Find the average rate of change for the

Problem 44A Velocity The distance of a particle from some fixed point is given by s(t) = t2 + 5t + 2, where t is time measured in seconds. Find the average velocity of the particle over the following intervals. a. 4 to 6 seconds ________________ b. 4 to 5 seconds ________________ c. Find the instantaneous velocity of the particle when t = 4.

Problem 1E Find the average rate of change for the function over the given interval. y = x2 + 2x between x = 1 and x = 3

Problem 2E Find the average rate of change for the function over the given interval. y = ?4x2 ? 6 between x = 2 and x = 6

Problem 3E Find the average rate of change for the function over the given interval. y = ?3x3 + 2x2 ? 4x + 1 between x = ?2 and x = 1

Problem 5E Find the average rate of change for the function over the given interval. between x = 1 and x = 4

Problem 6E Find the average rate of change for the function over the given interval. between x = 1 and x = 2

Problem 7E Find the average rate of change for the function over the given interval. y = ex between x = ?2 and x = 0

Problem 4E Find the average rate of change for the function over the given interval. y = 2x3 ? 4x2 + 6x between x = ?1 and x = 4

Problem 8E Find the average rate of change for the function over the given interval. y = ln x between x = 2 and x = 4

Problem 9E Suppose the position of an object moving in a straight line is given by s(t) = t2 + 5t + 2. Find the instantaneous velocity at each time. t = 6

Problem 11E Suppose the position of an object moving in a straight line is given by s(t) = 5t2 ? 2t ? 7. Find the instantaneous velocity at each time. t = 2

Problem 13E Suppose the position of an object moving in a straight line is given by s(t) = t3 + 2t + 9. Find the instantaneous velocity at each time. t = 1

Problem 10E Suppose the position of an object moving in a straight line is given by s(t) = t2 + 5t + 2. Find the instantaneous velocity at each time. t = 1

Problem 14E Suppose the position of an object moving in a straight line is given by s(t) = t3 + 2t + 9. Find the instantaneous velocity at each time. t = 4

Problem 16E Find the instantaneous rate of change for the function at the given value. s(t) = ?4t2 ? 6 at t = 2

Problem 12E Suppose the position of an object moving in a straight line is given by s(t) = 5t2 ? 2t ? 7. Find the instantaneous velocity at each time. t = 3

Problem 17E Find the instantaneous rate of change for the function at the given value. g(t) = 1 ? t2 at t = ?1

Problem 15E Find the instantaneous rate of change for the function at the given value. f(x) = x2 + 2x at x = 0

Problem 18E Find the instantaneous rate of change for the function at the given value. F(x) = x2 + 2 at x = 0

Problem 23E Explain the difference between the average rate of change of y as x changes from a to b, and the instantaneous rate of change of y at x = a.

Problem 24E If the instantaneous rate of change of f(x) with respect to x is positive when x = 1, is f increasing or decreasing there?

Problem 19E Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. f(x) = xx at x = 2

Problem 20E Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. f(x) = xx at x = 3

Problem 25A Profit Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 2x2 ? 5x + 6. Find the average rate of change of profit for the following changes in x. a. 2 to 4 ________________ b. 2 to 3 ________________ c. Find and interpret the instantaneous rate of change of profit with respect to the number of items produced when x = 2. (This number is called the marginal profit at x = 2.) ________________ d. Find the marginal profit at x = 4.

Problem 22E Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. f(x) = xlnx at x = 3

Problem 21E Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the function at the given value. f(x) = xln x at x = 2

Problem 31A Gasoline Prices In 2008, the price of gasoline in the United States inexplicably spiked and then dropped. The average monthly price (in cents) per gallon of unleaded regular gasoline for 2008 is shown in the following chart. Find the average rate of change per month in the average price per gallon for each time period. Source: U.S. Energy Information Administration. a. From January to July (the peak) ________________ b. From July to December ________________ c. From January to December

Problem 30A Sales The graph shows annual sales (in thousands of dollars) of a Nintendo game at a particular store. Find the average annual rate of change in sales for the following changes in years. a. 1 to 4 ________________ b. 4 to 7 ________________ c. 7 to 12 ________________ d. What do your answers for parts a–c tell you about the sales of this product? ________________ e. Give an example of another product that might have such a sales curve.

Problem 29A Interest If $1000 is invested in an account that pays 5% compounded continuously, the total amount, A(t), in the account after t years is A(t) = 1000e0.05t. a. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5). ________________ b. Find the average rate of change per year of the total amount in the account for the second five years of the investment (from t = 5 to t = 10). ________________ c. Estimate the instantaneous rate of change for t = 5.

Problem 27A Demand Suppose customers in a hardware store are willing to buy N(p) boxes of nails at pdollars per box, as given by N(p) = 80 ? 5p2, 1 ? p ? 4. a. Find the average rate of change of demand for a change inprice from $2 to $3. ________________ b. Find and interpret the instantaneous rate of change ofdemand when the price is $2. ________________ c. Find the instantaneous rate of change of demand when the price is $3. ________________ d. As the price is increased from $2 to $3, how is demandchanging? Is the change to be expected? Explain.

Problem 28A Interest If $1000 is invested in an account that pays 5% com-pounded annually, the total amount, A(t), in the account after t years is A(t) = 1000(1.05)t. a. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5). ________________ b. Find the average rate of change per year of the total amount in the account for the second five years of the investment (from t = 5 to t = 10). ________________ c. Estimate the instantaneous rate of change for t = 5.

Problem 26A Revenue The revenue (in thousands of dollars) from producing x units of an item is R(x) = 10x ? 0.002x2. a. Find the average rate of change of revenue when production is increased from 1000 to 1001 units. ________________ b. Find and interpret the instantaneous rate of change of revenue with respect to the number of items produced when 1000 units are produced. (This number is called the marginal revenue at x= 1000.) ________________ c. Find the additional revenue if production is increased from 1000 to 1001 units. ________________ d. Compare your answers for parts a and c. What do you find? How do these answers compare with your answer to part b?

Problem 32A Medicare Trust Fund The graph shows the money remaining in the Medicare Trust Fund at the end of the fiscal year. Source: Social Security Administration. Find the approximate average rate of change in the trust fund for each time period. a. From 2000 to 2008 (the peak) ________________ b. From 2008 to 2018

Problem 33A Flu Epidemic Epidemiologists in College Station, Texas, estimate that t days after the flu begins to spread in town, the percent of the population infected by the flu is approximated by p(t) = t2 + t for 0 ?t ? 5. a. Find the average rate of change of p with respect to t over the interval from 1 to 4 days. ________________ b. Find and interpret the instantaneous rate of change of p with respect to t at t = 3.

Problem 37A Thermic Effect of Food The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food (in kJ/hr) for a particular person is F(t) = ?10.28 + 175.9te?t/1.3, where t is the number of hours that have elapsed since eating a meal. Source: American Journal of Clinical Nutrition. a. Graph the function on[0,6] by [–20, 100]. ________________ b. Find the average rate of change of the thermic effect of food during the first hour after eating. ________________ c. Use a graphing calculator to find the instantaneous rate of change of the thermic effect of food exactly 1 hour after eating. ________________ d. Use a graphing calculator to estimate when the function stops increasing and begins to decrease.

Problem 34A World Population Growth The future size of the world population depends on how soon it reaches replacement-level fertility, the point at which each woman bears on average about 2.1 children. The graph shows projections for reaching that point in different years. Source: Population Reference Bureau. Ultimate World Population Size Under Different Assumptions a. Estimate the average rate of change in population for each projection from 1990 to 2050. Which projection shows the smallest rate of change in world population? ________________ b. Estimate the average rate of change in population from 2090 to 2130 for each projection. Interpret your answer.

Problem 38A Mass of Bighorn Yearlings The body mass of yearling bighorn sheep on Ram Mountain in Alberta, Canada, can be estimated by M(t) = 27.5 + 0.3t ? 0.001t2 where M(t) is measured in kilograms and t is days since May 25. Source: Canadian Journal of Zoology. a. Find the average rate of change of the weight of a bighorn yearling between 105 and 115 days past May 25. ________________ b. Find the instantaneous rate of change of weight for a bighorn yearling sheep whose age is 105 days past May 25. ________________ c. Graph the function M(t) on [5, 125] by [25, 65]. ________________ d. Does the behavior of the function past 125 days accurately model the mass of the sheep? Why or why not?

Problem 35A Bacteria Population The graph shows the population in millions of bacteria t minutes after an antibiotic is introduced into a culture. Find and interpret the average rate of change of population with respect to time for the following time intervals. a. 1 to 2 ________________ b. 2 to 3 ________________ c. 3 to 4 ________________ d. 4 to 5 ________________ e. How long after the antibiotic was introduced did the population begin to decrease? ________________ f. At what time did the rate of decrease of the population slow down?

Problem 39A Immigration The following graph shows how immigration (in thousands) to the United States has varied over the past century. Source: Homeland Security. a. Find the average annual rate of change in immigration for the first half of the century (from 1905 to 1955). ________________ b. Find the average annual rate of change in immigration for the second half of the century (from 1955 to 2005). ________________ c. Find the average annual rate of change in immigration for the entire century (from 1905 to 2005). ________________ d. Average your answers to parts a and b, and compare the result with your answer from part c. Will these always be equal for any two time periods? ________________ e. If the annual average rate of change for the entire century continues, predict the number of immigrants in 2009. Compare your answer to the actual number of 1,130,818 immigrants.

Problem 41A Temperature The graph shows the temperature T in degrees Celsius as a function of the altitude h in feet when an inversion layer is over Southern California. (An inversion layer is formed when air at a higher altitude, say 3000 ft, is warmer than air at sea level, even though air normally is cooler with increasing altitude.) Estimate and interpret the average rate of change in temperature for the following changes in altitude. a. 1000 to 3000 ft ________________ b. 1000 to 5000 ft ________________ c. 3000 to 9000 ft ________________ d. 1000 to 9000 ft ________________ e. At what altitude at or below 7000 ft is the temperature highest? Lowest? How would your answer change if 7000 ft is changed to 10,000 ft? ________________ f. At what altitude is the temperature the same as it is at 1000 ft?

Problem 42A Velocity A car is moving along a straight test track. The position in feet of the car, s(t), at various times t is measured, with the following results. t (sec) 0 2 4 6 8 10 s (t) (ft) 0 10 14 20 30 36 Find and interpret the average velocities for the following changes in t. a. 0 to 2 seconds ________________ b. 2 to 4 seconds ________________ c. 4 to 6 seconds ________________ d. 6 to 8 seconds ________________ e. Estimate the instantaneous velocity at 4 seconds i. by finding the average velocity between 2 and 6 seconds, and ii. by averaging the answers for the average velocity in the two seconds before and the two seconds after (that is, the answers to parts b and c). ________________ f. Estimate the instantaneous velocity at 6 seconds using the two methods in part e. ________________ g. Notice in parts e and f that your two answers are the same. Discuss whether this will always be the case, and why or why not.

Problem 43A Velocity Consider the example at the beginning of this section regarding the car traveling from San Francisco. a. Estimate the instantaneous velocity at 1 hour. Assume that the velocity changes at a steady rate from one half-hour to the next. ________________ b. Estimate the instantaneous velocity at 2 hours.

Problem 36A Molars The crown length (as shown below) of first molars in fetuses is related to the postconception age of the tooth as L(t) = ?0.01t2 + 0.788t ? 7.048, where L(t) is the crown length, in millimeters, of the molar t weeks after conception. Source: American Journal of Physical Anthropology. a. Find the average rate of growth in crown length during weeks 22 through 28. ________________ b. Find the instantaneous rate of growth in crown length when the tooth is exactly 22 weeks of age. ________________ c. Graph the function on [0, 50] by [0, 9]. Does a function that increases and then begins to decrease make sense for this particular application? What do you suppose is happening during the first 11 weeks? Does this function accurately model crown length during those weeks?

Problem 40A Drug Use The following chart shows how the percentage of eighth graders, tenth graders, and twelfth graders who have used marijuana in their lifetime has varied in recent years. Source: National Institute of Health. a. Find the average annual rate of change in the percent of eighth graders who have used marijuana in their lifetime over the three-year period 2003–2006 and the three-year period 2006–2009. Then calculate the annual rate of change for 2003–2009. ________________ b. Repeat part a for tenth graders. ________________ c. Repeat part a for twelfth graders. ________________ d. Discuss any similarities and differences between your answers to parts a through c, as well as possible reasons for these differences and similarities.