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Exponential Functions

Exponential Functions. Exponential Function. f(x) = a x for any positive number a other than one. Examples. What are the domain and range of y = 2(3 x ) – 4?. What are the roots of 0 =5 – 2.5 x ?. Properties of Powers (Review). When multiplying like bases , add exponents.

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Exponential Functions

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  1. Exponential Functions

  2. Exponential Function f(x) = ax for any positive number a other than one.

  3. Examples • What are the domain and range of y = 2(3x) – 4? • What are the roots of 0 =5 – 2.5x?

  4. Properties of Powers (Review) • When multiplying like bases, add exponents. ax● ay = ax+y • When dividing like bases, subtract exponents. • When raising a power to a power, multiply exponents. • (ax)y=axy

  5. Properties of Powers (Review) • When you have a monomial or a fraction raised to a power (with no add. or sub.), raise everything to that power. or

  6. Half-Life & Exponential Growth/Decay • The half-life of a substance is the time it takes for half of a substance to exist. • Mirrors the behavior of Exponential Growth & Decay functions. • Exponential Growth:y = kax, if a > 1 • k is the initial amount present • a is the rate at which the amount is growing • Exponential Decay:y = kax, 0 < a < 1 • k is the initial amount present • a is the rate at which the amount is growing

  7. Example • Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining? After 20 days: After 40 days: IN GENERAL: Models the mass of the substance after t days. Therefore, let graph, and find intersection. t ≈ 46.44 days

  8. Exponential Growth/Decay Example: A population initially contains 56.5 grams of a substance. If it is increasing at a rate of 15% per week, approximately how many weeks will it take for the population to reach 281.4 grams?

  9. Exponential Growth Example: How long will it take a population to triple if it is increasing at a rate of 2.75%?

  10. ecan be approximated by: The Number e • Many real-life phenomena are best modeled using the number e • e ≈ 2.718281828 • Interest compounding continuously: • I = Pert, where P = initial investment, • r = interest rate (decimal) • t = time in years

  11. Example Compounding Interest • A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded a.) quarterly b.) monthly c.) continuously

  12. Suggested HW • Sec. 1.3 (#5, 7, 11, 19, 21-31 odd) • 1.3 Web Assign Due Monday night

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