Ch 4 - Logarithmic and Exponential Functions - Overview

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Ch 4 - Logarithmic and Exponential Functions -
Overview

• 4.1 - Inverse Functions
• 4.2 - Logarithmic and Exponential Functions
• 4.3 - Derivatives of Logarithmic and Exponential
  Functions
• 4.4 - Derivatives of Inverse Trigonometric
  Functions
• 4.5 - LHopitals Rule Indeterminate Forms

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4.1 - Inverse Functions(page 242-250)
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Steps For Finding a Functions Inverse

• 1. Change f(x) to y
• 2. Switch x and y
• 3. Solve for y
• 4. Replace y with

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Example 3(page 244)
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Determining Whether Two Functions are Inverses
Two functions are inverses if the meet the
following definition.
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Determining Whether Two Functions are Inverses -
Example
Determine whether f and g are inverse functions
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Horizontal Line Test(page 245)

• The Horizontal Line Test is used to determine
  whether a function would have an inverse over its
  natural domain.
• If a horizontal line is drawn anywhere through
  the graph of a function and the horizontal line
  does not intersect the graph in more that one
  point, then the function passes the horizontal
  line test.
• When a function passes the horizontal line test,
  the function referred to as one-to-one function.
  The function is also said to be invertible.

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Horizontal Line Test(page 245)
Functions not passing the horizontal line test
must have their domains restricted in order to
work with their inverses.
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Graphs of Inverse Functions(page 246)
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Increasing or Decreasing Functions Have
Inverses(page 246)

• If the graph of a function f is always increasing
  or always decreasing over the domain of f, then
  the function f has an inverse over its entire
  natural domain.
• The derivative of a function (slopes of the
  tangent lines) determines whether a function is
  increasing or decreasing over an interval.
• So, the following theorem suggest that we can
  determine whether or not a function has an
  inverse over its entire domain (passes the
  horizontal line test).

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Example 8(page 247)
for all x.
So, even though we know that f has an inverse,
we can not Produce a formula for it.
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Restricting the Domain to Make Functions
Invertible(page 247)
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Chapter 3 Review Item
Differentiability Implies Continuity.
BUT
Continuity DOES NOT Imply Differentiability
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Continuity and Differentiability of Inverse
Functions(page 248)
If a function is differentiable over an interval,
then it is continuous over that interval.
If a function is continuous over an interval, it
is not necessarily differentiable. ( Corner
point, Point of vertical tangency, or Point of
discontinuity.