Let f(t) be the number of US billionaires in year t. (a) Express the following

In Problems 14, decide whether the graph is concave up, concave down, or neither.

In Problems 14, decide whether the graph is concave up, concave down, or neither.

In Problems 14, decide whether the graph is concave up, concave down, or neither.

In Problems 14, decide whether the graph is concave up, concave down, or neither.

Table 1.12 gives values of a function w = f(t). Is this function increasing or decreasing? Is the graph of this function concave up or concave down? Table 1.12 t 0 4 8 12 16 20 24 w 100 58 32 24 20 18 17

Graph a function f(x) which is increasing everywhere and concave up for negative x and concave down for positive x.

For which pairs of consecutive points in Figure 1.33 is the function graphed: (a) Increasing and concave up? (b) Increasing and concave down? (c) Decreasing and concave up? (d) Decreasing and concave down? A B C D E F G H I x Figure 1.33

Find the average rate of change of f(x) = 2x2 between x = 1 and x = 3.

Find the average rate of change of f(x) = 3x2 + 4 between x = 2 and x = 1. Illustrate your answer graphically. 1

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. 1

Table 1.13 gives the net sales of The Gap, Inc, which operates nearly 3000 clothing stores. 29 (a) Find the change in net sales between 2008 and 2010. (b) Find the average rate of change in net sales between 2005 and 2008. Give units and interpret your answer. (c) From 2005 to 2010, were there any one-year intervals during which the average rate of change was positive? If so, when? Table 1.13 Gap net sales, in millions of dollars Year 2005 2006 2007 2008 2009 2010 Sales 16,019 15,923 15,763 14,526 14,197 14,660 1

Table 1.14 shows world bicycle production.30 (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. Table 1.14 World bicycle production, in millions Year 1950 1960 1970 1980 1990 2000 Bicycles 11 20 36 62 92 101 1

Table 1.15 shows attendance at NFL football games.31 (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). Table 1.15 Attendance at NFL football games, in millions of fans Year 2003 2004 2005 2006 2007 Attendance 21.64 21.71 21.79 22.20 22.26 1

Figure 1.34 shows the total value of US imports, in billions of dollars.32 (a) Was the value of the imports higher in 1985 or in 2003? Approximately howmuch higher? (b) Estimate the average rate of change of US imports between 1985 and 2003. Give units and interpret your answer in terms of imports. 1

Figure 1.35 shows a particles distance from a point. What is the particles average velocity from t = 0 to t = 3? 2 4 1 3 5 s(t) t (sec) distance (meters) Figure 1.35 1

At time t in seconds, a particles distance s(t), in cm, from a point is given in the table. What is the average velocity of the particle from t = 3 to t = 10? t 0 3 6 10 13 s(t) 0 72 92 144 180 1

Table 1.16 shows world population, P, in billions of people, world passenger automobile production, A, in millions of cars, and world cell phone subscribers, C, inmillions of subscribers.33 (a) Find the average rate of change, with units, for each of P, A, and C between 1995 and 2005. (b) Between 1995 and 2005, which increased faster: (i) Population or the number of automobiles? (ii) Population or the number of cell phone subscribers? Table 1.16 Year 1995 2000 2005 P (billions) 5.68 6.07 6.45 A (millions) 36.1 41.3 45.9 C (millions) 91 740 2168 1

Table 1.17 gives sales of Pepsico, which operates two major businesses: beverages (including Pepsi) and snack foods.34 (a) Find the change in sales between 2003 and 2010. (b) Find the average rate of change in sales between 2003 and 2010. Give units and interpret your answer. Table 1.17 Pepsico sales, in millions of dollars Year 2003 2004 2005 2006 2007 2008 2009 2010 Sales 26,971 29,261 32,562 35,137 39,474 43,251 43,230 57,838 1

Figure 1.36 shows a particles distance from a point. What is the particles average velocity from t = 1 to t = 3? 2 4 1 3 5 7 s(t) t (sec) distance (meters) Figure 1.36 2

Do you expect the average rate of change (in units per year) of each of the following to be positive or negative? Explain your reasoning. (a) Number of acres of rain forest in the world. (b) Population of the world. (c) Number of polio cases each year in the US, since 1950. (d) Height of a sand dune that is being eroded. (e) Cost of living in the US. 2

Table 1.18 shows the production of tobacco in the US.35 (a) What is the average rate of change in tobacco production between 2003 and 2010? Give units and interpret your answer in terms of tobacco production. (b) During this seven-year period, is there any interval during which the average rate of change was positive? If so, when? Table 1.18 Tobacco production, in millions of pounds Year 2003 2004 2005 2006 2007 2008 2009 2010 Production 802 882 645 728 788 801 823 719 2

Figure 1.37 shows the length, L, in cm, of a sturgeon (a type of fish) as a function of the time, t, in years.36 (a) Is the function increasing or decreasing? Is the graph concave up or concave down? (b) Estimate the average rate of growth of the sturgeon between t = 5 and t = 15. Give units and interpret your answer in terms of the sturgeon. 2

Table 1.19 shows the total US labor force, L. Find the average rate of change between 1940 and 2000; between 1940 and 1960; between 1980 and 2000. Give units and interpret your answers in terms of the labor force.37 Table 1.19 US labor force, in thousands of workers Year 1940 1960 1980 2000 L 47,520 65,778 99,303 136,891 2

The total world marine catch38 of fish, in metric tons, was 17 million in 1950 and 99 million in 2001. Whatwas the average rate of change in the marine catch during this period? Give units and interpret your answer. 2

Table 1.20 gives the revenues, R, ofGeneralMotors, formerly the worlds largest auto manufacturer.39 (a) Find the change in revenues between 2003 and 2008. (b) Find the average rate of change in revenues between 2003 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 negative? If so, which? Table 1.20 GM revenues, billions of dollars Year 2003 2004 2005 2006 2007 2008 R 184.0 192.9 193.1 205.6 181.1 149.0 2

The number of US households with cable television40 was 12,168,450 in 1977 and 73,365,880 in 2003. Estimate the average rate of change in the number of US households with cable television during this 26-year period. Give units and interpret your answer. 2

Figure 1.38 shows the amount of nicotine N = f(t), in mg, in a persons bloodstream as a function of the time, t, in hours, since the last cigarette. (a) Is the average rate of change in nicotine level positive or negative? Explain. (b) Find the average rate of change in the nicotine level between t = 0 and t = 3. Give units and interpret your answer in terms of nicotine. 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 t (hours) N (mg) Figure 1.38 2

Table 1.21 shows the concentration, c, of creatinine in the bloodstream of a dog.41 (a) Including units, find the average rate at which the concentration is changing between the (i) 6th and 8th minutes. (ii) 8th and 10th minutes. (b) Explain the sign and relative magnitudes of your results in terms of creatinine. Table 1.21 t (minutes) 2 4 6 8 10 c ( mg/ml) 0.439 0.383 0.336 0.298 0.266 2

The population of the world reached 1 billion in 1804, 2 billion in 1927, 3 billion in 1960, 4 billion in 1974, 5 billion in 1987 and 6 billion in 1999. Find the average rate of change of the population of the world, in people per minute, during each of these intervals (that is, from 1804 to 1927, 1927 to 1960, etc.). 3

In terms of the muscle, interpret the (a) Verticalintercept (b) Horizontal intercept 3

(a) Find the change inmuscle contraction velocity when the load changes from 1 kg to 3 kg. Give units. (b) Find the average rate of change in the contraction velocity between 1 kg and 3 kg. Give units. 3

Table 1.22 gives the sales, S, of IntelCorporation, a leading manufacturer of integrated circuits.42 (a) Find the change in sales between 2005 and 2010. (b) Find the average rate of change in sales between 2005 and 2010. Give units and interpret your answer. 3

Let f(t) be the number of US billionaires in year t. (a) Express the following statements43 in terms of f. (i) In 1985 there were 13 US billionaires. (ii) In 1990 there were 99 US billionaires. (b) Find the average yearly increase in the number of US billionaires between 1985 and 1990. Express this using f. (c) Assuming the yearly increase remains constant, find a formula predicting the number of US billionaires in year t. 3

Figure 1.40 shows the position of an object at time t. (a) Draw a line on the graph whose slope represents the average velocity between t = 2 and t = 8. (b) Is average velocity greater between t = 0 and t = 3 or between t = 3 and t = 6? (c) Is average velocity positive or negative between t = 6 and t = 9? 1 2 3 4 5 6 7 8 9 t (sec) s (ft) Figure 1.40 3

In an experiment, a lizard is encouraged to run as fast as possible. Figure 1.41 shows the distance run in meters as a function of the time in seconds.44 (a) If the lizard were running faster and faster, what would be the concavity of the graph? Does this match what you see? (b) Estimate the average velocity of the lizard during this 0.8 second experiment. 0.2 0.4 0.6 0.8 1 2 time (seconds) distance (meters) Figure 1.41 3

Values of F(t), G(t), and H(t) are in Table 1.23. Which graph is concave up and which is concave down? Which function is linear? Table 1.23 t F(t) G(t) H(t) 10 15 15 15 20 22 18 17 30 28 21 20 40 33 24 24 50 37 27 29 60 40 30 35 3

Experiments suggest that the male maximum heart rate (the most times a males 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? 3

A car starts slowly and then speeds up. Eventually the car slows down and stops. Graph the distance that the car has traveled against time. 3

In Problems 3942, find the relative, or percent, change. B changes from 12,000 to 15,000 4

In Problems 3942, find the relative, or percent, change. S changes from 400 to 450 4

In Problems 3942, find the relative, or percent, change. W changes from 0.3 to 0.05 4

In Problems 3942, find the relative, or percent, change. R changes from 50 to 47 4

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. 4

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. 4

An increase in class size from 5 to 10; an increase in class size from 30 to 50. 4

An increase in sales from $100,000 to $500,000; an increase in sales from $20,000,000 to $20,500,000. 4

Find the relative change of a population if it changes (a) From1000to2000 (b) From2000to1000 (c) From1,000,000to1,001,000 4

On Black Monday, October 28, 1929, the stock market on Wall Street crashed. The Dow Jones average dropped from 298.94 to 260.64 in one day. What was the relative change in the index? 4

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. 5

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 1.2447 to estimate the inflation rate for each of years 20072012. Table 1.24 Year 2007 2008 2009 2010 2011 2012 CPI 202.416 211.08 211.143 216.687 220.223 226.655 5

During 2008 the US economy stopped growing and began to shrink. Table 1.2548 gives quarterly data on the US Gross Domestic Product (GDP), which measures the size of the economy. (a) Estimate 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? Table 1.25 t (years since 2008) 0 0.25 0.5 0.75 1.0 GDP (trillion dollars) 14.15 14.29 14.41 14.2 14.09 5

An alternative to petroleum-based diesel fuel, biodiesel, is derived from renewable resources such as food crops, algae, and animal oils. The table shows the recent annual percent growth in US biodiesel consumption.49 Year 2005 2006 2007 2008 2009 % growth over previous yr 237 186.6 37.2 11.7 7.3 (a) Find the largest time interval over which the percentage growth in the US consumption of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case. (b) Find the largest time interval over which the actual US consumption of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case. 5

Hydroelectric power is electric power generated by the force of moving water. Figure 1.42 shows50 the annual percent growth in hydroelectric power consumption by the US industrial sector between 2004 and 2009. (a) Find the largest time interval over which the percentage growth in the US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case. (b) Find the largest time interval over which the actual US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case. 2005 2007 2009 50 40 30 20 10 10 year percent growth over previous year Figure 1.42 5

Solar panels are arrays of photovoltaic cells that convert solar radiation into electricity. The table shows the annual percent change in the US price per watt of a solar panel.51 Year 2004 2005 2006 2007 2008 % growth over previous yr 5.7 6.7 9.7 3.7 3.6 (a) Find the largest time interval over which the percentage growth in the US price per watt of a solar panel was an increasing function of time. Interpret what increasing means, practically speaking, in this case. (b) Find the largest time interval over which the actual price per watt of a solar panel was an increasing function of time. Interpret what increasing means, practically speaking, in this case. 5

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