Lesson%201.3:%20Exponential%20Functions

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Lesson 1.3 Exponential Functions

• AP Calculus
• Mrs. Mongold

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Review of Exponent Rules

• ax ay axy
• (ax)y axy
• ax bx (ab)x

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Exponential Function

• Let a be a positive real number other than 1.
  Then f(x) ax is the exponential function with
  base a.
• Domain of f(x) ax is (-8, 8)
• Range of f(x) ax is (0, 8)

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Quick Examples

• y 2x

• y 2-x

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Exponential Growth

• y Pax
• P initial amount
• a 1 rate as a decimal
• x of times compounded
• F(x) P(1r)x

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Example

• Find growth of 100 invested in 2009 at an
  interest rate of 5.5 in 2013 which was
  compounded annually.
• Find amount in 2020

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Example

• Predict the world population in 2020 using an
  exponential model and the table

Year Population (millions) Ratio
2005 4936
2006 5023
2007 5111
2008 5201
2009 5329
2010 5422
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Exponential Decay

• The function ykax
• kgt0 and 0ltalt1

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Half-life example

• A certain radioactive substance has a half-life
  of 20 days. If there are 5 grams present
  initially when will there be only 1 gram
  remaining?

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The number e

• e2.718281828
• We can define e to be the number that the
  function f(x) (1 1/x)x approaches as x
  approaches infinity.
• yex and ye-x are frequently used as models for
  exponential growth and decay.

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Example

• yPert is used for interest compounded continually

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Homework

• Pgs 24-25/1-6 all, 7-17 odd, 23, 25, 29, 30, 33