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Activity 36

Activity 36. Exponential Functions (Section 5.1, pp. 384-392). Exponential Functions:. Let a > 0 be a positive number with a ≠ 1. The exponential function with base a is defined by f(x) = a x for all real numbers x.

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Activity 36

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  1. Activity 36 Exponential Functions (Section 5.1, pp. 384-392)

  2. Exponential Functions: Let a > 0 be a positive number with a ≠ 1. The exponential function with base a is defined by f(x) = ax for all real numbers x.

  3. In Activity 4 (Section P.4), we defined ax for a > 0 and x a rational number. So, what does, for instance, 5√2 mean? When x is irrational, we successively approximate x by rational numbers. For instance, as √2 ≈ 1.41421 . . . we successively approximate 5√2 with 51.4, 51.41, 51.414, 51.4142, 51.41421, . . . In practice, we simply use our calculator and find out 5√2 ≈ 9.73851 . . .

  4. Example 1 Let f(x) = 2x. Evaluate the following:

  5. Example 2: Draw the graph of each function: 0 1 1 2 -1 1/2 2 4 -2 1/4 3 8

  6. Notice that Consequently, the graph of g is the graph of f flipped over the y – axis.

  7. Example 3: Use the graph of f(x) = 3x to sketch the graph of each function:

  8. The Natural Exponential Function: The natural exponential function is the exponential function f(x) = ex with base e ≈ 2.71828. It is often referred to as the exponential function.

  9. Example 4: Sketch the graph of

  10. Example 5: When a certain drug is administered to a patient, the number of milligrams remaining in the patient’s bloodstream after t hours is modeled by D(t) = 50 e−0.2t. How many milligrams of the drug remain in the patient’s bloodstream after 3 hours?

  11. Compound Interest: Compound interest is calculated by the formula: • where • 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

  12. Example 6: Suppose you invest $2, 000 at an annual rate of 12% (r = 0.12) compounded quarterly (n = 4). How much money would you have one year later? What if the investment was compounded monthly (n = 12)?

  13. Compounding Continuously Continuously Compounded interest is calculated by the formula: where A(t) = amount after t years P = principal r = interest rate per year t = number of years

  14. Example 7: Suppose you invest $2,000 at an annual rate of 12% compounded continuously. How much money would you have after one years?

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