Chapter 2

For Activities 1 through \(6,\) use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=3 x-2\) is \(\frac{d f}{d x}=3\).

For Activities 1 through \(4, \quad\) numerically estimate the slope of the line tangent to the graph of the function \(f\) at the given input value. Show the numerical estimation table with at least four estimates. \(f(x)=2^{x} ; x=2\), estimate to the nearest tenth

Flight Distance The function \(p\) gives the number of miles from an airport that a plane has flown after \(t\) hours. a. What are the units on \(p^{\prime}(1.5)\) ? b. What common word is used for \(p^{\prime}(1.5) ?\)

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval. Apple Stock Prices During a media event at which CEO Steve Jobs spoke, Apple shares opened at \(\$ 156.86\) and dropped to \(\$ 151.80\) fifty minutes into Jobs's keynote address.

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=15 x+32\) is \(\frac{d f}{d x}=15\).

Numerically estimate the slope of the line tangent to the graph of the function \(f\) at the given input value. Show the numerical estimation table with at least four estimates. \(f(x)=-x^{2}+4 x ; x=3,\) estimate to the nearest tenth

Mutual Fund Value The function \(B\) gives the balance, in dollars, in a mutual fund \(t\) years after the initial investment. Assume that no deposits or withdrawals are made during the investment period. a. What are the units on \(B^{\prime}(12)\) ? b. What is the financial interpretation of \(B^{\prime}(12) ?\)

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval. China Internet Users The number of Internet users in China grew from 12 million in 2000 to 103 million in \(2005 .\) (Source: BDA [China], The Strategis Group and China Daily, July \(22,2005 .\)

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=3 x^{2}\) is \(f^{\prime}(x)=6 x\).

Numerically estimate the slope of the line tangent to the graph of the function \(f\) at the given input value. Show the numerical estimation table with at least four estimates. \(f(x)=2 \sqrt{x}, x=1,\) estimate to the nearest hundredth

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval. Iowa Community College Tuition The average community college tuition in Iowa during \(2000-2001\) was \(\$ 1,856 .\) In \(2009-2010,\) the average community college tuition in Iowa was \(\$ 3,660 .\)

Airline Profit The function \(f\) gives the weekly profit, in thousand dollars, that an airline makes on its flights from Boston to Washington D.C. when the ticket price is \(p\) dollars. Interpret the following: a. \(f(65)=15\) b. \(f^{\prime}(65)=1.5\) c. \(f^{\prime}(90)=-2\)

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=-3 x^{2}-5 x\) is \(f^{\prime}(x)=-6 x-5\).

Numerically estimate the slope of the line tangent to the graph of the function \(f\) at the given input value. Show the numerical estimation table with at least four estimates. \(f(x)=5 \ln x ; x=5,\) estimate to the nearest hundredth

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval. Unemployment Rate The unemployment rate was \(9.4 \%\) in July 2009, up from \(5.4 \%\) in January \(2003 .\)

Airline Sales The function \(t\) gives the number of oneway tickets from Boston to Washington D.C. that a certain airline sells in 1 week when the price of each ticket is \(p\) dollars. Interpret the following: a. \(t(115)=1750\) b. \(t^{\prime}(115)=220\) c. \(t^{\prime}(125)=22\)

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=x^{3}\) is \(f^{\prime}(x)=3 x^{2}\).

Calculate and write a sentence interpreting each of the following descriptions of change: a. change b. percentage change c. average rate of change Airline Profit AirTran posted a profit of \(\$ 17.6\) million at the end of 2009 compared with a loss of \(\$ 121.6\) million in 2008 .

Typing Speed The function \(w\) gives the number of words per minute (wpm) that a student in a keyboard class can type after \(t\) weeks in the course. a. Is it possible for \(w(2)\) to be negative? Explain. b. What are the units on \(\left.\frac{d w}{d t}\right|_{t=2}\) ? c. Is it possible for \(\left.\frac{d w}{d t}\right|_{t=2}\) to be negative? Explain.

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=2 x^{0.5}\) is \(f^{\prime}(x)=x^{-0.5}\).

Bank Account The future value of a certain savings account with no activity besides compounding of interest is modeled as $$ F(t)=1500\left(1.0407^{t}\right) \text { dollars } $$ where \(t\) is the number of years since \(\$ 1500\) was invested. a. Numerically estimate to the nearest cent the rate of change of the future value when \(t=10 .\) b. Calculate the percentage rate of change of the future value when \(t=10\).

Calculate and write a sentence interpreting each of the following descriptions of change: a. change b. percentage change c. average rate of change Airline Revenue For the second quarter of 2009 , AirTran posted revenue of \(\$ 603.7\) million compared with revenue of \(\$ 693.4\) million during the second quarter of 2008 .

Corn Crop \(\quad\) The function \(C\) gives the number of bushels of corn produced on a tract of farmland that is treated with \(f\) pounds of nitrogen per acre. a. Is it possible for \(C(90)\) to be negative? Explain. b. What are the units on \(\left.\frac{d C}{d f}\right|_{f=90}\) ? c. Is it possible for \(\left.\frac{d C}{d f}\right|_{f=90}\) to be negative? Explain.

For Activities 7 through 10 a. use the limit definition of the derivative (algebraic method) to write an expression for the rate-of-change function of the given function. b. evaluate the rate of change as indicated. \(f(x)=4 x^{2} ; f^{\prime}(2)\)

Swim Time The time it takes an average athlete to swim 100 meters freestyle at age \(x\) years can be modeled as \(t(x)=0.181 x^{2}-8.463 x+147.376\) seconds (Source: Based on data from Swimming World, August 1992 ) a. Numerically estimate to the nearest tenth the rate of change of the time for a 13 -year-old swimmer to swim 100 meters freestyle. b. Determine the percentage rate of change of swim time for a 13-year-old. c. Is a 13-year-old swimmer's time improving or getting worse as the swimmer gets older?

Calculate and write a sentence interpreting each of the following descriptions of change: a. change b. percentage change c. average rate of change ACT Scores The percentage of students meeting national mathematics benchmarks on the ACT increased from \(40 \%\) in 2004 to \(43 \%\) in 2008.

Shirt Profit The function \(P\) gives the profit in dollars that a fraternity makes selling \(x\) T-shirts. a. Is it possible for \(P(30)\) to be negative? Explain. b. Is it possible for \(P^{\prime}(100)\) to be negative? Explain. c. If \(P^{\prime}(200)=-1.5,\) is the fraternity losing money? Explain.

Electronics Sales (1990s) Annual U.S. factory sales of consumer electronics goods to dealers from 1990 through 2001 can be modeled as $$ s(t)=0.0388 t^{3}-0.495 t^{2}+5.698 t+43.6 $$ where output is measured in billion dollars and \(t\) is the number of years since \(1990 .\) (Sources: Based on dara from Statistical Abstract, 2001 ; and Consumer Electronics Association) a. Numerically estimate to the nearest tenth the derivative of \(s\) when \(t=10\). b. Interpret the answer to part \(a\).

a. use the limit definition of the derivative (algebraic method) to write an expression for the rate-of-change function of the given function. b. evaluate the rate of change as indicated. \(\quad s(t)=-2.3 t^{2} ; s^{\prime}(1.5)\)

Political Membership \(\quad\) The function \(m\) gives the number of members in a political organization \(t\) years after its founding. a. What are the units on \(m^{\prime}(10)\) ? b. Is it possible for \(m^{\prime}(10)\) to be negative? Explain.

Electronics Sales (1990s) Annual U.S. factory sales of consumer electronics goods to dealers from 1990 through 2001 can be modeled as $$ s(t)=0.0388 t^{3}-0.495 t^{2}+5.698 t+43.6 $$ where output is measured in billion dollars and \(t\) is the number of years since \(1990 .\) (Sources: Based on data from Statistical Abstract, 2001 ; and Consumer Electronics Association a. Numerically estimate to the nearest tenth the derivative of \(s\) when \(t=10\) b. Interpret the answer to part \(a\).

a. use the limit definition of the derivative (algebraic method) to write an expression for the rate-of-change function of the given function. b. evaluate the rate of change as indicated. \(\quad g(t)=4 t^{2}-3 ;\left.\frac{d g}{d t}\right|_{t=4}\)

October Madness The scatter plot shows the number of shares traded each day during October of \(1987 .\) The behavior of the graph on October 19 and 20 has been referred to as "October Madness." a. Calculate the percentage change and average rate of change in the number of shares traded per trading day between October 1 (when 193.2 million shares were traded) and October 30,1987 (when 303.4 shares were traded). b. Draw a secant line whose slope is the average rate of change between October 1 and October \(30,1987 .\) c. Write a sentence describing how the number of shares traded changed throughout the month. How well does the average rate of change calculated in part \(a\) reflect what occurred throughout the month?

Sketch a possible graph of \(t\) with input \(x\), given that \(t(3)=7\) \(t(4.4)=t(8)=0\) \(t^{\prime}(6.2)=0\) the graph of \(t\) has no concavity changes.

Mountain Bike Profit For a certain brand of bicycle, \(P(x)=1.02^{x}\) Canadian dollars gives the profit from the sale of \(x\) mountain bikes. On June \(27,2009, P\) Canadian dollars were worth \(C(P)=\frac{P}{1.1525}\) American dollars. Assume that this conversion applies today. a. Write a function for profit in American dollars from the sale of \(x\) mountain bikes. b. What is the profit in Canadian and in American dollars from the sale of 400 mountain bikes? c. Numerically estimate the rate of change in profit to the nearest cent in both Canadian dollars and American dollars.

a. use the limit definition of the derivative (algebraic method) to write an expression for the rate-of-change function of the given function. b. evaluate the rate of change as indicated. \(\quad m(p)=4 p+p^{2} ;\left.\frac{d m}{d p}\right|_{p=-2}\)

Weekly Sales The average weekly sales for Abercrombic and Fitch between 2004 and 2008 are given below. Average Weekly Sales for Abercrombie and Fitch \begin{tabular}{|c|c|} \hline Year & Thousand Dollars \\ \hline 2004 & 38.87 \\ \hline 2005 & 53.56 \\ \hline 2006 & 63.81 \\ \hline 2007 & 72.12 \\ \hline 2008 & 68.08 \\ \hline \end{tabular} (Source: Based on datz from the 2009 ANF Yearly Report) a. What behavior suggested by a scatter plot of the data indicates that a quadratic model is appropriate? b. Align the input so that \(t=0\) in \(2000 .\) Find a quadratic model for the data. c. Numerically estimate the derivative of the model from part \(b\) in 2007 to the nearest hundred dollars. d. Interpret the answer to part \(c\).

Falling Object An object is dropped off a building. Ignoring air resistance, the height above the ground \(t\) seconds after being dropped is given by $$ h(t)=-16 t^{2}+100 \text { feet } $$ a. Use the limit definition of the derivative to find a rate-of-change equation for the height. b. Use the answer to part \(a\) to determine how rapidly the object is falling after 1 second.

Weight Loss \(\quad\) The function \(w\) gives a person's weight \(t\) weeks after she begins a diet. Write a sentence of interpretation for each of the following statements: a. \(w(0)=167\) and \(w(12)=142\) b. \(w^{\prime}(1)=-2\) and \(w^{\prime}(9)=-1\) c. \(\left.\frac{d w}{d t}\right|_{t=12}=0\) and \(\left.\frac{d w}{d t}\right|_{t=15}=0.25\) d. Sketch a possible graph of \(w\).

Park City Population (Historic) Park City, Utah was settled as a mining community in 1870 and experienced growth until the late \(1950 \mathrm{~s}\) when the price of silver dropped. In the past 40 years, Park City has experienced new growth as a thriving ski resort. The population data for selected years between 1900 and 2009 are given below. Park City, Utah \begin{tabular}{|c|c|} \hline Year & Population \\ \hline 1900 & 3759 \\ \hline 1930 & 4281 \\ \hline 1940 & 3739 \\ \hline 1950 & 2254 \\ \hline 1970 & 1193 \\ \hline 1980 & 2823 \\ \hline 1990 & 4468 \\ \hline 2000 & 7341 \\ \hline 2009 & 11983 \\ \hline \end{tabular} (Source: Riley Moffart, Popularion History of Western U.S. Citio \(d\) Towne, \(1850-1990\), Lanham: Scarecrow, 1996 , 309 ; and U.S. Bureat of the Census) a. What behavior of a scatter plot of the data indicates that a cubic model is appropriate? b. Align the input so that \(t=0\) in \(1900 .\) Find a cubic model for the data. c. Numerically estimate the derivative of the model in 2008 to the nearest hundred. d. Interpret the answer to part \(c\).

Distance Clinton County, Michigan, is mostly flat farmland partitioned by straight roads (often gravel) that run either north/south or east/west. A tractor driven north on Lowell Road from the Schafers farm's mailbox is $$ f(t)=0.28 t+0.6 \text { miles } $$ north of Howe Road \(t\) minutes after leaving the farm's mailbox. a. How far is the Schafers' mailbox from Howe Road? b. Use the limit definition of the derivative to show that the tractor is moving at a constant speed. c. How quickly (in miles per hour) is the tractor moving?

Fuel Efficiency The function \(g\) gives the fuel efficiency, in miles per gallon, of a car traveling \(v\) miles per hour. Write a sentence of interpretation for each of the following statements. a. \(g(55)=32.5\) and \(g^{\prime}(55)=-0.25\) b. \(g^{\prime}(45)=0.15\) and \(g^{\prime}(51)=0\) c. Sketch a possible graph of \(g\).

Coal Prices The average price paid by the synfuel industry for a short ton of coal between 2002 and 2005 can be modeled as $$ p(t)=1.2 t^{2}-6.1 t+39.5 \text { dollars } $$ where \(t\) is the number of years since the beginning of \(2000 .\) a. Use the limit definition of the derivative to develop a formula for the rate of change of the price of coal used by the synthetic fuel industry. b. How quickly was the price of coal used by the synthetic fuel industry growing in the middle of \(2003 ?\)

Doubling Time The function \(D\) gives the time, in years, that it takes for an investment to double if interest is continuously compounded at \(r \% .\) a. What are the units on \(D^{\prime}(9)\) ? b. Why does it make sense that \(\left.\frac{d D}{d r}\right|_{r=a}\) is negative for every positive \(a\) ? c. Write a sentence of interpretation for each of the following statements: i. \(D(9)=7.7\) ii. \(D^{\prime}(5)=-2.77\) iii. \(\left.\frac{d D}{d r}\right|_{r=12}=-0.48\) iv. \(D(16)=5.79\)

Ice Cream Sales The table lists average monthly sales for an ice cream company. Ice Cream Sales \begin{tabular}{|c|c|c|c|} \hline Month & Monthly Sales (thou. dollars) & Month & Monthly Sales (thou. dollars) \\ \hline Jan & 50 & July & 167 \\ \hline Feb & 60 & Aug & 159 \\ \hline Mar & 77 & Sept & 108 \\ \hline Apr & 96 & Oct & 75 \\ \hline May & 137 & Nov & 61 \\ \hline June & 158 & Dec & 54 \\ \hline \end{tabular} a. Write a sine model for the ice cream data. b. Use the model to estimate the average rate of change in monthly sales between September and November. c. Numerically estimate to the nearest thousand dollars the rate of change of monthly sales in October. d. Write a sentence of interpretation for the answer to parts \(b\) and \(c\)

Swim Time The time it takes an average athlete to swim 100 meters freestyle at age \(x\) years can be modeled as $$ t(x)=0.181 x^{2}-8.463 x+147.376 \text { seconds } $$ (Source: Based on data from Swimming World, August 1992 ) a. Calculate the swim time when \(x=13\). b. Use the algebraic method to develop a formula for the derivative of \(t\) c. How quickly is the time to swim 100 meters freestyle changing for an average 13 -year-old athlete? Interpret the result.

Unemployment The relation \(u\) gives the number of people unemployed in a country \(t\) months after the election of a new president. a. Is \(u\) a function? Why or why not? b. Interpret the following facts about \(u(t)\) in statements describing the unemployment situation: $$ \text { i. } u(0)=3,000,000 $$ ii. \(u(12)=2,800,000\) iii. \(u^{\prime}(24)=0\) iv. \(\left.\frac{d u}{d t}\right|_{t=36}=100,000\) v. \(u^{\prime}(48)=-200,000\) c. On the basis of the information in part \(b\), sketch a possible graph of the number of people unemployed during the first 48 months of the president's term. Label numbers and units on the axes.

Airline Fuel The amount of airline fuel consumed by Southwest Airlines each year between 2004 and 2008 can be modeled as $$ f(t)=-0.009 t^{2}+0.12 t+1.19 \text { billion gallons } $$ where \(t\) is the number of years since 2004 (Source: Based on data from Bureau of Transportation Statistics) a. Calculate the amount of fuel consumed in \(2007 .\) b. Use the algebraic method to develop a formula for the derivative of \(f\) c. How quickly was the amount of fuel used by Southwest Airlines changing in \(2007 ?\) Interpret the result.

Single-Mom Births The function \(s\) gives the percentage of all births to single mothers in the United States in year \(t\) from 1940 through \(2000 .\) Using the following information, sketch a graph of \(s\). (Sources: Based on data from L. Usdansky, "Single Motherhood: Stereotypes vs. Statistics," New York Times, February 11 , \(1996,\) Section \(4,\) page \(\mathrm{E} 4 ;\) and on data from Statistical Abstract, 1998) \- \(s(1940) \approx 4\) \- \(s^{\prime}(t)\) is never zero. \- \(s(1970)=12\) \- \(s(2000)\) is approximately 21 percentage points more than \(s(1970)\). \- The average rate of change of \(s\) between 1970 and 1980 is 0.6 percentage point per year. \- Lines tangent to the graph of \(s\) lie below the graph at all points between 1940 and 1990 and above the graph between 1990 and \(2000 .\)

Flu Shots The percentage of adults who said they got a flu shot before the winter of year \(t\) is given by $$ S(t)=-0.18 t^{2}+5.24 t+9 \text { percent } $$ where \(t\) is the number of years since 2000 , data from \(2004 \leq t \leq 2009 .\) (Source: Based on data in USA Today, p. \(1 \mathrm{~A}, 5 / 18 / 2009)\) a. Find the derivative formula using the algebraic method. b. Evaluate the derivative of \(s\) in \(2007 .\) Interpret the result.