Chapter 1

(a) Make a table of values for \(y=e^{x}\) using \(x=\) \(0,1,2,3 .\) (b) Plot the points found in part (a). Does the graph look like an exponential growth or decay function? (c) Make a table of values for \(y=e^{-x}\) using \(x=\) \(0,1,2,3 .\) (d) Plot the points found in part (c). Does the graph look like an exponential growth or decay function?

Find the average rate of change of \(f(x)=3 x^{2}+4\) between \(x=-2\) and \(x=1\). Illustrate your answer graphically.

A company rents cars at $$\$ 40$$ a day and 15 cents a mile. Its competitor's cars are $$\$ 50$$ a day and 10 cents a mile. (a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled. (b) On the same axes, graph both functions. (c) How should you decide which company is cheaper?

A person breathes in and out every three seconds. The volume of air in the person's lungs varies between a minimum of 2 liters and a maximum of 4 liters. Which of the following is the best formula for the volume of air in the person's lungs as a function of time? (a) \(y=2+2 \sin \left(\frac{\pi}{3} t\right)\) (b) \(y=3+\sin \left(\frac{2 \pi}{3} t\right)\) (c) \(y=2+2 \sin \left(\frac{2 \pi}{3} t\right)\) (d) \(y=3+\sin \left(\frac{\pi}{3} t\right)\)

Write a formula representing the function. The energy, \(E\), expended by a swimming dolphin is proportional to the cube of the speed, \(v\), of the dolphin.

Use the variable \(u\) for the inside function to express each of the following as a composite function: (a) \(y=\left(5 t^{2}-2\right)^{6}\) (b) \(P=12 e^{-0.6 t}\) (c) \(C=12 \ln \left(q^{3}+1\right)\)

The exponential function \(y(x)=C e^{\mathrm{ax}}\) satisfies the conditions \(y(0)=2\) and \(y(1)=1 .\) Find the constants \(C\) and \(\alpha .\) What is \(y(2) ?\)

Solve for \(t\) using natural logarithms. $$2 P=P e^{0.3 t}$$

A company producing jigsaw puzzles has fixed costs of $$\$ 6000$$ and variable costs of $$\$ 2$$ per puzzle. The company sells the puzzles for $$\$ 5$$ each. (a) Find formulas for the cost function, the revenue function, and the profit function. (b) Sketch a graph of \(R(q)\) and \(C(q)\) on the same axes. What is the break- even point, \(q_{0}\), for the company?

Graph \(y=100 e^{-0.4 x}\). Describe what you see.

When a deposit of $$\$ 1000$$ is made into an account paying \(8 \%\) interest, compounded annually, the balance, $$\$ B$$, in the account after \(t\) years is given by \(B=1000(1.08)^{t}\). Find the average rate of change in the balance over the interval \(t=0\) to \(t=5\). Give units and interpret your answer in terms of the balance in the account.

Delta Cephei is one of the most visible stars in the night sky. Its brightness has periods of \(5.4\) days, the average brightness is \(4.0\) and its brightness varies by \(\pm 0.35 .\) Find a formula that models the brightness of Delta Cephei as a function of time, \(t\), with \(t=0\) at peak brightness.

Write a formula representing the function. The average velocity, \(v\), for a trip over a fixed distance, \(d\), is inversely proportional to the time of travel, \(t .\)

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+1)-m(z)$$

Air pressure, \(P\), decreases exponentially with the height, \(h\), in meters above sea level: $$ P=P_{0} e^{-0.00012 h} $$ where \(P_{0}\) is the air pressure at sea level. (a) At the top of Mount McKinley, height 6194 meters (about 20,320 feet), what is the air pressure, as a percent of the pressure at sea level? (b) The maximum cruising altitude of an ordinary commercial jet is around 12,000 meters (about 39,000 feet). At that height, what is the air pressure, as a percent of the sea level value?

Solve for \(t\) using natural logarithms. $$5 e^{3 t}=8 e^{2 t}$$

Production costs for manufacturing running shoes consist of a fixed overhead of $$\$ 650,000$$ plus variable costs of $$\$ 20$$ per pair of shoes. Each pair of shoes sells for $$\$ 70$$. (a) Find the total cost, \(C(q)\), the total revenue, \(R(q)\), and the total profit, \(\pi(q)\), as a function of the number of pairs of shoes produced, \(q .\) (b) Find the marginal cost, marginal revenue, and marginal profit. (c) How many pairs of shoes must be produced and sold for the company to make a profit?

Find a possible formula for the function represented by the data. $$ \begin{array}{c|c|c|c|c} \hline x & 0 & 1 & 2 & 3 \\ \hline f(x) & 4.30 & 6.02 & 8.43 & 11.80 \\ \hline \end{array} $$

Table \(1.11\) shows world bicycle production. \(^{28}\) (a) Find the change in bicycle production between 1950 and 2000 . Give units. (b) Find the average rate of change in bicycle production between 1950 and 2000. Give units and interpret your answer in terms of bicycle production. $$ \begin{array}{l} \text { Table 1.11 World bicycle production, in millions }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Bicycles } & 11 & 20 & 36 & 62 & 92 & 101 \\ \hline \end{array} \end{array} $$

Annual revenue \(R\) from McDonald's restaurants worldwide can be estimated by \(R=19.1+1.8 t\), where \(R\) is in billion dollars and \(t\) is in years since January \(1,2005 .{ }^{13}\) (a) What is the slope of this function? Include units. Interpret the slope in terms of McDonald's revenue. (b) What is the vertical intercept of this function? Include units. Interpret the vertical intercept in terms of McDonald's revenue. (c) What annual revenue does the function predict for \(2010 ?\) (d) When is annual revenue predicted to hit 30 billion dollars?

Write a formula representing the function. The gravitational force, \(F\), between two bodies is inversely proportional to the square of the distance \(d\) between them.

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+h)-m(z)$$

The antidepressant fluoxetine (or Prozac) has a half-life of about 3 days. What percentage of a dose remains in the body after one day? After one week?

Solve for \(t\) using natural logarithms. $$7 \cdot 3^{t}=5 \cdot 2^{t}$$

The cost \(C\), in millions of dollars, of producing \(q\) items is given by \(C=5.7+0.002 q\). Interpret the \(5.7\) and the \(0.002\) in terms of production. Give units.

Find a possible formula for the function represented by the data. $$ \begin{array}{c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 \\ \hline g(t) & 5.50 & 4.40 & 3.52 & 2.82 \\ \hline \end{array} $$

Table \(1.12\) gives the net sales of The Gap, Inc, which operates nearly 3000 clothing stores. \({ }^{29}\) (a) Find the change in net sales between 2005 and 2008 . (b) Find the average rate of change in net sales between 2005 and 2008. Give units and interpret your answer. (c) From 2003 to 2008 , were there any one-year intervals during which the average rate of change was positive? If so, when? $$ \begin{array}{l} \text { Table } 1.12 \text { Gap net sales, in millions of dollars }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Sales } & 15,854 & 16,267 & 16,019 & 15,923 & 15,763 & 14,526 \\\ \hline \end{array} \end{array} $$

Let \(y\) be the percent increase in annual US national production during a year when the unemployment rate changes by \(u\) percent. (For example, \(u=2\) if unemployment increases from \(4 \%\) to \(6 \%\).) Okun's law states that $$ y=3.5-2 u $$ (a) What is the meaning of the number \(3.5\) in Okun's law? (b) What is the effect on national production of a year when unemployment rises from \(5 \%\) to \(8 \%\) ? (c) What change in the unemployment rate corresponds to a year when production is the same as the year before? (d) What is the meaning of the coefficient \(-2\) in Okun's law?

The number of species of lizards, \(N\), found on an island off Baja California is proportional to the fourth root of the area, \(A\), of the island \({ }^{66}\) Write a formula for \(N\) as a function of \(A\). Graph this function. Is it increasing or decreasing? Is the graph concave up or concave down? What does this tell you about lizards and island area?

Simplify the quantities using \(m(z)=z^{2}\). $$m(z)-m(z-h)$$

A firm decides to increase output at a constant relative rate from its current level of 20,000 to 30,000 units during the next five years. Calculate the annual percent rate of increase required to achieve this growth.

Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=5(1.07)^{t}$$

The table shows the cost of manufacturing various quantities of an item and the revenue obtained from their sale. $$ \begin{array}{r|r|r|r|r|r|r|r|r|r} \hline \text { Quantity } & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\ \hline \text { Cost (\$) } & 120 & 400 & 600 & 780 & 1000 & 1320 & 1800 & 2500 & 3400 \\ \hline \text { Revenue (\$) } & 0 & 300 & 600 & 900 & 1200 & 1500 & 1800 & 2100 & 2400 \\ \hline \end{array} $$ (a) What range of production levels appears to be profitable? (b) Calculate the profit or loss for each of the quantities shown. Estimate the most profitable production level.

Table \(1.13\) shows attendance at NFL football games. (a) Find the average rate of change in the attendance from 2003 to 2007 . Give units. (b) Find the annual increase in the attendance for each year from 2003 to 2007 . (Your answer should be four numbers.) (c) Show that the average rate of change found in part (a) is the average of the four yearly changes found in part (b). $$ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline \text { Attendance } & 21.64 & 21.71 & 21.79 & 22.20 & 22.26 \\ \hline \end{array} $$

When a patient with a rapid heart rate takes a drug, the heart rate plunges dramatically and then slowly rises again as the drug wears off. Sketch the heart rate against time from the moment the drug is administered.

Which of the following tables could represent linear functions? $$ \begin{array}{l} \text { (a) }\\\ \begin{array}{l|c|c|c|c} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 27 & 25 & 23 & 21 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (b) }\\\ \begin{array}{l|l|l|l|l} \hline t & 15 & 20 & 25 & 30 \\ \hline s & 62 & 72 & 82 & 92 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (c) }\\\ \begin{array}{c|c|c|c|c} \hline u & 1 & 2 & 3 & 4 \\ \hline w & 5 & 10 & 18 & 28 \\ \hline \end{array} \end{array} $$

The surface area of a mammal, \(S\), satisfies the equation \(S=k M^{2 / 3}\), where \(M\) is the body mass, and the constant of proportionality \(k\) depends on the body shape of the mammal. A human of body mass 70 kilograms has surface area \(18,600 \mathrm{~cm}^{2} .\) Find the constant of proportionality for humans. Find the surface area of a human with body mass 60 kilograms.

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+h)-m(z-h)$$

The half-life of a radioactive substance is 12 days. There are \(10.32\) grams initially. (a) Write an equation for the amount, \(A\), of the substance as a function of time. (b) When is the substance reduced to 1 gram?

Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=7.7(0.92)^{t}$$

(a) Give an example of a possible company where the fixed costs are zero (or very small). (b) Give an example of a possible company where the marginal cost is zero (or very small).

One of the main contaminants of a nuclear accident, such as that at Chernobyl, is strontium-90, which decays exponentially at a rate of approximately \(2.5 \%\) per year. (a) Write the percent of strontium-90 remaining, \(P\), as a function of years, \(t\), since the nuclear accident. [Hint: \(100 \%\) of the contaminant remains at \(t=0 .]\) (b) Graph \(P\) against \(t\) (c) Estimate the half-life of strontium-90. (d) After the Chernobyl disaster, it was predicted that the region would not be safe for human habitation for 100 years. Estimate the percent of original strontium-90 remaining at this time.

Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=15 e^{-0.06 t}$$

A $$\$ 15,000$$ robot depreciates linearly to zero in 10 years. (a) Find a formula for its value as a function of time. (b) How much is the robot worth three years after it is purchased?

If the world's population increased exponentially from \(4.453\) billion in 1980 to \(5.937\) billion in 1998 and continued to increase at the same percentage rate between 1998 and 2008 , calculate what the world's population would have been in 2008 . How does this compare to the actual population of \(6.771\) billion?

Table \(1.14\) gives sales of Pepsico, which operates two major businesses: beverages (including Pepsi) and snack foods. \(^{32}\) (a) Find the change in sales between 2003 and 2008 . (b) Find the average rate of change in sales between 2003 and 2008. Give units and interpret your answer. $$ \begin{array}{l} \text { Table } 1.14 \text { Pepsico sales, in millions of dollars }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Sales } & 26,971 & 29,261 & 32,562 & 35,137 & 39,474 & 45,251 \\\ \hline \end{array} \end{array} $$

A company's pricing schedule in Table \(1.3\) is designed to encourage large orders. (A gross is 12 dozen.) Find a formula for: (a) \(q\) as a linear function of \(p\). (b) \(p\) as a linear function of \(q\). $$ \begin{array}{c|c|c|c|c} \hline q \text { (order size, gross) } & 3 & 4 & 5 & 6 \\ \hline p \text { (price/dozen) } & 15 & 12 & 9 & 6 \\ \hline \end{array} $$

Kleiber's Law states that the metabolic needs (such as calorie requirements) of a mammal are proportional to its body weight raised to the \(0.75\) power. \(^{67}\) Surprisingly, the daily diets of mammals conform to this relation well. Assuming Kleiber's Law holds: (a) Write a formula for \(C\), daily calorie consumption, as a function of body weight, \(W\). (b) Sketch a graph of this function. (You do not need scales on the axes.) (c) If a human weighing 150 pounds needs to consume 1800 calories a day, estimate the daily calorie requirement of a horse weighing 700 lbs and of a rabbit weighing 9 lbs. (d) On a per-pound basis, which animal requires more calories: a mouse or an elephant?

Represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=3.2 e^{0.03 t}$$

A new bus worth $$\$ 100,000$$ in 2010 depreciates linearly to $$\$ 20,000$$ in 2030 . (a) Find a formula for the value of the bus, \(V\), as a function of time, \(t\), in years since 2010 . (b) What is the value of the bus in 2015 ? (c) Find and interpret the vertical and horizontal intercepts of the graph of the function. (d) What is the domain of the function?