Chapter 2

Identify a function that has the given characteristics. Then sketch the function. $$ \begin{array}{l}{f(-2)=f(4)=0 ; f^{\prime}(1)=0, f^{\prime}(x)<0} \\ {\text { for } x<1 ; f^{\prime}(x)>0 \text { for } x>1}\end{array} $$

Managing a Store You are managing a store and have been adjusting the price of an item. You have found that you make a profit of \(\$ 50\) when 10 units are sold, \(\$ 60\) when 12 units are sold, and \(\$ 65\) when 14 units are sold. (a) Fit these data to the model \(P=a x^{2}+b x+c .\) (b) Use a graphing utility to graph \(P .\) (c) Find the point on the graph at which the marginal profit is zero. Interpret this point in the context of the problem.

Political Fundraiser A politician raises funds by selling tickets to a dinner for \(\$ 500 .\) The politician pays \(\$ 150\) for each dinner and has fixed costs of \(\$ 7000\) to rent a dining hall and wait staff. Write the profit \(P\) as a function of \(x,\) the number of dinners sold. Show that the derivative of the profit function is a constant and is equal to the increase in profit from each dinner sold.

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=t^{2} \sqrt{t-2} $$

Use a graphing utility to graph \(f\) on the interval \([-2,2] .\) Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{1}{4} x^{3} $$

Demand Function Given \(f(x)=x+1,\) which function would most likely represent a demand function? Explain your reasoning. Use a graphing utility to graph each function, and use each graph as part of your explanation. (a) \(p=f(x) \quad\) (b) \(p=x f(x) \quad\) (c) \(p=-f(x)+5\)

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\sqrt{x}(x-2)^{2} $$

Use a graphing utility to graph \(f\) on the interval \([-2,2] .\) Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{1}{2} x^{2} $$

Cost The cost of producing \(x\) units of a product is given by \(C=x^{3}-15 x^{2}+87 x-73, \quad 4 \leq x \leq 9.\) (a) Use a graphing utility to graph the marginal cost function and the average cost function, \(C / x,\) in the same viewing window. (b) Find the point of intersection of the graphs of \(d C / d x\) and \(C / x .\) Does this point have any significance?

Use a graphing utility to graph \(f\) and \(f^{\prime}\) over the given interval. Determine any points at which the graph of \(f\) has horizontal tangents. $$f(x)=4.1 x^{3}-12 x^{2}+2.5 x\quad [0,3] $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{6-5 x}{x^{2}-1}\right)^{2} $$

Use a graphing utility to graph \(f\) on the interval \([-2,2] .\) Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=-\frac{1}{2} x^{3} $$

MAKE A DECISION: INVENTORY REPLENISHMENT The ordering and transportation cost \(C\) per unit (in thousands of dollars) of the components used in manufacturing a product is given by $$C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad 1 \leq x\( where \)x\( is the order size (in hundreds). Find the rate of change of \)C\( with respect to \)x\( for each order size. What do these rates of change imply about increasing the size of an order? Of the given order sizes, which would you choose? Explain. (a) \)x=10 \quad\( (b) \)x=15 \quad\( (c) \)x=20$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) over the given interval. Determine any points at which the graph of \(f\) has horizontal tangents. $$ f(x)=x^{3}-1.4 x^{2}-0.96 x+1.44 \quad[-2,2] $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{4 x^{2}}{3-x}\right)^{3} $$

Use a graphing utility to graph \(f\) on the interval \([-2,2] .\) Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=-\frac{3}{2} x^{2} $$

Inventory Replenishment The ordering and transportation cost \(C\) per unit for the components used in manufacturing a product is \(C=\left(375,000+6 x^{2}\right) / x, \quad x \geq 1\) where \(C\) is measured in dollars and \(x\) is the order size. Find the rate of change of \(C\) with respect to \(x\) when \((a) x=200,\) (b) \(x=250,\) and (c) \(x=300 .\) Interpret the meaning of these values.

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(t)=\frac{36}{(3-t)^{2}}} & {(0,4)}\end{array}$$

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{2}-4 x $$

Consumer Awareness The prices per pound of lean and extra lean ground beef in the United States from 1998 to 2005 can be modeled by $$P=\frac{1.755-0.2079 t+0.00673 t^{2}}{1-0.1282 t+0.00434 t^{2}}, \quad 8 \leq t \leq 15$$ where \(t\) is the year, with \(t=8\) corresponding to \(1998 .\) Find \(d P / d t\) and evaluate it for \(t=8,10,12,\) and \(14 .\) Interpret the meaning of these values.

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } f(x)=g(x)+c, \text { then } f^{\prime}(x)=g^{\prime}(x) $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {s(x)=\frac{1}{\sqrt{x^{2}-3 x+4}}} & {\left(3, \frac{1}{2}\right)}\end{array}$$

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=2+6 x-x^{2} $$

Sales Analysis The monthly sales of memberships \(M\) at a newly built fitness center are modeled by $$M(t)=\frac{300 t}{t^{2}+1}+8$$ where \(t\) is the number of months since the center opened. (a) Find \(M^{\prime}(t)\). (b) Find \(M(3)\) and \(M^{\prime}(3)\) and interpret the results. (c) Find \(M(24)\) and \(M^{\prime}(24)\) and interpret the results.

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(t)=\left(t^{2}-9\right) \sqrt{t+2}} & {(-1,-8)}\end{array}$$

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{3}-3 x $$

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=2 g(x)+h(x) $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\y=\frac{2 x}{\sqrt{x+1}} & {(3,3)}\end{array}$$

Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{3}-6 x^{2} $$

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=3-g(x) $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\f(x)=\frac{x+1}{\sqrt{2 x-3}} & {(2,3)}\end{array}$$

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=g(x)+h(x) $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\y=\frac{x}{\sqrt{25+x^{2}}} & {(0,0)}\end{array}$$

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$

Compound Interest You deposit \(\$ 1000\) in an account with an annual interest rate of \(r\) (in decimal form) compounded monthly. At the end of 5 years, the balance is $$A=1000\left(1+\frac{r}{12}\right)^{60}.$$ Find the rates of change of \(A\) with respect to \(r\) when (a) \(r=0.08,\) (b) \(r=0.10,\) and (c) \(r=0.12\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.

Research Project Use your school's library, the Internet, or some other reference source to find information on a company that is noted for its philanthropy and community commitment. (One such business is described above.) Write a short paper about the company.

Environment An environmental study indicates that the average daily level \(P\) of a certain pollutant in the air, in parts per million, can be modeled by the equation $$P=0.25 \sqrt{0.5 n^{2}+5 n+25}$$ where \(n\) is the number of residents of the community, in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is \(12,000\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A tangent line to a graph can intersect the graph at more than one point.

Biology The number \(N\) of bacteria in a culture after \(t\) days is modeled by $$N=400\left[1-\frac{3}{\left(t^{2}+2\right)^{2}}\right]$$ Complete the table. What can you conclude? $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} \\ \hline d N / d t & {} & {} & {} & {} & {} \\ \hline\end{array} $$

Writing Use a graphing utility to graph the two function \(f(x)=x^{2}+1\) and \(g(x)=|x|+1\) in the same viewin window. Use the zoom and trace features to analyze the graphs near the point \((0,1) .\) What do you observe? Whic function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.

Depreciation The value \(V\) of a machine \(t\) years after it is purchased is inversely proportional to the square root of \(t+1 .\) The initial value of the machine is \(\$ 10,000 .\) (a) Write \(V\) as a function of \(t .\) (b) Find the rate of depreciation when \(t=1\) (c) Find the rate of depreciation when \(t=3\)

Credit Card Rate The average annual rate \(r\) (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by \(r=\sqrt{-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Find the derivative of this model. Which differentiation rule(s) did you use? (b) Use a graphing utility to graph the derivative on the interval \(0 \leq t \leq 5\). (c) Use the trace feature to find the years during which the finance rate was changing the most. (d) Use the trace feature to find the years during which the finance rate was changing the least.

True or False? determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(1-x)^{1 / 2}, \text { then } y^{\prime}=\frac{1}{2}(1-x)^{-1 / 2} $$

True or False? determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y\) is a differentiable function of \(u, u\) is a differentiable function of \(v,\) and \(v\) is a differentiable function of \(x,\) then \(\frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d v} \cdot \frac{d v}{d x}\)