1. Slide 4.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Natural Exponential Function Learn to develop a compound-interest formula. Learn to understand the number e. Learn to graph exponential functions. Learn to evaluate exponential functions. SECTION 4.2 1 2 3 4

3. Slide 4.2- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Terminology Interest A fee charged for borrowing a lender’s money is called the interest, denoted by I. Principal The original amount of money borrowed is called the principal, or initial amount, denoted by P. Time Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest, within a specified period. This period is called the time of the loan and is denoted by t.

4. Slide 4.2- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Terminology Interest Rate The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r. Unless stated otherwise, it is assumed to be for one year; that is, r is an annual interest rate. Simple Interest The amount of interest computed only on the principal is called simple interest.

5. Slide 4.2- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SIMPLE INTEREST FORMULA The simple interest I on a principal P at a rate r (expressed as a decimal) per year for t years is

6. Slide 4.2- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Calculating Simple Interest Juanita has deposited 8000 dollars in a bank for five years at a simple interest rate of 6%. a.How much interest will she receive? b.How much money will she receive at the end of five years? Solution a. Use the simple interest formula with P = 8000, r = 0.06, and t = 5.

7. Slide 4.2- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Calculating Simple Interest Solution continued b. The total amount due her in five years is the sum of the original principal and the interest earned:

8. Slide 4.2- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COMPOUND-INTEREST FORMULA A = amount after t years P = principal r = annual interest rate (expressed as a decimal) n = number of times interest is compounded each year t = number of years

9. Slide 4.2- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Different Compounding Periods to Compare Future Values One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future value A after one year if the interest is compounded (i)Annually. (ii)Semiannually. (iii)Quarterly. (iv)Monthly. (v)Daily.

10. Slide 4.2- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (i) Annually Solution In each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, is changing. Since t = 1, nt = n(1) = n.

11. Slide 4.2- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (iii) Quarterly (ii) Semiannually

12. Slide 4.2- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (iv) Monthly (v) Daily

13. Slide 4.2- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE VALUE OF e The number e, an irrational number, is sometimes called the Euler constant. The value of e to 15 places is e = 2.718281828459045. Mathematically speaking, e is the fixed number that the expression approaches as h gets larger and larger.

14. Slide 4.2- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CONTINUOUS COMPOUND-INTEREST FORMULA A = amount after t years P = principal r = annual interest rate (expressed as a decimal) t = number of years

15. Slide 4.2- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Calculating Continuous Compound Interest Find the amount when a principal of 8300 dollars is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months. Solution Convert eight years and three months to 8.25 years. P = $8300 and r = 0.075.

16. Slide 4.2- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Calculating the Amount of Repaying a Loan How much money did the government owe DeHaven’s descendants for 213 years on a 450,000-dollar loan at the interest rate of 6%? Solution a. With simple interest.

17. Slide 4.2- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Calculating the Amount of Repaying a Loan Solution continued b. With interest compounded yearly. c. With interest compounded quarterly.

18. Slide 4.2- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Calculating the Amount of Repaying a Loan Solution continued d. With interest compounded continuously. Notice the dramatic difference of more than $14 billion between quarterly and continuously compounding. Notice also the dramatic difference between simple interest and interest compounded yearly.

19. Slide 4.2- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE NATURAL EXPONENTIAL FUNCTION with base e is so prevalent in the sciences that it is often referred to as the exponential function or the natural exponential function. The exponential function

20. Slide 4.2- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE NATURAL EXPONENTIAL FUNCTION

21. Slide 4.2- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph Use transformations to sketch the graph of Solution Shift the graph of f (x) = e x, 1 unit right and 2 units up.

22. Slide 4.2- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MODEL FOR EXPONENTIAL GROWTH OR DECAY A(t) = amount at time t A 0 = A(0), the initial amount k = relative rate of growth (k > 0) or decay (k < 0) t = time

23. Slide 4.2- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Modeling Exponential Growth and Decay In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1 percent. Using the model on the previous slide, estimate the population of the world in the following years. a.2030 b.1990

24. Slide 4.2- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Bacterial Growth a. Use year 2000 as t = 0 Solution The model predicts there will be 11.26 billion people in the world in the year 2030.

25. Slide 4.2- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Bacterial Growth b. Use year 2000 as t = 0 Solution The model predicts that the world had 4.86 billion people in 1990 (actual was 5.28 billion).