Chapter 11

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \int_{1}^{2} \frac{(2+\ln x)^{3}}{x} d x $$

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{5}{e^{-5 x}+7} d x $$

Find the demand function \(x=f(p)\) that satisfies the initial conditions. $$ \frac{d x}{d p}=-\frac{400}{(0.02 p-1)^{3}}, \quad x=10,000 \text { when } p=\$ 100 $$

Find the particular solution \(y=f(x)\) that satisfies the differential equation and initial condition. $$ f^{\prime}(x)=\frac{x^{2}-5}{x^{2}}, x>0 ; \quad f(1)=2 $$

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{1}^{3}(4 x-3) d x $$

Find the equation of the function \(f\) whose graph passes through the point. $$ f^{\prime}(x)=\frac{x^{2}+4 x+3}{x-1} ; \quad(2,4) $$

Gardening An evergreen nursery usually sells a type of shrub after 5 years of growth and shaping. The growth rate during those 5 years is approximated by \(\frac{d h}{d t}=\frac{17.6 t}{\sqrt{17.6 t^{2}+1}}\) where \(t\) is time in years and \(h\) is height in inches. The seedlings are 6 inches tall when planted \((t=0)\). (a) Find the height function. (b) How tall are the shrubs when they are sold?

Lorenz Curve Economists use Lorenz curves to illustrate the distribution of income in a country. Letting \(x\) represent the percent of families in a country and \(y\) the percent of total income, the model \(y=x\) would represent a country in which each family had the same income. The Lorenz curve, \(y=f(x)\), represents the actual income distribution. The area between these two models, for

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{2}(x+4) d x $$

Cash Flow The rate of disbursement \(d Q / d t\) of a \(\$ 4\) million federal grant is proportional to the square of \(100-t\), where \(t\) is the time (in days, \(0 \leq t \leq 100\) ) and \(Q\) is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 50 days. Assume that the entire grant will be disbursed after 100 days.

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{2}(2-x) \sqrt{x} d x $$

A population of bacteria is growing at the rate of \(\frac{d P}{d t}=\frac{3000}{1+0.25 t}\) where \(t\) is the time in days. When \(t=0\), the population is \(1000 .\) (a) Write an equation that models the population \(P\) in terms of the time \(t\). (b) What is the population after 3 days? (c) After how many days will the population be \(12,000 ?\)

(a) use the marginal propensity to consume, \(d Q / d x\), to write \(Q\) as a function of \(x\), where \(x\) is the income (in dollars) and \(Q\) is the income consumed (in dollars). Assume that \(100 \%\) of the income is consumed for families that have annual incomes of \(\$ 25,000\) or less. (b) Use the result of part (a) and a spreadsheet to complete the table showing the income consumed and the income saved, \(x-Q\), for various incomes. (c) Use a graphing utility to represent graphically the income consumed and saved. $$ \begin{array}{|l|l|l|l|l|} \hline x & 25,000 & 50,000 & 100,000 & 150,000 \\ \hline Q & & & & \\ \hline x-Q & & & & \\ \hline \end{array} $$ $$ \frac{d Q}{d x}=\frac{0.95}{(x-24,999)^{0.05}}, \quad x \geq 25,000 $$

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{2}(2-x) \sqrt{x} d x $$

Because of an insufficient oxygen supply, the trout population in a lake is dying. The population's rate of change can be modeled by \(\frac{d P}{d t}=-125 e^{-t / 20}\) where \(t\) is the time in days. When \(t=0\), the population is \(2500 .\) (a) Write an equation that models the population \(P\) in terms of the time \(t\). (b) What is the population after 15 days? (c) According to this model, how long will it take for the entire trout population to die?

(a) use the marginal propensity to consume, \(d Q / d x\), to write \(Q\) as a function of \(x\), where \(x\) is the income (in dollars) and \(Q\) is the income consumed (in dollars). Assume that \(100 \%\) of the income is consumed for families that have annual incomes of \(\$ 25,000\) or less. (b) Use the result of part (a) and a spreadsheet to complete the table showing the income consumed and the income saved, \(x-Q\), for various incomes. (c) Use a graphing utility to represent graphically the income consumed and saved. $$ \begin{array}{|l|l|l|l|l|} \hline x & 25,000 & 50,000 & 100,000 & 150,000 \\ \hline Q & & & & \\ \hline x-Q & & & & \\ \hline \end{array} $$ $$ \frac{d Q}{d x}=\frac{0.93}{(x-24,999)^{0.07}}, \quad x \geq 25,000 $$

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{2}^{4} \frac{3 x^{2}}{x^{3}-1} d x $$

The marginal price for the demand of a product can be modeled by \(d p / d x=0.1 e^{-x / 500}\), where \(x\) is the quantity demanded. When the demand is 600 units, the price is \(\$ 30\). (a) Find the demand function, \(p=f(x)\). (b) Use a graphing utility to graph the demand function. Does price increase or decrease as demand increases? (c) Use the zoom and trace features of the graphing utility to find the quantity demanded when the price is \(\$ 22\).

Use a symbolic integration utility to find the indefinite integral. Verify the result by differentiating. $$ \int \frac{1}{\sqrt{x}+\sqrt{x+1}} d x $$

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=2, \quad f^{\prime}(2)=5, \quad f(2)=10 $$

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{\ln 6} \frac{e^{x}}{2} d x $$

The marginal revenue for the sale of a product can be modeled by \(\frac{d R}{d x}=50-0.02 x+\frac{100}{x+1}\) where \(x\) is the quantity demanded. (a) Find the revenue function. (b) Use a graphing utility to graph the revenue function. (c) Find the revenue when 1500 units are sold. (d) Use the zoom and trace features of the graphing utility to find the number of units sold when the revenue is \(\$ 60,230\)

Use a symbolic integration utility to find the indefinite integral. Verify the result by differentiating. $$ \int \frac{x}{\sqrt{3 x+2}} d x $$

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=x^{2}, \quad f^{\prime}(0)=6, \quad f(0)=3 $$

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. $$ y=3 x^{2}+1, \quad y=0, \quad x=0, \quad \text { and } \quad x=2 $$

From 2000 through 2005, the average salary for public school nurses \(S\) (in dollars) in the United States changed at the rate of \(\frac{d S}{d t}=1724.1 e^{-t / 4.2}\) where \(t=0\) corresponds to \(2000 .\) In 2005, the average salary for public school nurses was \(\$ 40,520 .\) (Source: Educational Research Service) (a) Write a model that gives the average salary for public school nurses per year. (b) Use the model to find the average salary for public school nurses in 2002 .

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=x^{-2 / 3}, \quad f^{\prime}(8)=6, \quad f(0)=0 $$

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. $$ y=1+\sqrt{x}, \quad y=0, \quad x=0, \quad \text { and } \quad x=4 $$

The rate of change in sales for The Yankee Candle Company from 1998 through 2005 can be modeled by \(\frac{d S}{d t}=0.528 t+\frac{597.2099}{t}\) where \(S\) is the sales (in millions) and \(t=8\) corresponds to \(1998 .\) In 1999, the sales for The Yankee Candle Company were \$256.6 million. (Source: The Yankee Candle Company) (a) Find a model for sales from 1998 through \(2005 .\) (b) Find The Yankee Candle Company's sales in 2004 .

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=x^{-3 / 2}, \quad f^{\prime}(1)=2, \quad f(9)=-4 $$

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. $$ y=4 / x \quad y=0, \quad x=1, \quad \text { and } x=3 $$

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

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. $$ y=e^{x}, \quad y=0, \quad x=0, \quad \text { and } \quad x=2 $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=4-x^{2} \quad[-2,2] $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=2 e^{x} \quad[-1,1] $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=e^{x / 4} \quad[0,4] $$

Find the profit function for the given marginal profit and initial condition. $$ \frac{d P}{d x}=-18 x+1650 \quad P(15)=\$ 22,725 $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=\frac{1}{(x-3)^{2}} \quad[0,2] $$

Find the profit function for the given marginal profit and initial condition. $$ \frac{d P}{d x}=-40 x+250 \quad P(5)=\$ 650 $$

Find the profit function for the given marginal profit and initial condition. $$ \frac{d P}{d x}=-24 x+805 \quad P(12)=\$ 8000 $$

Find the profit function for the given marginal profit and initial condition. $$ \frac{d P}{d x}=-30 x+920 \quad P(8)=\$ 6500 $$

State whether the function is even, odd, or neither. $$ f(x)=3 x^{4} $$

Use \(a(t)=-32\) feet per second per second as the acceleration due to gravity. The Grand Canyon is 6000 feet deep at the deepest part. A rock is dropped from this height. Express the height \(s\) of the rock as a function of the time \(t\) (in seconds). How long will it take the rock to hit the canyon floor?

State whether the function is even, odd, or neither. $$ g(x)=x^{3}-2 x $$

State whether the function is even, odd, or neither. $$ g(t)=2 t^{5}-3 t^{2} $$

Cost A company produces a product for which the marginal cost of producing \(x\) units is modeled by \(d C / d x=2 x-12\), and the fixed costs are \(\$ 125\). (a) Find the total cost function and the average cost function. (b) Find the total cost of producing 50 units. (c) In part (b), how much of the total cost is fixed? How much is variable? Give examples of fixed costs associated with the manufacturing of a product. Give examples of variable costs.

State whether the function is even, odd, or neither. $$ f(t)=5 t^{4}+1 $$

Tree Growth An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by \(d h / d t=1.5 t+5\), where \(t\) is the time in years and \(h\) is the height in centimeters. The seedlings are 12 centimeters tall when planted \((t=0)\). (a) Find the height after \(t\) years. (b) How tall are the shrubs when they are sold?

Use the value \(\int_{0}^{1} x^{2} d x=\frac{1}{3}\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-1}^{0} x^{2} d x\) (b) \(\int_{-1}^{1} x^{2} d x\) (c) \(\int_{0}^{1}-x^{2} d x\)

The growth rate of Horry County in South Carolina can be modeled by \(d P / d t=105.46 t+2642.7\), where \(t\) is the time in years, with \(t=0\) corresponding to 1970 . The county's population was 226,992 in \(2005 .\) (Source: U.S. Census Bureau) (a) Find the model for Horry County's population. (b) Use the model to predict the population in \(2012 .\) Does your answer seem reasonable? Explain your reasoning.