Chapter 7

Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 1^{+}} \frac{5}{1-x} $$

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

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) \(p=f(x)\) (b) \(p=x f(x)\) (c) \(p=-f(x)+5\)

A union contract guarantees a \(9 \%\) yearly increase for 5 years. For a current salary of $$\$ 28,500$$, the salaries for the next 5 years are given by \(S=28,500(1.09)^{[t]}\) where \(t=0\) represents the present year. (a) Use the greatest integer function of a graphing utility to graph the salary function, and discuss its continuity. (b) Find the salary during the fifth year (when \(t=5\) ).

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. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -\frac{3}{2} & -1 & -\frac{1}{2} & 0 & \frac{1}{2} & 1 & \frac{3}{2} & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{1}{4} x^{3} $$

Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow-2^{-}} \frac{1}{x+2} $$

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

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?

The number of units in inventory in a small company is \(N=25\left(2 \| \frac{t+2}{2} \rrbracket-t\right), \quad 0 \leq t \leq 12\) where the real number \(t\) is the time in months. (a) Use the greatest integer function of a graphing utility to graph this function, and discuss its continuity. (b) How often must the company replenish its inventory?

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. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -\frac{3}{2} & -1 & -\frac{1}{2} & 0 & \frac{1}{2} & 1 & \frac{3}{2} & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{1}{2} x^{2} $$

Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0^{-}} \frac{x+1}{x} $$

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

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) \(x=10\) (b) \(x=15\) (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. $$ f(x)=4.1 x^{3}-12 x^{2}+2.5 x \quad[0,3] $$

You have purchased a franchise. You have determined a linear model for your revenue as a function of time. Is the model a continuous function? Would your actual revenue be a continuous function of time? Explain your reasoning.

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. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -\frac{3}{2} & -1 & -\frac{1}{2} & 0 & \frac{1}{2} & 1 & \frac{3}{2} & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ $$ f(x)=-\frac{1}{2} x^{3} $$

Use a graphing utility to estimate the limit $$ \lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x^{2}-4 x+4} $$

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

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.

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. $$ f(x)=x^{3}-1.4 x^{2}-0.96 x+1.44 \quad[-2,2] $$

The gestation period of rabbits is about 29 to 35 days. Therefore, the population of a form (rabbits' home) can increase dramatically in a short period of time. The table gives the population of a form, where \(t\) is the time in months and \(N\) is the rabbit population. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline N & 2 & 8 & 10 & 14 & 10 & 15 & 12 \\ \hline \end{array} $$ (a) Use a graphing utility to graph the population as a function of time. (b) Find any points of discontinuity in the function. Explain your reasoning.

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. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -\frac{3}{2} & -1 & -\frac{1}{2} & 0 & \frac{1}{2} & 1 & \frac{3}{2} & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ $$ f(x)=-\frac{3}{2} x^{2} $$

Use a graphing utility to estimate the limit $$ \lim _{x \rightarrow 1} \frac{x^{2}+6 x-7}{x^{3}-x^{2}+2 x-2} $$

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.

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 \(f ?\) $$ f(x)=x^{2}-4 x $$

Use a graphing utility to estimate the limit $$ \lim _{x \rightarrow-4} \frac{x^{3}+4 x^{2}+x+4}{2 x^{2}+7 x-4} $$

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.

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

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 \(f ?\) $$ f(x)=2+6 x-x^{2} $$

Use a graphing utility to estimate the limit $$ \lim _{x \rightarrow-2} \frac{4 x^{3}+7 x^{2}+x+6}{3 x^{2}-x-14} $$

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. $$ f(t)=\left(t^{2}-9\right) \sqrt{t+2} \quad(-1,-8) $$

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 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 \(f ?\) $$ f(x)=x^{3}-3 x $$

The cost (in dollars) of removing \(p \%\) of the pollutants from the water in a small lake is given by \(C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100\) where \(C\) is the cost and \(p\) is the percent of pollutants. (a) Find the cost of removing \(50 \%\) of the pollutants. (b) What percent of the pollutants can be removed for \(\$ 100,000 ?\) (c) Evaluate \(\lim _{p \rightarrow 100^{-}} C .\) Explain your results.

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 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 \(f ?\) $$ f(x)=x^{3}-6 x^{2} $$

You deposit in an account that is compounded quarterly at an annual rate of \(r\) (in decimal form). The balance \(A\) after 10 years is \(A=2000\left(1+\frac{r}{4}\right)^{40}\) (a) Complete the table. $$ \begin{array}{|l|l|l|l|l|l|} \hline r & 0.059 & 0.0599 & 0.06 & 0.0601 & 0.061 \\ \hline A & & & & & \\ \hline \end{array} $$ (b) Does the limit of \(A\) exist as the interest rate approaches \(6 \%\) ? If so, what is the limit?

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

The slope of the graph of \(y=x^{2}\) is different at every point on the graph of \(f\).

Consider a certificate of deposit that pays \(10 \%\) (annual percentage rate) on an initial deposit of \(\$ 1000\). The balance \(A\) after 10 years is \(A=1000(1+0.1 x)^{10 / x}\) where \(x\) is the length of the compounding period (in years). (a) Use a graphing utility to graph \(A\), where \(0 \leq x \leq 1\). (b) Use the zoom and trace features to estimate the balance for quarterly compounding and daily compounding. (c) Use the zoom and trace features to estimate \(\lim _{x \rightarrow 0^{+}} A\) What do you think this limit represents? Explain your reasoning.

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

The limit of \(f(x)=(1+x)^{1 / x}\) is a natural base for many business applications, as you will see in Section \(10.2\). \(\lim _{x \rightarrow 0}(1+x)^{1 / x}=e \approx 2.718\) (a) Show the reasonableness of this limit by completing the table. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -0.01 & -0.001 & -0.0001 & 0 & 0.0001 & 0.001 & 0.01 \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph \(f\) and to confirm the answer in part (a). (c) Find the domain and range of the function.

You deposit 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) \(r=0.12\).

If a function is differentiable at a point, then it is continuous at that point.

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 .

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

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 (a) Write \(V\) as a function of \(t\). (b) Find the rate of depreciation when \(t=1\). (c) Find the rate of depreciation when \(t=3\).

Given that the value of the machine \(t\) years after it is purchased is inversely proportional to the cube root of \(t+1\).

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.

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