Chapter 8

A car is traveling at a rate of 66 feet per second ( 45 miles per hour) when the brakes are applied. The position function for the car is given by \(s=-8.25 t^{2}+66 t\), where \(s\) is measured in feet and \(t\) is measured in seconds. Create a table showing the position, velocity, and acceleration for each given value of \(t\). What can you conclude?

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=x \sqrt{x+3} $$

A retailer has determined the cost \(C\) for ordering and storing \(x\) units of a product to be modeled by \(C=3 x+\frac{20,000}{x}, 0

The numbers (in thousands) of cases \(y\) of HIV/AIDS reported in the years 2001 through 2005 can be modeled by \(y^{2}-1141.6=24.9099 t^{3}-183.045 t^{2}+452.79 t\) where \(t\) represents the year, with \(t=1\) corresponding to 2001. (a) Use a graphing utility to graph the model and describe the results. (b) Use the graph to estimate the year during which the number of reported cases was increasing at the greatest rate. (c) Complete the table to estimate the year during which the number of reported cases was increasing at the greatest rate. Compare this estimate with your answer in part (b).

In Exercises, use a graphing utility to graph \(f_{t} f^{\prime}\), and \(f^{\prime \prime}\) in the same viewing window. What is the relationship among the degree of \(f\) and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives? $$ f(x)=x^{2}-6 x+6 $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=x \sqrt{9-x} $$

The quantity demanded \(x\) for a product is inversely proportional to the cube of the price \(p\) for \(p>1\). When the price is \(\$ 10\) per unit, the quantity demanded is eight units. The initial cost is \(\$ 100\) and the cost per unit is \$4. What price will yield a maximum profit?

In Exercises, use a graphing utility to graph \(f, f^{\prime}\), and \(f^{\prime \prime}\) in the same viewing window. What is the relationship among the degree of \(f\) and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives? $$ f(x)=3 x^{3}-9 x $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=\frac{4}{1+x^{2}} $$

When soft drinks were sold for \(\$ 1.00\) per can at football games, approximately 6000 cans were sold. When the price was raised to \(\$ 1.20\) per can, the quantity demanded dropped to 5600 . The initial cost is \(\$ 5000\) and the cost per unit is \(\$ 0.50\). Assuming that the demand function is linear, use the table feature of a graphing utility to determine the price that will yield a maximum profit.

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=\frac{2}{x^{2}-1} $$

Coughing forces the trachea (windpipe) to contract, which in turn affects the velocity of the air through the trachea. The velocity of the air during coughing can be modeled by \(v=k(R-r) r^{2}, 0 \leq r

In Exercises, sketch a graph of a function \(f\) having the given characteristics. \begin{aligned} &f(2)=f(4)=0 \\ &f^{\prime}(x)<0 \text { if } x<3 \\ &f^{\prime}(3)=0 \\ &f^{\prime}(x)>0 \text { if } x>3 \\ &f^{\prime}(x)>0 \end{aligned}

The resident population \(P\) (in millions) of the United States from 1790 through 2000 can be modeled by \(P=0.00000583 t^{3}+0.005003 t^{2}+0.13776 t+4.658\) \(-10 \leq t \leq 200\), where \(t=0\) corresponds to 1800 (a) Make a conjecture about the maximum and minimum populations in the U.S. from 1790 to 2000 . (b) Analytically find the maximum and minimum populations over the interval. (c) Write a brief paragraph comparing your conjecture with your results in part (b).

The table shows the retail values \(y\) (in billions of dollars) of motor homes sold in the United States for 2000 to 2005, where \(t\) is the year, with \(t=0\) corresponding to 2000. (Source: Recreation Vehicle Industry Association) \begin{tabular}{|l|l|l|l|l|l|l|} \hline\(t\) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline\(y\) & \(9.5\) & \(8.6\) & \(11.0\) & \(12.1\) & \(14.7\) & \(14.4\) \\ \hline \end{tabular} (a) Use a graphing utility to find a cubic model for the total retail value \(y(t)\) of the motor homes. (b) Use a graphing utility to graph the model and plot the data in the same viewing window. How well does the model fit the data? (c) Find the first and second derivatives of the function. (d) Show that the retail value of motor homes was increasing from 2001 to 2004 (e) Find the year when the retail value was increasing at the greatest rate by solving \(y^{\prime \prime}(t)=0\). (f) Explain the relationship among your answers for parts (c), (d), and (e).

An object is thrown upward from the top of a 64 -foot building with an initial velocity of 48 feet per second. (a) Write the position, velocity, and acceleration functions of the object. (b) When will the object hit the ground? (c) When is the velocity of the object zero? (d) How high does the object go? (e) Use a graphing utility to graph the position, velocity, and acceleration functions in the same viewing window. Write a short paragraph that describes the relationship among these functions.

In Exercises, sketch a graph of a function \(f\) having the given characteristics. $$ \begin{aligned} &f(0)=f(2)=0 \\ &f^{\prime}(x)>0 \text { if } x<1 \\ &f^{\prime}(1)=0 \\ &f^{\prime}(x)<0 \text { if } x>1 \\ &f^{\prime}(x)<0 \end{aligned} $$

$$ \begin{aligned} &f(0)=f(2)=0 \\ &f^{\prime}(x)<0 \text { if } x<1 \\ &f^{\prime}(1)=0 \\ &f^{\prime}(x)>0 \text { if } x>1 \\ &f^{\prime \prime}(x)>0 \end{aligned} $$

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(x+1)(x+2)(x+3)(x+4), \text { then } \frac{d^{5} y}{d x^{5}}=0 $$

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } f^{\prime}(c) \text { and } g^{\prime}(c) \text { are zero and } h(x)=f(x) g(x) \text { , then } h^{\prime}(c)=0 . $$

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The second derivative represents the rate of change of the first derivative.

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f^{\prime}(x)=2 x+5 $$

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f(x)=3 x^{2}-2 $$

Develop a general rule for \([x f(x)]^{(n)}\) where \(f\) is a differentiable function of \(x\).

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f^{\prime}(x)=-x^{2}+2 x-1 $$

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f^{\prime}(x)=x^{2}+x-6 $$

In Exercises, identify the point of diminishing returns for the input output function. For each function, \(R\) is the revenue and \(x\) is the amount spent on advertising. Use a graphing utility to verify your results. $$ R=\frac{1}{50,000}\left(600 x^{2}-x^{3}\right), \quad 0 \leq x \leq 400 $$

In Exercises , identify the point of diminishing returns for the inputoutput function. For each function, \(R\) is the revenue and \(x\) is the amount spent on advertising. Use a graphing utility to verify your results. $$ R=-\frac{4}{5}\left(x^{3}-9 x^{2}-27\right), \quad 0 \leq x \leq 5 $$

In Exercises, you are given the total cost of producing \(x\) units. Find the production level that minimizes the average cost per unit. Use a graphing utility to verify your results. $$ C=0.5 x^{2}+15 x+5000 $$

In Exercises, consider a college student who works from 7 P.M. to 11 P.M. assembling mechanical components. The number \(N\) of components assembled after \(t\) hours is given by the function. At what time is the student assembling components at the greatest rate? $$ N=-0.12 t^{3}+0.54 t^{2}+8.22 t, \quad 0 \leq t \leq 4 $$

In Exercises, consider a college student who works from 7 P.M. to 11 P.M. assembling mechanical components. The number \(N\) of components assembled after \(t\) hours is given by the function. At what time is the student assembling components at the greatest rate? $$ N=\frac{20 t^{2}}{4+t^{2}}, \quad 0 \leq t \leq 4 $$

In Exercises, find the time \(t\) in years when the annual sales \(x\) of a new product are increasing at the greatest rate. Use a graphing utility to verify your results. $$ x=\frac{10,000 t^{2}}{9+t^{2}} $$

In Exercises, find the time \(t\) in years when the annual sales \(x\) of a new product are increasing at the greatest rate. Use a graphing utility to verify your results.$$ x=\frac{500,000 t^{2}}{36+t^{2}} $$

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ f(x)=\frac{1}{2} x^{3}-x^{2}+3 x-5, \quad[0,3] $$

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ f(x)=-\frac{1}{20} x^{5}-\frac{1}{12} x^{2}-\frac{1}{3} x+1, \quad[-2,2] $$

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ \begin{aligned} &f(x)=\frac{2}{x^{2}+1}, \quad[-3,3]\\\ &\text { 72. } f(x)=\frac{x^{2}}{x^{2}+1}, \quad[-3,3] \end{aligned} $$

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ f(x)=\frac{x^{2}}{x^{2}+1}, \quad[-3,3] $$

A manufacturer has determined that the total cost \(C\) of operating a factory is \(C=0.5 x^{2}+10 x+7200\), where \(x\) is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is \(C / x\).)

The cost \(C\) for ordering and storing \(x\) units is \(C=2 x+300,000 / x .\) What order size will produce a minimum cost?

Phishing Phishing is a criminal activity used by an individual or group to fraudulently acquire information by masquerading as a trustworthy person or business in an electronic communication. Criminals create spoof sites on the Internet to trick victims into giving them information. The sites are designed to copy the exact look and feel of a "real" site. A model for the number of reported spoof sites from November 2005 through October 2006 is \(f(t)=88.253 t^{3}-1116.16 t^{2}+4541.4 t+4161,0 \leq t \leq 11\) where \(t\) represents the number of months since November 2005.

Let \(S\) represent monthly sales of a new digital audio player. Write a statement describing \(S^{\prime}\) and \(S^{\prime \prime}\) for each of the following. (a) The rate of change of sales is increasing. (b) Sales are increasing, but at a greater rate. (c) The rate of change of sales is steady. (d) Sales are steady. (e) Sales are declining, but at a lower rate. (f) Sales have bottomed out and have begun to rise.

The spread of a virus can be modeled by \(N=-t^{3}+12 t^{2}, \quad 0 \leq t \leq 12\) where \(N\) is the number of people infected (in hundreds), and \(t\) is the time (in weeks). (a) What is the maximum number of people projected to be infected? (b) When will the virus be spreading most rapidly? (c) Use a graphing utility to graph the model and to verify your results.