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Higher DerivativesConcavity
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 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?
3
1/ 2( ) 17P t t
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? 4
Consider the derivative of this function … it can tell us things
about the original function
Consider the derivative of this function … it can tell us things
about the original function
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.5
'( ) ''( )xD f x f x
2
2
2( )
x
d yf x
dx D
Find Some Derivatives
Find the second and third derivatives of the following functions
6
3 2( ) 4 2f x x x
2/32y x2
( )1
xf x
x
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
7
3 2( ) 2 7 9s t t t t
Concavity of a Graph
Concave down• Opens down
Concave up• Opens up
8
Point of Inflection where function changes from
concave down to concave up
Point of Inflection where function changes from
concave down to concave up
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 allx 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
11
3 2( ) 4 2f x x x
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
13
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