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

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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

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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) > 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

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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) < 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

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