Chapter 11

Find the indefinite integral. $$\int v^{2}(1-v)^{6} d v$$

Find the indefinite integral. $$\int \frac{(\sqrt{x}-1)^{2}}{x^{2}} d x$$

Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of $$ 3.5 e^{0.05 \mathrm{y}} $$ billion metric tons/year, \(t\) yr from 1980 (which corresponds to \(t=0\) ). Had it not been for the energy crisis, the rate of production of coal since 1980 might have been only $$ 3.5 e^{0.011} $$ billion metric tons/year, \(t\) yr from 1980 . Determine how much additional coal was produced between 1980 and the end of the century as an alternate energy source.

A car moves along a straight road in s?ch a way that its velocity (in feet/second) at any time \(t\) (in seconds) is given by $$ v(t)=3 t \sqrt{16-t^{2}} \quad(0 \leq t \leq 4) $$ Find the distance traveled by the car in the \(4 \mathrm{sec}\) from \(t=0\) to \(t=4\)

To test air purifiers, engineers ran a purifier in a smoke-filled \(10-\mathrm{ft} \times 20\) -ft room. While conducting a test for a certain brand of air purifier, it was determined that the amount of smoke in the room was decreasing at the rate of $$ \begin{aligned} R(t)=& 0.00032 t^{4}-0.01872 t^{3}+0.3948 t^{2} \\ &-3.83 t+17.63 \quad(0 \leq t \leq 20) \end{aligned} $$ percent of the (original) amount of the smoke per minute, \(i\) min after the start of the test. How much smoke was left in the room \(5 \mathrm{~min}\) after the start of the test? Ten min after the start of the test?

Find the indefinite integral. $$\int x^{3}\left(x^{2}+1\right)^{3 / 2} d x$$

Find the indefinite integral. $$\int(x+1)^{2}\left(1-\frac{1}{x}\right) d x$$

Mobile-phone ad spending between \(2005(t=1)\) and \(2011(t=7)\) is projected to be $$ S(t)=0.86 t^{0.96} \quad(1 \leq t \leq 7) $$ where \(S(t)\) is measured in billions of dollars and \(t\) is measured in years. What is the projected average spending per year on mobile-phone spending between 2005 and 2011 ?

The number of television set-top boxes shipped worldwide from the beginning of 2003 until the beginning of 2009 is projected to be $$ \begin{array}{r} f(t)=-0.05556 t^{3}+0.262 t^{2}+17.46 t+63.4 \\ (0 \leq t \leq 6) \end{array} $$ million units/year, where \(t\) is measured in years, with \(t=0\) corresponding to 2003 . If the projection held true, how many set-top boxes were expected to be shipped from the beginning of 2003 until the beginning of \(2009 ?\)

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=5(2 x-1)^{4} ;(1,3)$$

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=2 x+1 ; f(1)=3$$

In an endeavor to curb population growth in a Southeast Asian island state, the government has decided to launch an extensive propaganda campaign. Without curbs, the government expects the rate of population growth to have been $$ 60 e^{0.02 t} $$ thousand people/year, \(t\) yr from now, over the next 5 yr. However, successful implementation of the proposed campaign is expected to result in a population growth rate of $$ -t^{2}+60 $$ thousand people/year, \(t\) yr from now, over the next 5 yr. Assuming that the campaign is mounted, how many fewer people will there be in that country 5 yr from now than there would have been if no curbs had been imposed?

The increase in carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of \(\mathrm{CO}_{2}\) in the atmosphere from 1958 through 2007 is approximated by \(A(t)=0.010716 t^{2}+0.8212 t+313.4 \quad(1 \leq t \leq 50)\) where \(A(t)\) is measured in parts per million volume (ppmv) and \(t\) in years, with \(t=1\) corresponding to 1958 . Find the average rate of increase of the average amount of \(\mathrm{CO}_{2}\) in the atmosphere from 1958 through 2007 .

PropucnoN The production of oil (in millions of barrels per day) extracted from oil sands in Canada is projected to grow according to the function $$ P(t)=\frac{4.76}{1+4.11 e^{-0.22 r}} \quad(0 \leq t \leq 15) $$ where \(t\) is measured in years, with \(t=0\) corresponding to \(2005 .\) What is the total production of oil from oil sands over the years from 2005 until \(2020(t=15)\) ?

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=\frac{3 x^{2}}{2 \sqrt{x^{3}-1}} ;(1,1)$$

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=3 x^{2}-6 x ; f(2)=4$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) and \(g\) are continuous on \([a, b]\) and either \(f(x) \geq g(x)\) for all \(x\) in \([a, b]\) or \(f(x) \leq g(x)\) for all \(x\) in \([a, b]\), then the area of the region bounded by the graphs of \(f\) and \(g\) and the vertical lines \(x=a\) and \(x=b\) is given by \(\int_{a}^{b}\lfloor f(x)-g(x) \mid d x\).

The White House wants to cut the gasoline usage from 140 billion gallons per year in 2007 to 128 billion gallons per year in 2017 . But estimates by the Department of Energy's Energy Information Agency suggest that won't happen. In fact, the agency's projection of gasoline usage from the beginning of 2007 until the beginning of 2017 is given by $$ A(t)=0.014 t^{2}+1.93 t+140 \quad(0 \leq t \leq 10) $$ where \(A(t)\) is measured in billions of gallons/year and \(t\) is in years, with \(t=0\) corresponding to 2007 . a. According to the agency's projection, what will be gasoline consumption at the beginning of \(2017 ?\) b. What will be the average consumption/year over the period from the beginning of 2007 until the beginning of 2017 ?

The population aged \(65 \mathrm{yr}\) old and older (in millions) from 2000 to 2050 is projected to be $$ f(t)=\frac{85}{1+1.859 e^{-0.06 t}} \quad(0 \leq t \leq 5) $$ where \(t\) is measured in decades, with \(t=0\) corresponding to 2000 . By how much will the population aged \(65 \mathrm{yr}\) and older increase from the beginning of 2000 until the beginning of 2030 ?

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=-2 x e^{-x^{2}+1} ;(1,0)$$

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=3 x^{2}+4 x-1 ; f(2)=9$$

The number of U.S. citizens aged 65 yr and older from 1900 through 2050 is estimated to be growing at the rate of $$ R(t)=0.063 t^{2}-0.48 t+3.87 \quad(0 \leq t \leq 15) $$ million people/decade, where \(t\) is measured in decades, with \(t=0\) corresponding to \(1900 .\) Show that the average rate of growth of U.S. citizens aged \(65 \mathrm{yr}\) and older between 2000 and 2050 will be growing at twice the rate of that between 1950 and 2000 .

Consider an artery of length \(L \mathrm{~cm}\) and radius \(R \mathrm{~cm} .\) Using Poiseuille's law (page 131 ), it can be shown that the rate at which blood flows through the artery (measured in cubic centimeters/second) is given by $$ V=\int_{0}^{R} \frac{k}{L} x\left(R^{2}-x^{2}\right) d x $$ where \(k\) is a constant. Find an expression for \(V\) that does not involve an integral.

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=1-\frac{2 x}{x^{2}+1} ;(0,2)$$

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=\frac{1}{\sqrt{x}} ; f(4)=2$$

The total vacancy rate (in percent) of offices in Manhattan from 2000 through 2006 is approximated by the function $$ \begin{array}{r} R(t)=0.032 t^{4}-0.26 t^{3}-0.478 t^{2}+5.82 t+3.8 \\ (0 \leq t \leq 6) \end{array} $$ where \(t\) is measured in years, with \(t=0\) corresponding to 2000 . What was the average vacancy rate of offices in Manhattan over the period from 2000 through \(2006 ?\)

The number of cable telephone subscribers stood at \(3.2\) million at the beginning of \(2004(t=0)\). For the next \(5 \mathrm{yr}\), the number was projected to grow at the rate of $$ R(t)=3.36(t+1)^{0.05} \quad(0 \leq t \leq 5) $$ million subscribers/year. If the projection held true, how many cable telephone subscribers were there at the beginning of \(2008(t=4)\) ?

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=1+\frac{1}{x^{2}} ; f(1)=2$$

The sales of Universal Instruments in the first \(t\) yr of its operation are approximated by the function $$ S(t)=t \sqrt{0.2 t^{2}+4} $$ where \(S(t)\) is measured in millions of dollars. What were Universal's average yearly sales over its first 5 yr of operation?

Find the area of the region bounded by the graph of the function \(f(x)=(x+1) / \sqrt{x}\), the \(x\) -axis, and the lines \(x=a\) and \(x=b\) where \(a\) and \(b\) are, respectively, the \(x\) -coordinates of the relative minimum point and the inflection point of \(f\)

The number of viewers of a weekly TV newsmagazine show, introduced in the 2003 season, has been increasing at the rate of $$ 3\left(2+\frac{1}{2} t\right)^{-1 / 3} \quad(1 \leq t \leq 6) $$ million viewers/year in its \(t\) th year on the air. The number of viewers of the program during its first year on the air is given by \(9(5 / 2)^{2 / 3}\) million. Find how many viewers were expected in the 2008 season.

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=e^{x}-2 x ; f(0)=2$$

Show that the area of a region \(R\) bounded above by the graph of a function \(f\) and below by the graph of a function \(g\) from \(x=a\) to \(x=b\) is given by $$ \int_{a}^{b}[f(x)-g(x)] d x $$ Hint: The validity of the formula was verified earlier for the case when both \(f\) and \(g\) were nonnegative. Now, let \(f\) and \(g\) be two functions such that \(f(x) \geq g(x)\) for \(a \leq x \leq b\). Then, there exists some nonnegative constant \(c\) such that the curves \(y=f(x)+c\) and \(\bar{y}=\) \(g(x)+c\) are translated in the \(y\) -direction in such a way that the region \(R^{r}\) has the same area as the region \(R\) (see the accompanying figures). Show that the area of \(R^{\prime}\) is given by $$ \int_{a}^{b}\\{[f(x)+c]-[g(x)+c]\\} d x=\int_{a}^{b}[f(x)-g(x)] d x $$

The manager of TeleStar Cable Service estimates that the total number of subscribers to the service in a certain city \(t\) yr from now will be $$ N(t)=-\frac{40,000}{\sqrt{1+0.2 t}}+50,000 $$ Find the average number of cable television subscribers over the next \(5 \mathrm{yr}\) if this prediction holds true.

Determine whether the statement is true or false. Give a reason for your answer. \(\int_{-1}^{1} \frac{1}{x^{3}} d x=-\left.\frac{1}{2 x^{2}}\right|_{-1} ^{1}=-\frac{1}{2}-\left(-\frac{1}{2}\right)=0\)

The registrar of Kellogg University estimates that the total student enrollment in the Continuing Education division will grow at the rate of $$ N^{\prime}(t)=2000(1+0.2 t)^{-3 / 2} $$ students/year, \(t\) yr from now. If the current student enrollment is 1000 , find an expression giving the total student enrollment \(t\) yr from now. What will be the student enrollment 5 yr from now?

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=\frac{x+1}{x} ; f(1)=1$$

The concentration of a certain drug in a patient's bloodstream \(t\) hr after injection is $$ C(t)=\frac{0.2 t}{t^{2}+1} $$ \(\mathrm{mg} / \mathrm{cm}^{3}\). Determine the average concentration of the drug in the patient's bloodstream over the first \(4 \mathrm{hr}\) after the drug is injected.

The number of people watching TV on mobile phones is expected to grow at the rate of $$ N^{\prime}(t)=\frac{5.4145}{\sqrt{1+0.91 t}} \quad(0 \leq t \leq 4) $$ million/year. The number of people watching TV on mobile phones at the beginning of \(2007(t=0)\) was \(11.9\) million. a. Find an expression giving the number of people watching TV on mobile phones in year \(t .\) b. According to this projection, how many people will be watching TV on mobile phones at the beginning of \(2011 ?\)

Find \(f(x)\) by solving the initial value problem. $$f^{\prime}(x)=1+e^{x}+\frac{1}{x} ; f(1)=3+e$$

The price of a certain commodity in dollars/unit at time \(t\) (measured in weeks) is given by $$ p=18-3 e^{-2 t}-6 e^{-\sqrt{/ 3}} $$ What is the average price of the commodity over the 5 -wk period from \(t=0\) to \(t=5\) ?

Determine whether the statement is true or false. Give a reason for your answer. \(\int_{0}^{2}(1-x) d x\) gives the area of the region under the graph of \(f(x)=1-x\) on the interval \([0,2]\).

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=\frac{1}{2} x^{-1 / 2} ;(2, \sqrt{2})$$ 60\. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

Determine whether the statement is true or false. Give a reason for your answer. The total revenue realized in selling the first 500 units of a product is given by $$ \int_{0}^{500} R^{\prime}(x) d x=R(500)-R(0) $$ where \(R(x)\) is the total revenue.

PopULATION GRowTH The population of a certain city is projected to grow at the rate of $$ r(t)=400\left(1+\frac{2 t}{24+t^{2}}\right) \quad(0 \leq t \leq 5) $$ people/year, \(t\) years from now. The current population is 60,000 . What will be the population 5 yr from now?

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(t)=t^{2}-2 t+3 ;(1,2)$$

When organic waste is dumped into a pond, the oxidization process that takes place reduces the pond's oxygen content. However, in time, nature will restore the oxygen content to its natural level. Suppose that the oxygen content \(t\) days after organic waste has been dumped into a pond is given by $$ f(t)=100\left(\frac{t^{2}+10 t+100}{t^{2}+20 t+100}\right) $$ percent of its normal level. Find the average content of oxygen in the pond over the first 10 days after organic waste has been dumped into it.

In calm waters, the oil spilling from the ruptured hull of a grounded tanker forms an oil slick that is circular in shape. If the radius \(r\) of the circle is increasing at the rate of $$ r^{\prime}(t)=\frac{30}{\sqrt{2 t+4}} $$ feet/minute \(t\) min after the rupture occurs, find an expression for the radius at any time \(t\). How large is the polluted area 16 min after the rupture occurred?

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=e^{x}+x ;(0,3)$$

During the construction of ? high-rise apartment building, a construction worker accidently drops a hammer that falls vertically a distance of \(h \mathrm{ft}\). The velocity of the hammer after falling a distance of \(x \mathrm{ft}\) is \(v=\sqrt{2 g x} \mathrm{ft} / \mathrm{sec}(0 \leq x \leq h) .\) Show that the aver- age velocity of the hammer over this path is \(\bar{v}=\frac{2}{3} \sqrt{2 g h}\) \(\mathrm{ft} / \mathrm{sec} .\)