Lesson 5 3

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```							Higher Derivatives
Concavity
2nd Derivative Test
Lesson 5.3
Just because the price of a stock is
increasing … does that make it a good
• When might it be a good buy?
What might that have to do with
derivatives?

2
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
Plot the derivative for 36 months

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

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

13

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