).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow 0^{-}} f(x)\), where $$ f(x)=\left\\{\begin{array}{ll} -x+1 & \text { if } x \leq 0 \\ 2 x+3 & \text { if } x>0 \end{array}\right. $$

Find the indicated limit. \(\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}}\)

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

In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\left(x^{3}+1\right)\left(x^{2}-2\right) ;(2,18)\)

Find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative. \(\lim _{h \rightarrow 0} \frac{3(2+h)^{2}-(2+h)-10}{h}\)

AIR TEMPERATURE The air temperature at a height of \(h \mathrm{ft}\) from the surface of the earth is \(T=f(h)\) degrees Fahrenheit. a. Give a physical interpretation of \(f^{\prime}(h) .\) Give units. b. Generally speaking, what do you expect the sign of \(f^{\prime}(h)\) to be? c. If you know that \(f^{\prime}(1000)=-0.05\), estimate the change in the air temperature if the altitude changes from \(1000 \mathrm{ft}\) to \(1001 \mathrm{ft}\).

In Exercises 39-44, determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} 2 x-4 & \text { if } x \leq 0 \\ 1 & \text { if } x>0 \end{array}\right. $$

Find the derivative of the function. \(g(u)=\frac{2 u^{2}}{\left(u^{2}+u\right)^{3}}\)

In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\frac{x^{2}}{x+1} ;\left(2, \frac{4}{3}\right)\)

Find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative. \(\lim _{t \rightarrow 0} \frac{1-(1+t)^{2}}{t(1+t)^{2}}\)

REVENUE OF A TRAVEL AGENCY Suppose that the total revenue realized by the Odyssey Travel Agency is \(R=f(x)\) thousand dollars if \(x\) thousand dollars are spent on advertising. a. What does $$ \frac{f(b)-f(a)}{b-a} \quad(0

Find the indicated limit. \(\lim _{x \rightarrow 3} \frac{x \sqrt{x^{2}+7}}{2 x-\sqrt{2 x+3}}\)

Find the derivative of the function. \(h(x)=\frac{\left(3 x^{2}+1\right)^{3}}{\left(x^{2}-1\right)^{4}}\)

In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\frac{x+1}{x^{2}+1} ;(1,1)\)

In Exercises 41-44, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=2 x^{2}-3 x+4 ;(2,6)\)

In Exercises 41-46, let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes time and \(f(x)\) denotes the population of seals at time \(x\).

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} x+5 & \text { if } x \leq 0 \\ -x^{2}+5 & \text { if } x>0 \end{array}\right. $$

In Exercises 41-48, find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a}[f(x)-g(x)]\)