CHAPTER 3. Exponential and Logarithmic Functions

Transcription

1 CHAPTER 3 Exponential and Logarithmic Functions

2 Section 3.1 (e-book 5.1-part 1) Exponential Functions and Their Graphs Definition 1: Let. An exponential function with base a is a function defined as Example 1: a) b). c) d) e) Remark 1: When, the function is also an exponential function since we can write:, with. Example 2: a) b) c) More Functions of Exponential Forms: Let be any function and. Then is also considered as an exponential function.

3 72 Example 3: a) b) c) Evaluating & Graphing Exponential Functions: An exponential function can be graphed by point plotting. We can always use a graphing calculator to graph of an exponential function. Example 4: Complete the table and then graph and on one set of axes. Discuss the graphs. a) b) x Exercise 1: Repeat the above example for functions and. Exercise 2: Sketch all the graphs in example 1 and exercise 1 using a graphing calculator.

4 73 The Number e: As you are familiar with the irrational number, there is another equally important irrational number which, in a natural way, occurs in many areas of mathematics. For more study of number e, see a calculus book. The Natural Exponential Function: The function is called the Natural Exponential Function. This function is one of the most important functions in entire field of mathematics. Exercise 3: Complete the table and graph the function without and with a calculator. x Properties of Exponential Functions observations: : By looking at the graphs, we make the following 1. The domain of an exponential function is and its range is. 2. The y-intercept is, there is no x-intercept. 3. The horizontal asymptote is x-axis, or line. There are No vertical asymptotes. 4. This function is increasing for and decreasing for 5. Exponential functions are one-to-one. This implies the next property: 6. a) If, then b) If, then

5 74 Example 5: Solving Exponential Equations (Applications of property 6 above): a) b) c) d) Applications: 1. Investment: Interest in a saving is calculated by one of the three methods: a) Simple Interest b) Compound Interest c) Continuous Compounding a) Simple Interest: In this method interest is calculated solely as a percentage of the principal sum. So if P dollars are deposited in an account earning interest at an annual rate of r, the amount of interest I and the balance A after t years are given, respectively, by or

6 75 Example 6: Suppose that $12,000 is deposited into a savings account with an annual interest rate of 2.5 %. Calculate the amount of money in the account at the end of the 6 th year if simple interest is used. b) Compound Interest: In this method a year is divided into k shorter periods. Interest is calculated not only on the initial principal but also the accumulated interest of prior periods. So, suppose P dollars are deposited in an account earning interest at an annual rate of r, compounded k times a year. If A is the amount in the account after t years, then it can be shown that & Example 7: Suppose that $12,000 is deposited into a savings account with an annual interest rate of 2.5 %. Calculate the amount of money in the account at the end of the 6 th year if the interest is compounded i) Monthly ii) Weekly iii) Daily Example 8: How much should be invested in an account now in order to have $10,000 in 5 years if the account pays 3% annual interest compounded quarterly?

7 c) Continuous Compounding: If the number of times k that the interest is compounded increases to, we say that the interest is compounded contiguously. So, suppose P dollars are deposited in an account earning interest at an annual rate of r, compounded continuously. If A is the amount in the account after t years it can be shown (needs calculus) that 76 Example 9: a) Suppose $18000 is deposited into an account earning interest at an annual rate of 5.75% compounded continuously. Find the amount in the account after 4 years. b) Repeat example 9 above with continuous compounding instead. b) (Graphical solution only) A mutual fund is paying 2.5 % compounded continuously. If you decide to invest $25,000 in this fund today, approximately how long will it take for your money to double?

8 77 Exercise: 1. Which functions are exponential functions? a) b) c) d) e) f) g) 2. Given, and, find 3. Simplify the following expressions: a) b) c) 4. Solve the exponential equations a) c) 5. Suppose that $25,000 is deposited into a savings account with an annual interest rate of 5.3 %. Calculate the amount of money in the account at the end of the 8 th year if i) The interest is compounded monthly. ii) The interest is compounded daily. iii) The interest is compounded continuously. 5. (Graphical solution only) A bank account is pays interest at 7.5 % compounded continuously. If you invest $50,000 in this fund now, approximately how long will it take for your money to increase to $75,000? What about the length of time to double your money?