Higher Derivatives Concavity 2nd Derivative Test

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Higher DerivativesConcavity2nd Derivative Test

• Lesson 5.3

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Think About It

• Just because the price of a stock is increasing
  does that make it a good buy?
• When might it be a good buy?
• When might it be a bad buy?
• What might that have to do with derivatives?

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Think About It

• It is important to knowthe rate of the rate of
  increase!
• The faster the rate of increase, the better.
• Suppose a stock price is modeled by
• What is the rate of increase for several months
  in the future?

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Think About It

• Plot the derivative for 36 months
• The stock is increasing at a decreasing rate
• Is that a good deal?
• What happens really long term?

Consider the derivative of this function it can
tell us things about the original function
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Higher Derivatives

• The derivative of the first derivative is called
  the second derivative
• Other notations
• Third derivative f '''(x), etc.
• Fourth derivative f (4)(x), etc.

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Find Some Derivatives

• Find the second and third derivatives of the
  following functions

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Velocity and Acceleration

• Consider a function which gives a car's distance
  from a starting point as a function of time
• The first derivative is the velocity function
• The rate of change of distance
• The second derivative is the acceleration
• The rate of change of velocity

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Concavity of a Graph

• Concave down
• Opens down
• Concave up
• Opens up

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Concavity of a Graph

• Concave down
• Decreasing slope
• Second derivativeis negative
• Concave up
• Increasing slope
• Second derivative is positive

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Test for Concavity

• Let f be function with derivatives f ' and f ''
• Derivatives exist for all points in (a, b)
• If f ''(x) gt 0 for allx in (a, b)
• Then f(x) concave up
• If f ''(x) lt 0 for all x in (a, b)
• Then f(x) concave down

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Test for Concavity

• Strategy
• Find c where f ''(c) 0
• This is the test point
• Check left and right of test point, c
• Where f ''(x) lt 0, f(x) concave down
• Where f ''(x) gt 0, f(x) concave up
• Try it

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Determining Max or Min

• Use second derivative test at critical points
• When f '(c) 0
• If f ''(c) gt 0
• This is a minimum
• If f ''(c) lt 0
• This is a maximum
• If f ''(c) 0
• You cannot tell one way or the other!

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Assignment

• Lesson 5.3
• Page 345
• Exercises 1 85 EOO