Accelerated Calculus
The Second Derivative
Name______________________
9/15/10
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)  x 2  4 x
a. Over what interval(s) is
f ( x ) increasing?
b. Over what interval(s) is
f ( x ) decreasing?
c. Use your answers to
part a & b to efficiently
sketch a graph of f ( x) .
Calculus
The Second Derivative
Before we begin to investigate and interpret 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:
Supposse are given a function f ( x) .
Using the fact that derivatives tell us information about a function’s behavior (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
While this tells us what is going on with the derivative, what about f ( x) ?
Let’s study this visually first:
Consider the graph of a function f ( x) , below:
a. Over what interval(s) is f ( x ) increasing?
b. Over what interval(s) is f ( x ) decreasing?
c. When f ( x ) increasing, what can we say
about f ( x) ?
d. When f ( x ) decreasing, what can we say
about f ( x) ?
Calculus
So, now we have a formal definition of concavity, using the idea of derivatives!
If the graph of f ( x) is concave up on an interval, then f ( x)  0 on that interval.
If the graph of f ( x) is concave down on an interval, then f ( x)  0 on that interval.
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:
Suppose y  s (t ) is the function describing the position of an object at time t, what does the
first derivative tell us?
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 may help…
Calculus
Now, let’s consider the following exercises:
1)
2) Use the set of axes below: