Calculus
Name______________________
The Second Derivative
10/3/2011
Review of the First Derivative
Suppose we are given a function f ( x) .
What can you conclude about the behavior of the function if:
f ( x)  0 on an interval, then f ( x ) is ______________________ on that interval
f ( x)  0 on an interval, then f ( x ) is ______________________ on that interval
So, here is a quick application:
Ex:
Given the function: f ( x)  x3  3x 2
a. Using the power rule, find f ( x) .
b. Using your result from part (a), determine over what interval(s) f ( x) is increasing
and over what interval(s) f ( x) is decreasing?
c. Use your answers to part a & b to efficiently sketch a fairly accurate graph of f ( x) .
Calculus
Now, let’s consider looking at another example, but this time thinking about the behavior of the
derivative of f ( x) .
y
Consider the graph of a function f ( x) , below:


a. Over what interval(s) is f ( x ) increasing?
Hint: look for interval(s) where the slopes of tangent lines
are getting more positive!


x









b. Over what interval(s) is f ( x ) decreasing?
Hint: look for interval(s) where the slopes of tangent lines
are getting more negative!



c. When f ( x ) increasing, what can we say about f ( x) ?
d. When f ( x ) decreasing, what can we say about f ( x) ?
The Second Derivative
Let’s consider the notation we will see when working with this “higher order” derivative:
f ( x )
d2y
dx 2
d  dy 
 
dx  dx 
Using similar language as we did when reviewing the first derivative, let us now consider the
following:
Suppose are given a function f ( x) .
Using the fact that derivatives tell us information about the behavior of a function (i.e.
increasing vs. decreasing), what can you conclude about the behavior of the derivative if:
f ( x)  0 on an interval, then f ( x ) is ______________________ on that interval
f ( x)  0 on an interval, then f ( x ) is ______________________ on that interval

Calculus
While this tells us what is going on with the derivative, what about describing what is going
on with the behavior of f ( x) over these intervals? In comes the idea of concavity, but now
defined using the idea of derivatives!
If the graph of f ( x) is concave up on an interval, then f ( x)  0 on that interval.
A function is concave up on an interval if, as you increase x-values, the slopes of the
tangent lines at those x-values increase (get more positive) – examine the following
diagram to the right!
If the graph of f ( x) is concave down on an interval, then f ( x)  0 on that interval.
A function is concave down on an interval if, as you increase x-values, the slopes of the
tangent lines at those x-values decrease (get more negative)—examine the following
diagram to the right!
When we look at applications of what the second derivative tells us in terms of “rates of
change”, let us first recall what the first derivative tells us:
Q1: Suppose y  s (t ) is the function describing the position of an object at time t, what does
the first derivative tell us in this context?
Q2: Using the idea of derivatives (as a “rate of change”), what does the second derivative tell us in
this context?
Here is a nice challenge question to consider (use your knowledge of “derivatives” and what
these derivatives tell us about graphs of functions):
Ex:
Hint: A sketch of what is going on with the behavior of this function will help…
HW: Pg. 97/#3, 4, 7, 10—12, 14, 16, 17, 21, 23