CHAPTER 5 SECTION 5.4 EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION

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CHAPTER 5SECTION 5.4EXPONENTIAL
FUNCTIONSDIFFERENTIATION AND INTEGRATION
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Definition of the Natural Exponential Function
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Recall
This means
and
Exponential and log functions are interchangeable.
Start with the base.
Change of Base Theorem
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Solve.
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Solve.
We cant take a log of -1.
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Theorem 5.10 Operations with Exponential Functions
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Properties of the Natural Exponential Function
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Theorem 5.11 Derivative of the Natural
Exponential Function
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5.4 Exponential Functions

• Example 3 Find dy/dx

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5.4 Exponential Functions

• Example 3 (concluded)

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Find each derivative
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5.4 Exponential Functions

• THEOREM 2
• or
• The derivative of e to some power is the product
  of e
• to that power and the derivative of the power.

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5.4 Exponential Functions

• Example 4 Differentiate each of the following
  with
• respect to x

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5.4 Exponential Functions

• Example 4 (concluded)

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Find each derivative
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Theorem
1. Find the slope of the line tangent to f (x) at
x 3.
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Theorem
1. Find the slope of the line tangent to f (x) at
x 3.
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4. Find extrema and inflection points for
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4. Find extrema and inflection points for
Crit s
Crit s
Cant ever work.
none
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Intervals
Test values
f (test pt)
f(x)
f (test pt)
f(x)
rel min
rel max
Inf pt
Inf pt
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5.4 Exponential Functions

• Example 7 Graph with x
  0. Analyze the graph using calculus.
• First, we find some values, plot the points, and
  sketch
• the graph.

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• Example 4 (continued)
• a) Derivatives. Since
• b) Critical values. Since
  the derivative
• for all real
  numbers x. Thus, the
• derivative exists for all real numbers, and the
  equation
• h??(x) 0 has no solution. There are no
  critical values.

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• Example 4 (continued)
• c) Increasing. Since the derivative
  for all real numbers x, we know
  that h is increasing over the entire real number
  line.
• d) Inflection Points. Since
  we know that the equation h???(x) 0
  has no solution. Thus there are no points of
  inflection.

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5.4 Exponential Functions

• Example 4 (concluded)
• e) Concavity. Since
  for all real numbers x, h is decreasing and
  the graph is concave down over the entire real
  number line.

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• Example 4 (continued)

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Theorem 5.12 Integration Rules for Exponential
Functions
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Theorem
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Theorem
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AP QUESTION
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Why is x -1/2 the only critical number???????
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AP QUESTION
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