College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Sixth Edition James StewartLothar RedlinSaleem Watson

2. Exponential and Logarithmic Functions 4

3. Exponential Functions 4.1

4. Chapter Overview • In this chapter, we study a new class of functions called exponential functions. • For example, f(x) = 2xis an exponential function (with base 2).

5. Chapter Overview • Notice how quickly the values of this function increase:f(3) = 23 = 8 f(10) = 210 = 1,024 f(30) = 230 = 1,073,741,824

6. Chapter Overview • Compare that with the function g(x) = x2where g(30) = 302 = 900. • The point is, when the variable is in the exponent, even a small change in the variable can cause a dramatic change in the value of the function.

7. Exponential Functions

8. Exponential Functions • In Section P.4, we defined axfor a > 0 and x a rational number. • However, we have not yet defined irrational powers. • So, what is meant by or 2π?

9. Exponential Functions • To define axwhen x is irrational, we approximate x by rational numbers. • For example, since is an irrational number, we successively approximate by these rational powers:

10. Exponential Functions • Intuitively, we can see that these rational powers of a are getting closer and closer to . • It can be shown using advanced mathematics that there is exactly one number that these powers approach. • We define to be this number.

11. Exponential Functions • For example, using a calculator, we find: • The more decimal places of we use in our calculation, the better our approximation of . • It can be proved that the Laws of Exponents are still true when the exponents are real numbers.

12. Exponential Function—Definition • The exponential function with base ais defined for all real numbers x by: f(x) = axwhere a > 0 and a≠ 1. • We assume a ≠ 1 because the function f(x) = 1x = 1 is just a constant function.

13. Exponential Functions • Here are some examples: f(x) = 2xg(x) = 3xh(x) = 10x

14. E.g. 1—Evaluating Exponential Functions • Let f(x) = 3x and evaluate the following: • f(5) • f(–⅔) • f(π) • f( ) • We use a calculator to obtain the values of f.

15. Example (a) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, 5, ENTER • Output: 243 • Thus, f(5) = 35 = 243

16. Example (b) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, (, (–), 2, ÷, 3, ), ENTER • Output: 0.4807498 • Thus, f(–⅔) = 3–⅔≈ 0.4807

17. Example (c) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, π, ENTER • Output: 31.5442807 • Thus, f(π) = 3π≈ 31.544

18. Example (d) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, √, 2, ENTER • Output: 4.7288043 • Thus, f( ) = ≈ 4.7288

19. Graphs of Exponential Functions

20. Graphs of Exponential Functions • We first graph exponential functions by plotting points. • We will see that these graphs have an easily recognizable shape.

21. E.g. 2—Graphing Exp. Functions by Plotting Points • Draw the graph of each function. • f(x) = 3x • g(x) = (⅓)x

22. E.g. 2—Graphing Exp. Functions by Plotting Points • First, we calculate values of f(x) and g(x).

23. E.g. 2—Graphing Exp. Functions by Plotting Points • Then, we the plot points to sketch the graphs.

24. E.g. 2—Graphing Exp. Functions by Plotting Points • Notice that: • So, we could have obtained the graph of gfrom the graph of fby reflecting in the y-axis.

25. Graphs of Exponential Functions • The figure shows the graphs of the family of exponential functions f(x) = ax for various values of the base a. • All these graphs pass through the point (0, 1) because a0 = 1 for a ≠ 0.

26. Graphs of Exponential Functions • You can see from the figure that there are two kinds of exponential functions: • If 0 < a < 1, the function decreases rapidly. • If a > 1, the function increases rapidly.

27. Graphs of Exponential Functions • The x-axis is a horizontal asymptote for the exponential function f(x) = ax. • This is because: • When a > 1, we have ax→ 0 as x → –∞. • When 0 < a < 1, we have ax→ 0 as x →∞.

28. Graphs of Exponential Functions • Also, ax> 0 for all . • So, the function f(x) = ax has domain and range (0, ∞). • These observations are summarized as follows.

29. Graphs of Exponential Functions • The exponential function f(x) = ax (a > 0, a≠ 1)has domain and range (0, ∞). • The line y = 0 (the x-axis) is a horizontal asymptote of f.

30. Graphs of Exponential Functions • The graph of f has one of these shapes.

31. E.g. 3—Identifying Graphs of Exponential Functions • Find the exponential function f(x) = ax whose graph is given.

32. Example (a) E.g. 3—Identifying Graphs • Since f(2) = a2 = 25, we see that the base is a = 5. • Thus, f(x) = 5x

33. Example (b) E.g. 3—Identifying Graphs • Since f(3) = a3 = 1/8 , we see that the base is a = ½ . • Thus, f(x) = (½)x

34. Graphs of Exponential Functions • In the next example, we see how to graph certain functions—not by plotting points—but by: • Taking the basic graphs of the exponential functions in Figure 2. • Applying the shifting and reflecting transformations of Section 2.5.

35. E.g. 4—Transformations of Exponential Functions • Use the graph of f(x) = 2x to sketch the graph of each function. • g(x) = 1 + 2x • h(x) = –2x • k(x) = 2x –1

36. Example (a) E.g. 4—Transformations • To obtain the graph of g(x) = 1 + 2x, we start with the graph of f(x) = 2x and shift it upward 1 unit. • Notice that the line y = 1 is now a horizontal asymptote.

37. Example (b) E.g. 4—Transformations • Again, we start with the graph of f(x) = 2x. • However, here, we reflect in the x-axis to get the graph of h(x) = –2x.

38. Example (c) E.g. 4—Transformations • This time, we start with the graph of f(x) = 2x and shift it to the right by 1 unit—to get the graph of k(x) = 2x–1.

39. E.g. 5—Comparing Exponential and Power Functions • Compare the rates of growth of the exponential function f(x) = 2x and the power function g(x) = x2 by drawing the graphs of both functions in these viewing rectangles. • [0, 3] by [0, 8] • [0, 6] by [ 0, 25] • [0, 20] by [0, 1000]

40. Example (a) E.g. 5—Exp. and Power Functions • The figure shows that the graph of g(x) = x2 catches up with, and becomes higher than, the graph of f(x) = 2x at x = 2.

41. Example (b) E.g. 5—Exp. and Power Functions • The larger viewing rectangle here shows that the graph of f(x) = 2x overtakes that of g(x) = x2 when x = 4.

42. Example (c) E.g. 5—Exp. and Power Functions • This figure gives a more global view and shows that, when x is large, f(x) = 2xis much larger than g(x) = x2.

43. Compound Interest

44. Compound Interest • Exponential functions occur in calculating compound interest. • Suppose an amount of money P, called the principal, is invested at an interest rate i per time period. • Then, after one time period, the interest is Pi, and the amount A of money is:A = P + Pi + P(1 + i)

45. Compound Interest • If the interest is reinvested, the new principal is P(1 + i), and the amount after another time period is: A = P(1 + i)(1 + i) = P(1 + i)2 • Similarly, after a third time period, the amount is: A = P(1 + i)3

46. Compound Interest • In general, after k periods, the amount is: A = P(1 + i)k • Notice that this is an exponential function with base 1 + i.

47. Compound Interest • If the annual interest rate is r and interest is compounded n times per year. • Then, in each time period, the interest rate is i =r/n, and there are nt time periods in t years. • This leads to the following formula for the amount after t years.

48. Compound Interest • Compound interestis calculated by the formulawhere: • A(t) = amount after t years • P = principal • r = interest rate per year • n = number of times interest is compounded per year • t = number of years

49. E.g. 6—Calculating Compound Interest • A sum of $1000 is invested at an interest rate of 12% per year. • Find the amounts in the account after 3 years if interest is compounded: • Annually • Semiannually • Quarterly • Monthly • Daily

50. E.g. 6—Calculating Compound Interest • We use the compound interest formula with: P = $1000, r = 0.12, t = 3