Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 1 Chapter 4 Exponential Functions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide Using Exponential Functions to Model Data

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 3 Exponential model, exponentially related, approximately exponentially related Definition An exponential model is an exponential function, or its graph, that describes the relationship between two quantities for an authentic situation. If all of the data points for a situation lie on an exponential curve, then we say the independent and dependent variables are exponentially related.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 4 Exponential model, exponentially related, approximately exponentially related Definition If no exponential curve contains all of the data points, but an exponential curve comes close to all of the data points (and perhaps contains some of them), then we say the variables are approximately exponentially related.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 5 Example: Modeling with an Exponential Function Suppose that a peach has 3 million bacteria on it at noon on Monday and that one bacterium divides into two bacteria every hour, on average.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 6 Example: Modeling with an Exponential Function Let B = f(t) be the number of bacteria (in millions) on the peach at t hours after noon on Monday. 1. Find an equation of f. 2. Predict the number of bacteria on the peach at noon on Tuesday.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 7 Solution 1. We complete a table of value of f based on the assumption that one bacterium divides into two bacteria every hour.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 8 Solution 1. As the value of t increases by 1, the value of B changes by greater and greater amounts, so it would not be appropriate to model the data by using a linear function. Note, though, that as the value of t increases by 1, the value of B is multiplied by 2, so we can model the situation by using an exponential model of the form f(t) = a(2) t. The B-intercept is (0, 3), so f(t) = 3(2) t.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 9 Solution 1. Use a graphing calculator table and graph to verify our work.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 10 Solution 2. Use t = 24 to represent noon on Tuesday. Substitute 24 for t in our equation of f: f(24) = 3(2) 24 = 50,331,648 According to the model, there would be 50,331,648 million bacteria. To omit writing “million,” we must add six zeroes to 50,331,648 – that is 50,331,648,000,000. There would be able 50 trillion bacteria at noon on Tuesday.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 11 Exponential Model y = ab t If y = ab t is an exponential model where y is a quantity at time t, then the coefficient a is the value of that quantity present at time t = 0. For example, the bacteria model f(t) = 3(2) t has coefficient 3, which represents the 3 million bacteria that were present at time t = 0.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 12 Example: Modeling with an Exponential Function A person invests $5000 in an account that earns 6% interest compounded annually. 1. Let V = f(t) be the value (in dollars) of the account at t years after the money is invested. Find an equation of f. 2. What will be the value after 10 years?

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 13 Solution 1. Each year, the investment value is equal to the previous year’s value (100% of it) plus 6% of the previous year’s value. So, the value is equal to 106% of the previous year’s value.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 14 Solution 1. As the value of t increases by 1, the value of V is multiplied by So, f is the exponential function f(t) = a(1.06) t. Since the value of the account at the start is $5000, we have a = So, f(t) = 5000(1.06) t.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 15 Solution 2. To find the value in 10 years, substitute 10 for t: f(10) = 5000(1.06) 10 ≈ The value will be $ in 10 years.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 16 Half-life Definition If a quantity decays exponentially, the half-life is the amount of time it takes for that quantity to be reduced to half.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 17 Example: Modeling with an Exponential Function The world’s worse nuclear accident occurred in Chernobyl, Ukraine, on April 26, Immediately afterward, 28 people died from acute radiation sickness. So far, about 25,000 people have died from the accident, mostly due to the release of the radioactive element cesium-137 (Source: Medicine Worldwide)

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 18 Example: Modeling with an Exponential Function Cesium-137 has a half-life of 30 years. Let P = f(t) be the percent of the cesium-137 that remains at t years since Find an equation of f. 2. Describe the meaning of the base of f. 3. What percent of the cesium-137 will remain in 2014?

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 19 Solution 1. We discuss two methods of finding an equation of f. Method 1 The table shows the values of P at various years t. The data can be modeled well with an exponential function. So,

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 20 Solution 1. We can use a graphing calculator table and graph to verify our equation. Write this equation in the form f(t) = ab t : Since we can write P = f(t) = 100(0.977) t

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 21 Solution 1. Method 2 Instead of recognizing a pattern from the table, we can find an equation of f by using the first two points on the table, (0,100) and (30, 50). Since the P-intercept is (0, 100), we have P = f(t) = 100b t

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 22 Solution 1. To find b, we substitute the coordinates of (30, 50) into the equation: So, an equation of f is f(t) = 100(0.977) t, the same equation we found using Method 1.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 23 Solution 2. The base of f is Each year, 97.7% of the previous year’s cesium-137 is present. In other words, the cesium-137 decays by 2.3% each year. 3. Since 2014 – 1986 = 28, substitute 28 for t: f(28) = 100(0.977) 28 ≈ In 2014, about 52.1% of the cesium-137 will remain.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 24 Meaning of the Base of an Exponential Model If f(t) = ab t, where a > 0, models a quantity at time t, then the percent rate of change is constant. In particular, If b > 1, then the quantity grows exponentially at a rate of b – 1 percent (in decimal form) per unit of time. If 0 < b < 1, then the quantity decays exponentially at a rate of 1 – b percent (in decimal form) per unit of time.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 25 Example: Modeling with an Exponential Function The number of severe near collisions on airplane runways has decayed approximately exponentially from 67 in 2000 to 16 in 2010 (Source: Federal Aviation Administration). Predict the number of severe near collisions in 2018.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 26 Solution Let n be the number of severe near collisions on airplane runways in the year that is t years since Known values of t and n are shown in the table below.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 27 Solution Because the variables t and n are approximately exponentially related, we want an equation in the form n = ab t. The n-intercept is (0, 67). So, the equation is of the form n = 67b t

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 28 Solution To find b, substitute (10, 16) into the equation and solve for b:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 29 Solution Substitute for b in the equation n = 67b t : n = 67(0.867) t Finally, to predict the number of severe near collisions in 2018, substitute 2018 – 2000 = 18 for t in the equation and solve for n: n = 67(0.867) 18 ≈ 5.13

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 30 Solution The model predicts that there will be about 5 severe near collisions in We use a graphing calculator table to check that each of the three ordered pairs (0, 67), (10, 16), and (18, 5.13) approximately satisfies the equation n = 67(0.867) t.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.5, Slide 31 Four-Step Modeling Process To find a model and then make estimates and predictions, 1. Create a scattergram of the data. Decide whether a line or an exponential curve comes close to the points. 2. Find the equation of your function. 3. Verify your equation by checking that the graph comes close to all of the data points. 4. Use your equation of the model to draw conclusions, make estimates, and/or make predictions.