Chapter 5

A cup of coffee at \(90^{\circ} \mathrm{C}\) is put into a \(20^{\circ} \mathrm{C}\) room when \(t=0 .\) The coffee's temperature is changing at a rate of \(r(t)=-7\left(0.9^{t}\right)^{\circ} \mathrm{C}\) per minute, with \(t\) in minutes. Estimate the coffee's temperature when \(t=10\)

Explain in words what the integral represents and give units. \(\int_{1}^{3} v(t) d t,\) where \(v(t)\) is velocity in meters/sec and \(t\) is time in seconds.

Find the area under the graph of \(f(x)=x^{2}+2\) between \(x=0\) and \(x=6.\)

Suppose that you travel 30 miles/hour for 2 hours, then 40 miles/hour for \(1 / 2\) hour, then 20 miles/hour for 4 hours. (a) What is the total distance you traveled? (b) Sketch a graph of the velocity function for this trip. (c) Represent the total distance traveled on your graph in part (b).

If the marginal cost function \(C^{\prime}(q)\) is measured in dollars per ton, and \(q\) gives the quantity in tons, what are the units of measurement for \(\int_{800}^{900} C^{\prime}(q) d q ?\) What does this integral represent?

Explain in words what the integral represents and give units. \(\int_{0}^{6} a(t) d t,\) where \(a(t)\) is acceleration in \(\mathrm{km} / \mathrm{hr}^{2}\) and \(t\) is time in hours.

Find the area under \(P=100(0.6)^{t}\) between \(t=0\) and \(t=8.\)

Find the average value of the function over the given interval. $$g(t)=1+t \text { over }[0,2]$$

The marginal cost of drilling an oil well depends on the depth at which you are drilling: drilling becomes more expensive, per meter, as you dig deeper into the earth. The fixed costs are 1,000,000 riyals (the riyal is the unit of currency of Saudi Arabia), and, if \(x\) is the depth in meters, the marginal costs are $$ C^{\prime}(x)=4000+10 x \quad \text { riyals } / \text { meter } $$ Find the total cost of drilling a 500 -meter well.

Explain in words what the integral represents and give units. \(\int_{2005}^{2011} f(t) d t,\) where \(f(t)\) is the rate at which world population is growing in year \(t,\) in billion people per year.

Find the total area between \(y=4-x^{2}\) and the \(x\) -axis for \(0 \leq x \leq 3.\)

Estimate $$\int_{0}^{6} 2^{x} d x$$ using a left-hand sum with \(n=2\).

Find the average value of the function over the given interval. $$g(t)=e^{t} \text { over }[0,10]$$

Explain in words what the integral represents and give units. \(\int_{0}^{5} s(x) d x,\) where \(s(x)\) is rate of change of salinity (salt concentration) in gm/liter per cm in sea water, and where \(x\) is depth below the surface of the water in \(\mathrm{cm} .\)

A car comes to a stop six seconds after the driver applies the brakes. While the brakes are on, the velocities recorded are in Table 5.5 $$\begin{array}{l|r|r|r|r} \hline \text { Time since brakes applied (sec) } & 0 & 2 & 4 & 6 \\ \hline \text { Velocity (ft)sec) } & 88 & 45 & 16 & 0 \\ \hline \end{array}$$ (a) Give lower and upper estimates for the distance the car traveled after the brakes were applied. (b) On a sketch of velocity against time, show the lower and upper estimates of part (a).

Find the area between \(y=x+5\) and \(y=2 x+1\) between \(x=0\) and \(x=2.\)

Estimate $$\int_{0}^{12} \frac{1}{x+1} d x$$ using a left-hand sum with \(n=3\).

Oil leaks out of a tanker at a rate of \(r=f(t)\) gallons per minute, where \(t\) is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

A car starts moving at time \(t=0\) and goes faster and faster. Its velocity is shown in the following table. Estimate how far the car travels during the 12 seconds. $$\begin{array}{c|c|c|c|c|c} \hline t \text { (seconds) } & 0 & 3 & 6 & 9 & 12 \\ \hline \text { Velocity (ft/sec) } & 0 & 10 & 25 & 45 & 75 \\ \hline \end{array}$$

Find the area enclosed by \(y=3 x\) and \(y=x^{2}.\)

Estimate $$\int_{0}^{1} e^{-x^{2}} d x$$ using \(n=5\) rectangles to form a (a) Left-hand sum (b) Right-hand sum

Pollution is removed from a lake on day \(t\) at a rate of \(f(t) \mathrm{kg} /\) day. (a) Explain the meaning of the statement \(f(12)=500\). (b) If \(\int_{5}^{15} f(t) d t=4000,\) give the units of the \(5,\) the \(15,\) and the 4000. (c) Give the meaning of \(\int_{5}^{15} f(t) d t=4000\).

Use the following table to estimate $$\int_{10}^{26} f(x) d x$$. $$\begin{array}{c|c|c|c|c|c} \hline x & 10 & 14 & 18 & 22 & 26 \\\\\hline f(x) & 100 & 88 & 72 & 50 & 28 \\\\\hline\end{array}$$

Annual coal production in the US (in billion tons per year) is given in the table. \(^{6}\) Estimate the total amount of coal produced in the US between 1997 and \(2009 .\) If \(r=f(t)\) is the rate of coal production \(t\) years since 1997, write an integral to represent the \(1997-2009\) coal production. $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 1997 & 1999 & 2001 & 2003 & 2005 & 2007 & 2009 \\ \hline \text { Rate } & 1.090 & 1.094 & 1.121 & 1.072 & 1.132 & 1.147 & 1.073 \\\ \hline \end{array}$$

Use the table to estimate $$\int_{0}^{40} f(x) d x .$$ What values of \(n\) and \(\Delta x\) did you use? $$\begin{array}{l|c|c|c|c|c} \hline x & 0 & 10 & 20 & 30 & 40 \\\\\hline f(x) & 350 & 410 & 435 & 450 & 460 \\\\\hline\end{array}$$

The following table gives the US emissions, \(H(t),\) of hydrofluorocarbons, or "super greenhouse gasses," in teragrams equivalent of carbon dioxide, with \(t\) in years since \(2000 .^{7}\) (a) What are the units and meaning of \(\int_{0}^{10} H(t) d t ?\) (b) Estimate \(\int_{0}^{10} H(t) d t\) $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2002 & 2004 & 2006 & 2008 & 2010 \\ \hline H(t) & 7104 & 7022 & 7163 & 7159 & 7048 & 6822 \\ \hline \end{array}$$

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff's data follow: $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Time since start (min) } & 0 & 15 & 30 & 45 & 60 & 75 & 90 \\ \hline \text { Speed (mph) } & 12 & 11 & 10 & 10 & 8 & 7 & 0 \\ \hline \end{array}$$ (a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour. (b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half.

Use the following table to estimate $$\int_{0}^{15} f(x) d x$$. $$\begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 3 & 6 & 9 & 12 & 15 \\\\\hline f(x) & 50 & 48 & 44 & 36 & 24 & 8 \\\\\hline\end{array}$$

World annual natural gas \(^{8}\) consumption, \(N,\) in millions of metric tons of oil equivalent, is approximated by \(N=\) \(1770+53 t,\) where \(t\) is in years since 1990 (a) How much natural gas was consumed in \(1990 ?\) In \(2010 ?\) (b) Estimate the total amount of natural gas consumed during the 20 -year period from 1990 to 2010 .

The marginal cost function for a company is given by $$ C^{\prime}(q)=q^{2}-16 q+70 \text { dollars/unit } $$ where \(q\) is the quantity produced. If \(C(0)=500\), find the total cost of producing 20 units. What is the fixed cost and what is the total variable cost for this quantity?

The following table gives world oil consumption, in billions of barrels per year. \(^{1}\) Estimate total oil consumption during this 25 -year period. $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1985 & 1990 & 1995 & 2000 & 2005 & 2010 \\\\\hline \text { Oil (bn barrels/yr) } & 20.9 & 23.3 & 25.6 & 28.0 & 30.7 & 31.7 \\\\\hline\end{array}$$

Use the following table to estimate $$\int_{3}^{4} W(t) d t .$$ What are \(n\) and \(\Delta t ?\) $$\begin{array}{c|c|c|c|c|c|c} \hline t & 3.0 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 \\\\\hline W(t) & 25 & 23 & 20 & 15 & 9 & 2 \\\\\hline\end{array}$$

Solar photovoltaic (PV) cells are the world's fastest growing energy source. In year \(t\) since \(2007,\) PV cells were manufactured worldwide at a rate of \(S=3.7 e^{0.61 t}\) gigawatts per year. \(^{9}\) Estimate the total solar energy generating capacity of the PV cells manufactured between 2007 and 2011.

The marginal cost function of producing \(q\) mountain bikes is $$ C^{\prime}(q)=\frac{600}{0.3 q+5} $$ (a) If the fixed cost in producing the bicycles is \(\$ 2000\), find the total cost to produce 30 bicycles. (b) If the bikes are sold for \(\$ 200\) each, what is the profit (or loss) on the first 30 bicycles? (c) Find the marginal profit on the \(31^{\text {st }}\) bicycle.

If \(t\) is measured in days since June 1 , the inventory \(I(t)\) for an item in a warchouse is given by $$I(t)=5000(0.9)^{t}$$ (a) Find the average inventory in the warehouse during the 90 days after June 1 (b) Graph \(I(t)\) and illustrate the average graphically.

The velocity of a car is \(f(t)=5 t\) meters/sec. Use a graph of \(f(t)\) to find the exact distance traveled by the car, in meters, from \(t=0\) to \(t=10\) seconds.

The marginal revenue function on sales of \(q\) units of a product is \(R^{\prime}(q)=200-12 \sqrt{q}\) dollars per unit. (a) Graph \(R^{\prime}(q)\) (b) Estimate the total revenue if sales are 100 units. (c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if sales are 101 units.

The population of the world \(t\) years after 2010 is predicted to be \(P=6.9 e^{0.012 t}\) billion. (a) What population is predicted in \(2020 ?\) (b) What is the predicted average population between 2010 and \(2020 ?\)

Filters at a water treatment plant become less effective over time. The rate at which pollution passes through the filters into a nearby lake is given in the following table. (a) Estimate the total quantity of pollution entering the lake during the 30 -day period. (b) Your answer to part (a) is only an estimate. Give bounds (lower and upper estimates) between which the true quantity of pollution must lie. (Assume the rate of pollution is continually increasing.) $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Day } & 0 & 6 & 12 & 18 & 24 & 30 \\ \hline \text { Ratc (kg/day) } & 7 & 8 & 10 & 13 & 18 & 35 \\ \hline \end{array}$$

The net worth, \(f(t)\), of a company is growing at a rate of \(f^{\prime}(t)=2000-12 t^{2}\) dollars per year, where \(t\) is in years since \(2005 .\) How is the net worth of the company expected to change between 2005 and \(2015 ?\) If the company is worth \(\$ 40.000\) in \(2005,\) what is it worth in \(2015 ?\)

A car initially going \(50 \mathrm{ft} / \mathrm{sec}\) brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds. (a) Graph the velocity from \(t=0\) to \(t=5\) (b) How far does the car travel? (c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant rate?

Your velocity is \(v(t)=\ln \left(t^{2}+1\right) \mathrm{ft} / \mathrm{sec}\) for \(t\) in seconds, \(0 \leq t \leq 3 .\) Find the distance traveled during this time.

Oil is pumped from a well at a rate of \(r(t)\) barrels per day, with \(t\) in days. Assume \(r^{\prime}(t)<0\) and \(t_{0}>0\) What does the value of \(\int_{0}^{t_{0}} r(t) d t\) tells us about the oil well?

Without calculation, what can you say about the relationship between the values of the two integrals: $$\int_{0}^{2} e^{x^{2}} d x \text { and } \int_{0}^{2} e^{t^{2}} d t ?$$

A forest fire covers 2000 acres at time \(t=0 .\) The fire is growing at a rate of \(8 \sqrt{t}\) acres per hour, where \(t\) is in hours. How many acres are covered 24 hours later?

If we know $$\int_{2}^{5} f(x) d x=4,$$ what is the value of $$3\left(\int_{2}^{5} f(x) d x\right)+1 ?$$

The number of hours, \(H,\) of daylight in Madrid as a function of date is approximated by the formula $$H=12+2.4 \sin (0.0172(t-80))$$ where \(t\) is the number of days since the start of the year. Find the average number of hours of daylight in Madrid: (a) in January (b) in June (c) over a year (d) Explain why the relative magnitudes of your answers to parts (a), (b), and (c) are reasonable.

Your velocity is given by \(v(t)=t^{2}+1\) in \(\mathrm{m} / \mathrm{sec},\) with \(t\) in Eeconds. Estimate the distance, \(s\), traveled between \(t=0\) and \(t=5 .\) Explain how you arrived at your estimate.

(a) Graph \(f(x)=x(x+2)(x-1)\) (b) Find the total area between the graph and the \(x\) -axis between \(x=-2\) and \(x=1\) (c) Find \(\int_{-2}^{1} f(x) d x\) and interpret it in terms of areas.

Water is pumped out of a holding tank at a rate of \(5-5 e^{-0.12 t}\) liters/minute, where \(t\) is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later?