Understanding Exponential Functions and Their Applications in Real Life
This activity explores the fundamental concepts of exponential functions, defined by f(x) = a^x, where a > 0 and a ≠ 1. We will delve into rational approximations of irrational numbers and evaluate functions through various examples, including the natural exponential function f(x) = e^x. Applications in real life, such as modeling drug dosage decay and calculating compound interest, will be discussed with relevant examples. Graphing these functions is also an integral part of understanding their behavior.
Understanding Exponential Functions and Their Applications in Real Life
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Presentation Transcript
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) = ax for all real numbers x.
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 . . .
Example 1 Let f(x) = 2x. Evaluate the following:
Example 2: Draw the graph of each function: 0 1 1 2 -1 1/2 2 4 -2 1/4 3 8
Notice that Consequently, the graph of g is the graph of f flipped over the y – axis.
Example 3: Use the graph of f(x) = 3x to sketch the graph of each function:
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.
Example 4: Sketch the graph of
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?
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
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)?
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
Example 7: Suppose you invest $2,000 at an annual rate of 12% compounded continuously. How much money would you have after one years?
