In lesson 2.10 we had seen how the first and second derivatives influence the shape of the graph of a function. In this lesson we will again look at these principles. Now, because we know basic differentiation rules, we will have a deeper understanding of why these facts are true. We will also use these principles along with our differentiation rules to understand the shapes of graphs.

The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If  is positive, then must be increasing. If is negative, then  must be decreasing. If is zero, then must be at a relative maximum or relative minimum.  tells us similar things about .  also gives us valuable information about . In particular it tells us when the function is concave up, concave down, or there is a point of inflection. This same type of information is indicated about  by  and so on.
 

increasing	+
decreasing	-
relative min. or max.	0
concave up	increasing	+
concave down	decreasing	-
point of inflection	relative min. or max.	0
	concave up	increasing	+
	concave down	decreasing	-
	point of inflection	relative min. or max.	0
		concave up	increasing
		concave down	decreasing
		point of inflection	relative min. or max.
			concave up
			concave down
			point of inflection

Increasing and Decreasing Functions Teaching Video part 1
Extrema and the First Derivative Test Teaching Video part 1
Concavity and the Second Derivative Test Teaching Video part 1

Relative Extrema and the First Derivative Test



Concavity and the Second Derivative Test



Rolle's Rule and the Mean Value Theorem

If you have the Journey Through Calculus CD, load and run Resources/Module 3/Increasing and Decreasing Functions/Increasing-Decreasing Detector.

If you have the Journey Through Calculus CD, load and run Resources/Module 3/Concavity/Introduction.

Gathering Information on a Function

We already know that a derivative tells the slope of the curve at any instant (at least where the function is differentiable). This slope then also tells us where the function is increasing or decreasing. If we take the second derivative we will have a formula for the instantaneous rate of change of the first derivative. If the second derivative is zero, then the function must have a point of inflection there. If the second derivative is positive then the rate of change of the first derivative is positive which means that the function is concave up.

Here is an example of using the first and second derivatives to gain clues about the shape of the original function.

Example Problem:

Analyze the function: .

	First, restate the function.
	Next, find the first derivative, so that we can find all of the critical points
	Set the first derivative equal to zero, factor and solve. These then are our critical points.
	Now we calculate the second derivative.
	Again, we set the second derivative equal to zero, factor, and solve. We will need to use the quadratic formula for solving the trinomial. These are our inflection points.

Now, we will set up a table with these key points as "dividers" of the domain.

Key Points and Intervals	f(x)	f'(x)	f''(x)	Analysis
interval:	-	+	-	Negative, increasing, and concave down
(0,0)	0	0	0
interval:	-	-	-	Negative, decreasing, and concave down
(.369,-.005)	-.005	-	0
interval:	-	-	+	Negative, decreasing, and concave up
(.571,-.0084)	-.0084	0	+
interval:	-	+	+	Negative, increasing, and concave up
(.773,-.004)	-.004	+	0
interval:	-	+	-	Negative, increasing, and concave down
(1,0)	0	0	0
interval:	+	+	+	Positive, increasing, and concave up

Check to see if each of the analyses in the last column agree with the graph of the function below.

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 4.3 . This module allows you to practice using information about f ', f '', and asymptotes to determine the shape of the graph of f.