Exponential Functions Section 7'2

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Exponential FunctionsSection 7.2

• Mat 231 Calculus I
• Saint Leo University

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What is an exponential Function?

• The class of elementary exponential functions
  have the form
• f(x) ax
• where a is positive real number, and a is not
  equal to 1

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Properties of Elementary Exponential Functions

• Domain The set of all real numbers.
• Range The set of all positive real numers.
• X-intercept none
• Y-intercept 1
• The graphs of all elementary exponential
  functions is a smooth continuous curve.
• Horizontal Asymptote y0

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Graphs of Elementary Exponential Functions
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Exponential functions at large

• In many of the sciences, we find that exponential
  functions model real life data very well. A
  general form of an exponential function is
• P(x) P0 akx D
• where

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Derivatives of Exponentials

• We recall that given f(x) ax , the definition
  of the derivative states that
• f(x)

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What is that limit?

• Lets investigate the limit with a 2 and then
  with a 3.
• What do we find?

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Derivative of Exponential Formula

• Given f(x) ax , we find that
• f(x) (lna)ax.
• In case of f(x) ex , we have that
• f(x) ex .

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Examples

• Do the problems on Page 431/30, 32, 34, 38 and
  42, 48

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Example 30

• Solution

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Example 32

• Solution

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Example 34

• Solution

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Example 38

• Solution

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Example 48

• Solution

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Integrals of Exponentials
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Examples

• Do the problems on Page 433/ 72, 74, 76, 78, 80

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Example 72

• Solution

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Example 74

• Solution

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Example 76

• Solution

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Example 78

• Solution

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Example 80

• Solution

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Homework and Project

• Homework 6 - Due April 3rd
• Read Section 7.2 on starting on Page 421 and
  work out the problems
• Page 431/3 6 (all), 29, 33, 35, 37, 43, 47,
  71 79 (odds)
• Project 3 Due April 10th
• Pg 432/59, 60