1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 4 Exponential and Logarithmic Functions

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 The Natural Exponential Function Define simple interest. Develop a compound interest formula. Understand the number e. Graph exponential functions. Evaluate exponential functions. SECTION 4.2 1 2 3 4 5

3. 3 © 2010 Pearson Education, Inc. All rights reserved SIMPLE INTEREST A fee charged for borrowing a lender’s money is called the interest, denoted by I. The original, or initial amount of money borrowed is the principal, denoted by P. The period of time during which the borrower pays back the principal plus the interest is the time, denoted by t.

4. 4 © 2010 Pearson Education, Inc. All rights reserved The interest rate is the percent charged for the use of the principal for the given period. The interest rate, denoted by r, is expressed as a decimal. Unless stated otherwise, the period is assumed to be one year; that is, r is an annual rate. The amount of interest computed only on the principal is called simple interest. SIMPLE INTEREST

5. 5 © 2010 Pearson Education, Inc. All rights reserved 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. 6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Calculating Simple Interest Juanita has deposited $8000 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 be in her account at the end of five years? Solution a. Use the simple interest formula with P = $8000, r = 0.06, and t = 5.

7. 7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Calculating Simple Interest Solution continued b. In five years, the amount A she will receive is the original principal plus the interest earned:

8. 8 © 2010 Pearson Education, Inc. All rights reserved Compound interest is the interest paid on both the principal and the accrued (previously earned) interest. Interest that is compounded annually is paid once a year. For interest compounded annually, the amount A in the account after t years is given by COMPOUND INTEREST

9. 9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Compound Interest Juanita deposits $8000 in a bank at the interest rate of 6% compounded annually for five years. a.How much money will she have in her account after five years? b.How much interest will she receive?

10. 10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Calculating Compound Interest Solution a. Here P = $8000, r = 0.06, and t = 5. b. Interest = A  P = $10,705.80  $8000 = $2705.80.

11. 11 © 2010 Pearson Education, Inc. All rights reserved 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

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Different Compounding Periods to Compare Future Values If $100 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.

13. 13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (i) Annual Compounding: Solution In the following computations, P = 100, r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, changes. Since t = 1, nt = n(1) = n.

14. 14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (iii) Quarterly Compounding: (ii) Semiannual Compounding:

15. 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Different Compounding Periods to Compare Future Values (iv) Monthly Compounding: (v) Daily Compounding:

16. 16 © 2010 Pearson Education, Inc. All rights reserved THE VALUE OF e The value of e to 15 places is e = 2.718281828459045. gets closer and closer to a fixed number. This irrational number is denoted by e and is sometimes called the Euler number. As h gets larger and larger,

17. 17 © 2010 Pearson Education, Inc. All rights reserved CONTINUOUS COMPOUND INTEREST FORMULA A = amount after t years P = principal r = annual interest rate (expressed as a decimal) t = number of years

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Calculating Continuous Compound Interest Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months. Solution P = $8300 and r = 0.075. Convert eight years and three months to 8.25 years.

19. 19 © 2010 Pearson Education, Inc. All rights reserved 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 loan at the interest rate of 6%? Solution a. With simple interest,

20. 20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Calculating the Amount of Repaying a Loan Solution continued b. With interest compounded yearly, c. With interest compounded quarterly,

21. 21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Calculating the Amount of Repaying a Loan Solution continued d. With interest compounded continuously, Notice the dramatic difference between quarterly and continuous compounding and the dramatic difference between simple interest and compound interest.

22. 22 © 2010 Pearson Education, Inc. All rights reserved 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

23. 23 © 2010 Pearson Education, Inc. All rights reserved THE NATURAL EXPONENTIAL FUNCTION

24. 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Use transformations to sketch the graph of Solution Start with the graph of y = e x.

25. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Use transformations to sketch the graph of Solution coninued Shift the graph of y = e x one unit right.

26. 26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Use transformations to sketch the graph of Solution continued Shift the graph of y = e x – 1 two units up.

27. 27 © 2010 Pearson Education, Inc. All rights reserved 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

28. 28 © 2010 Pearson Education, Inc. All rights reserved 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%. Using the model on the previous slide, estimate the population of the world in the following years. a.2030 b.1990

29. 29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 a. The year 2000 corresponds to t = 0. So A 0 = 6 (billion), k = 0.021, and 2030 corresponds to t = 30. Solution The model predicts that if the rate of growth is 2.1% per year, over 11.26 billion people will be in the world in 2030. Modeling Exponential Growth and Decay

30. 30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 b. The year 1990 corresponds to t =  10. Solution The model predicts that the world had over 4.86 billion people in 1990. (The actual population in 1990 was 5.28 billion.) Modeling Exponential Growth and Decay