Lesson 5 3

							Higher Derivatives
    Concavity
2nd Derivative Test
     Lesson 5.3
          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?


                                        2
          Think About It
It is important to know
the rate of the rate of increase!

The faster the rate of increase, the better.

Suppose a stock price is modeled by
             P(t )  17  t   1/ 2

• What is the rate of increase for several months
  in the future?                                3
          Think About It
Plot the derivative for 36 months

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




The stock is increasing at a decreasing
rate
• Is that a good deal?
• What happens really long term?                             4
       Higher Derivatives
The derivative of the first derivative is
called the second derivative
           Dx  f '( x)  f ''( x)
                      d2y
                                 D  f ( x) 
                                      2
Other notations
                      dx 2            x




Third derivative f '''(x), etc.

Fourth derivative f (4)(x), etc.
                                                5
   Find Some Derivatives
Find the second and third derivatives of the
following functions

    f ( x)  x  4 x  2
                 3      2




                            y  2 x   2/3

                 x
      f ( x) 
               1  x2


                                             6
 Velocity and Acceleration
Consider a function which gives a car's
distance from a starting point as a function
of time     s(t )  t  2t  7t  9
                     3    2




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
                                                7
   Concavity of a Graph

Concave down
• Opens down




Concave up
• Opens up         Point of Inflection
                    where function
                    changes from
                   concave down to
                     concave up
                                         8
    Concavity of a Graph
Concave down
• Decreasing slope
• Second derivative
 is negative


Concave up
• Increasing slope
• Second derivative is positive


                                  9
        Test for Concavity
Let f be function with derivatives f ' and f ''
• Derivatives exist for all points in (a, b)


If f ''(x) > 0 for all
x in (a, b)
• Then f(x) concave up


If f ''(x) < 0 for all x in (a, b)
• Then f(x) concave down
                                                  10
           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) < 0, f(x) concave down
  • Where f ''(x) > 0, f(x) concave up


  Try it      f ( x)  x  4 x  2
                        3     2


                                           11
   Determining Max or Min
Use second derivative test at critical points

When f '(c) = 0 …
If f ''(c) > 0
• This is a minimum
If f ''(c) < 0
• This is a maximum
If f ''(c) = 0
• You cannot tell one way or the other!     12
  Assignment

Lesson 5.3
Page 345
Exercises 1 – 85 EOO




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