Section 5.2 The Natural Exponential Function

Transcription

1 Section 5.2 The Natural Exponential Function Objectives Identify the characteristics of the natural exponential function f(x) = e x, including the domain, range, intercept, asymptote, end behavior, and its graph. Sketch the graph of natural exponential functions using transformations. Solve natural exponential equations by relating the bases. Solve continuous compound interest application problems. Determine the present value of an investment using continuous compound interest. Solve population growth application problems. Preliminaries Consider the natural exponential function f(x) = e x. List the following properties. Domain: Range: y-intercept: Asymptote: End behavior: and Graph: Continuous compound interest can be calculated by using the formula A = Pe rt Write down the meaning of each value in the formula. A: P: r: t: Page 204

2 Warm-up 6. Use your calculator to approximate the following values rounded to four decimal places. (A) e 3 (B) 12e Determine the transformations that are performed on a base function. (A) m(x) = 4(x + 2) 3 (B) p(x) = 3x 5 Page 205

3 Class Notes and Examples Each of the following functions were created using transformations of f(x) = e x. Determine the transformations that were performed. List the domain, horizontal asymptote, range, and y-intercept. Graph the given function. (Note: we will discuss an algebraic method to determine the x-intercept later in chapter 5.) (A) J(x) = e x + 2 Transformation(s): Domain: Horizontal Asymptote: Range: y-intercept: Graph: Page 206

4 (B) L(x) = 2 e x 3 Transformation(s): Domain: Horizontal Asymptote: Range: y-intercept: Graph: Page 207

5 5.2.2 Solve the following equations. Check your answers in the original equation. 3 (A) e 5 = e x 1 (B) e2x e 3 = ex Dmitry invests $3200 in a savings account that earns 4.6% interest compounded continuously. How much money would Dmitry have in the account after 3.5 years? Anna has a choice between two investment options for a $1000 gift she received. The first option earns 7.8% interest compounded continuously. The second option earns 7.9% interest compounded semi-annually. Which option would yield the greatest amount of money after 5 years? Page 208

6 5.2.5 Arturo wants to have $15,000 in 6 years, so he will place money into a savings account that pays 3.7 % interest compounded continuously. How much should Arturo invest now to have $15,000 in 6 years? Check your answer. What is the general exponential growth model? The population of a city can be measured by P(t) = 12,500e 0.02t, where t represents time in years after (A) What was the population in 1985? (B) What was the population in 2000? (C) What does the model predict the population to be in the year 2020? Page 209

7 5.2.7 An invasive beetle was discovered in a small Pacific island 15 years ago. It is estimated that there are 12,400 beetles on the island now, with a relative growth rate of 16%. (A) How many beetles were initially discovered 15 years ago? (B) How many beetles will there be after another 15 years? Page 210

8 Section 5.2 Self-Assessment (Answers on page 257) 1. (Multiple Choice) What is the range of y = e x + 12? (A) (0, ) (B) (, 0) (C) (, 12) (D) (12, ) (E) (, 12] 2. Kathryn invested $7500 in an investment account that earned 6.1% interest compounded continuously for 40 years. How much money is in the account after 40 years? 3. (Multiple Choice) Marco needs to have $1,000,000 in an investment account in 35 years. How much should be invested today at an interest rate of 8.2% compounded continuously to have $1,000,000 in 35 years? The minimum amount that should be invested is: (A) More than $68,000 (B) Between $65,000 and $68,000 (C) Between $62,000 and $65,000 (D) Between $59,000 and $62,000 (E) Between $56,000 and $59, Anna has a choice between two investment options for a $1000 gift she received. The first option earns 7.8% interest compounded continuously. The second option earns 7.9% interest compounded semi-annually. Calculate the amount of money earned in each investment and determine which option would yield the greatest amount of money after 5 years. 5. (Multiple Choice) A species of owl was introduced in an area 30 years ago. It is estimated that there are 6700 owls in the area now, and the population has a relative exponential growth rate of 6% per year. How many owls will there be 25 years from now? (A) (B) (C) (D) (E) Less than 29,000 owls Between 29,000 owls and 31,000 owls Between 31,000 owls and 33,000 owls Between 33,000 owls and 35,000 owls More than 35,000 owls Page 211