Math 1210 - Section 4.3 - How Derivatives Affect the Shape of a Graph
In Math 1050 you learned that if a graph is going up from left to right, then the function is ______________. And if the graph is going down from left to right, then the function is _____________. We can use the slope of the curve to help determine where a graph is increasing or decreasing.
Example 1- Given the graph of , use the slope (i.e. the derivative) to determine where the graph is increasing or decreasing.
If on , then is increasing on .
That is, if the derivative is _____________ , then the function is ______________ .
If on , then is decreasing on .
That is, if the derivative is ____________ , then the function is ______________ .
It’s important to point out that increasing and decreasing only apply to _______________. Now let’s look at the derivative (i.e. slopes) near extreme values. Note that the derivative ______________ signs on either side of the location of an extreme value.
near a maximum-
Change from _______ to _______ _______ to _______
near a minimum-
Change from _______ to _______ _______ to _______
*We don’t need to look at the exact function or know the exact function values to determine where the extreme values occur, we only need the sign of the derivative around the critical numbers. (i.e. around where or DNE)
We can summarize the First Derivative Test for Extreme Values with some pictures.
Example 2- Use a “sign” line to organize the information below to determine:
a) critical numbers b) where is increasing or decreasing c) the location of any extreme values. d) Then draw a rough sketch the function assuming is continuous.
I. when and when ; II.
at and ;
on and
Use the first derivative test to find any extreme values and where they occur. Then sketch the graph. Example 3-
What does the word concave mean? Can we apply the word "concave" to a graph?
We use the word concavity to describe when the graph is:
“Cupped” Upward
=
Concave ___ -OR- “Cupped” Downward=
Concave ______
We can describe concavity in terms of the 1st derivative using the following definitions:
A graph is concave up when is ___________. A graph is concave down when is _____________. Use following simple examples to understand the definitions.
Example 5-The graphs of and are given. Determine the concavity by visual inspection, then graph the derivatives to make sure the definitions support your answers.
There are more practical ways of describing concavity:
I. Where are the tangent lines with respect to the graph when the function is concave up? Concave down? Concave up- tangents are ______________ the graph
Concave down- tangents are ______________ the graph
II. Since concavity is determined by how the first derivative is changing (is increasing or decreasing?), is there something that would tell us how is changing? What describes whether is increasing or decreasing? _____
If is _______, then is increasing. (So, is concave up.) If is _______, then is decreasing. (So, is concave down.)
It is important to note that concavity describes the graph on an interval more so than at a single point. Please notice, for function in example 5 above, there is a point at which the concavity changes.
Point of Inflection- A point where the graph of a function is continuous AND the concavity _____________.
*To find possible points of inflection, we find where equals _______ or doesn’t exist.* (You could also call these critical numbers for the second derivative.)
Example 6- Find the possible points of inflection, then use a graph to determine which points are actual points of inflection.
a. b.
Example 7- Use the first and second derivatives and a “sign” line to graph the general shape of the function and find the extreme values of .
Consider example 7 above. a. What is the concavity at the point where the relative maximum occurs?
b. What is the concavity at the point where the relative minimum occurs?
We can summarize the tests we’ve learned with the following pictures
Example 8- Find the intervals on which the following functions are increasing or decreasing. Then use your calculator if necessary to graph the function and its derivative on the same set of axes to verify your work.
a.
b. on
This figure shows typical features of a function. Notice how the sign, value,
33-44.
a. Find the intervals of increase or decrease. b. Find the local max and min values. c. Find the intervals of concavity and the inflection points. d. Use the information to sketch the graph.
45-52.
a. Find the horizontal and vertical asymptotes. b. Find the intervals of increase or decrease.
c. Find the local max and min values. d. Find the intervals of concavity and the inflection points. e. Use the information from parts (a)-(d) to sketch the graph. CONFIRM your work with a graphing calculator.
#46.
To Summarize:
The first derivative tells us:
The second derivative tells us:
To find increasing, decreasing, concave up, concave down, extreme values, and draw a rough sketch of the graph:
1- Find the critical numbers for the first derivative.
2- Use the critical numbers, end points (if any), and a sign line to find intervals of increasing and decreasing. 3- Find “critical numbers” for the second derivative.
