Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

2 4.6 Applications and Models of Exponential Growth and Decay The Exponential Growth or Decay Function Growth Function Models Decay Function Models

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Exponential Growth or Decay Function In many situations that occur in ecology, biology, economics, and the social sciences, a quantity changes at a rate proportional to the amount present. In such cases the amount present at time t is a special function of t called an exponential growth or decay function.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Exponential Growth or Decay Function Let y 0 be the amount or number present at time t = 0. Then, under certain conditions, the amount present at any time t is modeled by where k is a constant.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Exponential Growth or Decay Function When k > 0, the function describes growth. In Section 4.2, we saw examples of exponential growth: compound interest and atmospheric carbon dioxide, for example. When k < 0, the function describes decay; one example of exponential decay is radioactivity.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE In Example 11, Section 4.2, we discussed the growth of atmospheric carbon dioxide over time. A function based on the data from the table was given in that example. Now we can see how to determine such a function from the data. YearCarbon Dioxide (ppm)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution (a) Find an exponential function that gives the amount of carbon dioxide y in year x. Recall that the graph of the data points showed exponential growth, so the equation will take the form y = y 0 e kt. We must find the values of y 0 and k. The data began with the year 1990, so to simplify our work we let 1990 correspond to x = 0, 1991 correspond to x = 1, and so on. Since y 0 is the initial amount, y 0 = 353 in 1990 when x = 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution Find an exponential function that gives the amount of carbon dioxide y in year x. Thus the equation is From the last pair of values in the table, we know that in 2275 the carbon dioxide level is expected to be 2000 ppm. The year 2275 corresponds to 2275 – 1990 = 285. Substitute 2000 for y and 285 for x and solve for k. (a)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Divide by 353. Take the logarithm on each side. In e x = x, for all x. Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Multiply by and rewrite. Use a calculator. A function that models the data is Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution (b) Let y = 2(280) = 560 in y = 353 e x, and find x. Divide by 353. Take the logarithm on each side. Estimate the year when future levels of carbon dioxide will be double the preindustrial level of 280 ppm.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE In e x = x, for all x. Multiply by and rewrite. Use a calculator. Since x = 0 corresponds to 1990, the preindustrial carbon dioxide level will double in the 75th year after 1990, or during 2065, according to this model. Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Example 2 FINDING DOUBLING TIME FOR MONEY How long will it take for the money in an account that accrues interest at a rate of 3%, compounded continuously, to double? Solution Continuous compounding formula Let A = 2P and r = Divide by P. Take the logarithm on each side.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 2 FINDING DOUBLING TIME FOR MONEY Solution Divide by 0.03 In e x = x Use a calculator. It will take about 23 yr for the amount to double. How long will it take for the money in an account that accrues interest at a rate of 3%, compounded continuously, to double?

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH According to the U.S. Census Bureau, the world population reached 6 billion people during 1999 and was growing exponentially. By the end of 2010, the population had grown to billion. The projected world population (in billions of people) t years after 2010, is given by the function

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH Based on this model, what will the world population be in 2020? Solution (a) Since t = 0 represents the year 2010, in 2020, t would be 2020 – 2010 = 10 yr. We must find  (t) when t is 10. Let t = 10. The population will be billion at the end of Given formula

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH Solution (b) Let f (t) = 9. In what year will the world population reach 9 billion? Divide by Take the logarithm on each side.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH In e x = x, for all x. Divide by and rewrite. Use a calculator. Thus, 34.8 yr after 2010, during the year 2044, world population will reach 9 billion. Solution (b) In what year will the world population reach 9 billion?

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Suppose 600 g of a radioactive substance are present initially and 3 yr later only 300 g remain. (a) Determine the exponential equation that models this decay. Solution Let y = 600 and t = 0. e 0 = 1 To express the situation as an exponential equation y = y 0 e kt, we use the given values first to find y 0 and then to find k.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Let y = 300 and t = 3. Divide by 600. Thus, y 0 = 600, which gives y = 600 e kt. Because the initial amount (600 g ) decays to half that amount (300 g ) in 3 yr, its half-life is 3 yr. Now we solve this exponential equation for k. Take logarithms on each side. Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Divide by 3. In e x = x, for all x. Use a calculator. Solution Thus, the exponential decay equation is y = 600 e –0.231t.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY To find the amount present after 6 yr, let t = 6. Solution (b) How much of the substance will be present after 6 yr? After 6 yr, 150 g of the substance will remain.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 Example 5 SOLVING A CARBON DATING PROBLEM Carbon-14, also known as radiocarbon, is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists can determine the age of the remains by comparing the amount of radiocarbon with the amount present in living plants and animals. This technique is called carbon dating. The amount of radiocarbon present after t years is given by where y 0 is the amount present in living plants and animals.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Example 5 SOLVING A CARBON DATING PROBLEM Find the half-life of carbon-14. Solution (a) If y 0 is the amount of radiocarbon present in a living thing, then y 0 is half this initial amount. Thus, we substitute and solve the given equation for t. Let y = y 0. Given equation

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 5 SOLVING A CARBON DATING PROBLEM Solution Divide by y 0. Take the logarithm on each side. Find the half-life of carbon-14. (a) In e x = x, for all x.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Example 5 SOLVING A CARBON DATING PROBLEM Solution Divide by – Use a calculator. The half-life is about 5700 yr. Find the half-life of carbon-14. (a)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Example 5 SOLVING A CARBON DATING PROBLEM Solution (b) Charcoal from an ancient fire pit on Java contained ¼ the carbon-14 of a living sample of the same size. Estimate the age of the charcoal. Solve again for t, this time letting the amount y = y 0. Given equation Let y = y 0.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Example 5 SOLVING A CARBON DATING PROBLEM Divide by y 0. Take the logarithm on each side. In e x = x; Divide by − Use a calculator. The charcoal is about 11,400 yr old. Solution

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Example 6 MODELING NEWTON’S LAW OF COOLING Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and the environment around it. The temperature  (t) of the body at time t in appropriate units after being introduced into an environment having constant temperature T 0 is where C and k are constants.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Example 6 MODELING NEWTON’S LAW OF COOLING A pot of coffee with a temperature of 100°C is set down in a room with a temperature of 20°C. The coffee cools to 60°C after 1 hr. Write an equation to model the data. (a) Solution We must find values of C and k in the given formula. From the information, when t = 0, T 0 = 20, and the temperature of the coffee is f (0) = 100. Also, when t = 1, f (1) = 60. Substitute the first pair of values into the function along with T 0 = 20.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 Example 6 MODELING NEWTON’S LAW OF COOLING Write an equation to model the data. (a) Solution Given formula Let t = 0,  (0) = 100, and T 0 = 20. e 0 = 1 Subtract 20. The function that models the data can now be given. Let T 0 = 20 and C = 80.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Example 6 MODELING NEWTON’S LAW OF COOLING Let t = 1 and  (1) = 60. Subtract 20. Divide by 80. Take the logarithm on each side. Solution Formula with T 0 = 20 and C = 80. Now use the remaining pair of values in this function to find k.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 Example 6 MODELING NEWTON’S LAW OF COOLING Thus, the model is In e x = x for all x. Solution Multiply by –1 and rewrite. Use a calculator.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 To find the temperature after ½ hr, let t = ½ in the model from part (a). Example 6 MODELING NEWTON’S LAW OF COOLING (b) Solution Model from part (a) Let t = ½. Find the temperature after half an hour. Use a calculator.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 To find how long it will take for the coffee to cool to 50°C, let  (t) = 50. Example 6 MODELING NEWTON’S LAW OF COOLING (c) Solution Let  (t) = 50. Subtract 20. How long will it take for the coffee to cool to 50°C? Divide by 80.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 Example 6 MODELING NEWTON’S LAW OF COOLING Solution How long will it take for the coffee to cool to 50°C? Take the logarithm on each side. In e x = x, for all x. (c)