Chapter 1

The population, \(P\), in millions, of Nicaragua was 5.4 million in 2004 and growing at an annual rate of \(3.4 \% .\) Let \(t\) be time in years since 2004 (a) Express \(P\) as a function in the form \(P=P_{0} a^{t}\) (b) Express \(P\) as an exponential function using base \(e\) (c) Compare the annual and continuous growth rates.

Plot graphs based on the following story: "As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumperto-bumper until we passed an accident, after which traffic flow went back to normal until I exited." Distance from my exit vs time on the highway

Experiments suggest that the male maximum heart rate (the most times a male's heart can safely beat in a minute) decreases by 9 beats per minute during the first 21 years of his life, and by 26 beats per minute during the first 33 years. \(^{45}\) If you model the maximum heart rate as a function of age, should you use a function that is increasing or decreasing? Concave up or concave down?

The demand for a product is given by \(p=90-10 q .\) Find the ratio \(\frac{\text { Relative change in demand }}{\text { Relative change in price }}\) if the price changes from \(p=50\) to \(p=51 .\) Interpret this ratio.

The gross world product is \(W=32.4(1.036)^{t},\) where \(W\) is in trillions of dollars and \(t\) is years since 2001 Find a formula for gross world product using a continuous growth rate.

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector. $$\begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \text { \% growth over previous yr } & -1.9 & -10 & -45.4 & 5.1 & 11 \\ \hline \end{array}$$ (a) According to the US Department of Energy, the US industrial sector consumed about 29 trillion BTUs of hydroelectric power in \(2006 .\) Approximately how much hydroelectric power (in trillion BTUs) did the US consume in \(2007 ?\) In \(2005 ?\) (b) Graph the points showing the annual US consumption of hydroelectric power, in trillion BTUs, for the years 2004 to \(2009 .\) Label the scales on the horizontal and vertical axes. (c) According to this data, when did the largest yearly decrease, in trillion BTUs, in the US consumption of hydroelectric power occur? What was this decrease?

A demand curve has equation \(q=100-5 p,\) where \(p\) is price in dollars. A \(\$ 2\) tax is imposed on consumers. Find the equation of the new demand curve. Sketch both curves.

The population of the world can be represented by \(P=\) \(7(1.0115)^{t},\) where \(P\) is in billions of people and \(t\) is years since \(2012 .\) Find a formula for the population of the world using a continuous growth rate.

(a) Use the Rule of 70 to predict the doubling time of an investment which is earning \(8 \%\) interest per year. (b) Find the doubling time exactly, and compare your answer to part (a).

Find the relative, or percent, change. \(B\) changes from 12,000 to 15,000

A supply curve has equation \(q=4 p-20,\) where \(p\) is price in dollars. A \(\$ 2\) tax is imposed on suppliers. Find the equation of the new supply curve. Sketch both curves.

A fishery stocks a pond with 1000 young trout. The number of trout \(t\) years later is given by \(P(t)=1000 e^{-0.5 t}\) (a) How many trout are left after six months? After 1 year? (b) Find \(P(3)\) and interpret it in terms of trout. (c) At what time are there 100 trout left? (d) Graph the number of trout against time, and describe how the population is changing. What might be causing this?

A business associate who owes you 3000 offers to pay you 2800 now, or else pay you three yearly installments of 1000 each, with the first installment paid now. If you use only financial reasons to make your decision, which option should you choose? Justify your answer, assuming a 6 \%$ interest rate per year, compounded continuously.

Find the relative, or percent, change. \(S\) changes from 400 to 450

A tax of $$ 8\( per unit is imposed on the supplier of an item. The original supply curve is \)q=0.5 p-25\( and the demand curve is \)q=165-0.5 p,\( where \)p$ is price in dollars. Find the equilibrium price and quantity before and after the tax is imposed.

The Hershey Company is the largest US producer of chocolate. In \(2011,\) annual net sales were 6.1 billion dollars and were increasing at a continuous rate of \(4.2 \%\) per year. \(^{65}\) (a) Write a formula for annual net sales, \(S,\) as a function of time, \(t,\) in years since 2011 (b) Estimate annual net sales in 2015 (c) Use a graph to estimate the year in which annual net sales are expected to pass 8 billion dollars and check your estimate using logarithms.

A person is to be paid 2000 for work done over a year. Three payment options are being considered. Option 1 is to pay the 2000 in full now. Option 2 is to pay \(\$ 1000\) now and \(\$ 1000\) in a year. Option 3 is to pay the full 2000 in a year. Assume an annual interest rate of \(5 \%\) a year, compounded continuously. (a) Without doing any calculations, which option is the best option financially for the worker? Explain. (b) Find the future value, in one year's time, of all three options. (c) Find the present value of all three options.

Find the relative, or percent, change. \(W\) changes from 0.3 to 0.05

The demand and supply curves for a product are given in terms of price, \(p,\) by $$q=2500-20 p \quad \text { and } \quad q=10 p-500$$ (a) Find the equilibrium price and quantity. Represent your answers on a graph. (b) A specific tax of $$ 6\( per unit is imposed on suppliers. Find the new equilibrium price and quantity. Represent your answers on the graph. (c) How much of the $$ 6\) tax is paid by consumers and how much by producers? (d) What is the total tax revenue received by the government?

During a recession a firm's revenue declines continuously so that the revenue, \(R\) (measured in millions of dollars), in \(t\) years' time is given by \(R=5 e^{-0.15 t}\) (a) Calculate the current revenue and the revenue in two years' time. (b) After how many years will the revenue decline to \(\$ 2.7\) million?

A company is considering whether to buy a new machine, which costs \(\$ 97.000 .\) The cash flows (adjusted for taxes and depreciation) that would be generated by the new machine are given in the following table: $$\begin{array}{c|c|c|c|c}\hline \text { Ycar } & 1 & 2 & 3 & 4 \\\\\hline \text { Cash flow } & 550.000 & \$ 40.000 & \$ 25,000 & 520,000 \\\\\hline\end{array}$$ (a) Find the total present value of the cash flows. Treat each year's cash flow as a lump sum at the end of the year and use an interest rate of \(7.5 \%\) per year, compounded annually. (b) Based on a comparison of the cost of the machine and the present value of the cash flows, would you recommend purchasing the machine?

Find the relative, or percent, change. \(R\) changes from 50 to 47

In Example \(8,\) the demand and supply curves are given by \(q=100-2 p\) and \(q=3 p-50,\) respectively; the equilibrium price is $$ 30\( and the equilibrium quantity is 40 units. A sales tax of \)5 \%$ is imposed on the consumer. (a) Find the equation of the new demand and supply curves. (b) Find the new equilibrium price and quantity. (c) How much is paid in taxes on each unit? How much of this is paid by the consumer and how much by the producer? (d) How much tax does the government collect?

The population of a city is 50,000 in 2008 and is growing at a continuous yearly rate of \(4.5 \%\) (a) Give the population of the city as a function of the number of years since \(2008 .\) Sketch a graph of the population against time. (b) What will be the city's population in the year \(2018 ?\) (c) Calculate the time for the population of the city to reach \(100,000 .\) This is called the doubling time of the population.

Which relative change is bigger in magnitude? Justify your answer. The change in the Dow Jones average from 164.6 to 77.9 in \(1931 ;\) the change in the Dow Jones average from 13261.8 to 8776.4 in 2008

For children and adults with diseases such as asthma, the number of respiratory deaths per year increases by \(0.33 \%\) when pollution particles increase by a microgram per cubic meter of \operatorname{air}^{66}. (a) Write a formula for the number of respiratory deaths per year as a function of quantity of pollution in the air. (Let \(Q_{0}\) be the number of deaths per year with no pollution.) (b) What quantity of air pollution results in twice as many respiratory deaths per year as there would be without pollution?

Which relative change is bigger in magnitude? Justify your answer. The change in the US population from 5.2 million to 7.2 million from 1800 to \(1810 ;\) the change in the US population from 151.3 to 179.3 from 1950 to \(1960 .\)

The concentration of the car exhaust fume nitrous oxide, \(\mathrm{NO}_{2},\) in the air near a busy road is a function of distance from the road. The concentration decays exponentially at a continuous rate of \(2.54 \%\) per meter. \(^{67}\) At what distance from the road is the concentration of \(\mathrm{NO}_{2}\) half what it is on the road?

Which relative change is bigger in magnitude? Justify your answer. An increase in class size from 5 to \(10 ;\) an increase in class size from 30 to 50 .

You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for 375 . The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the extended warranty are 150 at the end of the first year of the extension and 250 at the end of the second year of the extension. The interest rate is \(5 \%\) per year, compounded annually. Should you buy the extended warranty? Explain.

With time, \(t,\) in years since the start of \(1980,\) textbook prices have increased at \(6.7 \%\) per year while inflation has been \(3.3 \%\) per year. \(^{68}\) Assume both rates are continuous growth rates. (a) Find a formula for \(B(t),\) the price of a textbook in year \(t\) if it \(\operatorname{cost} \$ B_{0}\) in 1980 (b) Find a formula for \(P(t),\) the price of an item in year \(t\) if it cost \(\$ P_{0}\) in 1980 and its price rose according to inflation. (c) A textbook cost \(\$ 50\) in \(1980 .\) When is its price predicted to be double the price that would have resulted from inflation alone?

Which relative change is bigger in magnitude? Justify your answer. An increase in sales from \(\$ 100,000\) to \(\$ 500,000 ;\) an increase in sales from \(\$ 20,000,000\) to \(\$ 20,500,000\)

In \(2011,\) the populations of China and India were approximately 1.34 and 1.19 billion people \(^{69},\) respectively. However, due to central control the annual population growth rate of China was \(0.4 \%\) while the population of India was growing by \(1.37 \%\) each year. If these growth rates remain constant, when will the population of India exceed that of China?

Which relative change is bigger in magnitude? Justify your answer. Find the relative change of a population if it changes (a) From 1000 to \(2000 \quad\) (b) From 2000 to 1000 (c) From 1,000,000 to 1,001,000

In 2010 , there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US. \(^{70}\) The number of vehicles grew \(15.5 \%\) over the previous decade, while the population has been growing at \(9.7 \%\) per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?

On January \(27,2013,\) the cost to mail a letter in the US \(^{46}\) was raised from 45 cents to 46 cents. Find the relative change in the cost.

The US Consumer Price Index (CPI) is a measure of the cost of living. The inflation rate is the annual relative rate of change of the CPI. Use the January data in Table \(1.24^{47}\) to estimate the inflation rate for each of years \(2007-2012\) $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2007 & 2008 & 2009 & 2010 & 2011 & 2012 \\ \hline \text { CPI } & 202.416 & 211.08 & 211.143 & 216.687 & 220.223 & 226.655 \\ \hline \end{array}$$

During 2008 the US economy stopped growing and began to shrink. Table \(1.25^{48}\) gives quarterly data on the US Gross Domestic Product (GDP), which measures the size of the economy.(a) Fstimate the relative growth rate (percent per year) at the first four times in the table. (b) Economists often say an economy is in recession if the GDP decreases for two quarters in a row. Was the US in recession in \(2008 ?\) $$\begin{array}{c|c|c|c|c|c} \hline t \text { (years since 2008) } & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline \text { GDP (trillion dollars) } & 14.15 & 14.29 & 14.41 & 14.2 & 14.09 \\\ \hline \end{array}$$

(a) Write an equation for a graph obtained by vertically stretching the graph of \(y=x^{2}\) by a factor of \(2,\) followed by a vertical upward shift of 1 unit. Sketch it. (b) What is the equation if the order of the transformations (stretching and shifting) in part (a) is interchanged? (c) Are the two graphs the same? Explain the effect of reversing the order of transformations. (GRAPH CAN'T COPY)

The functions \(r=f(t)\) and \(V=g(r)\) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable \(t\) is in minutes, \(r\) is in feet, and \(V\) is in cubic feet. The inflation begins at \(t=0 .\) In each case, give a mathematical expression that represents the given statement. The volume of the balloon if its radius were twice as big.

School organizations raise money by selling candy door to door. When the price is \(\$ 1\) a school organization sells 2765 candies and when the price goes up to \(\$ 1.25\) the quantity of sold candy drops down to 2440 (a) Find the relative change in the price of candy. (b) Find the relative change in the quantity of candy sold. (c) Find and interpret the ratio \(\frac{\text { Relative change in quantity }}{\text { Relative change in price }}\)