Quote of the Day:
"If people do not believe that mathematics is simple, it is only
because they do not realize how complicated life is."
Objectives:
The student will test functions for concavity.
The student will test for points of inflection.
1. Collect homework.
2. Concavity
A. concave down - does not hold water
concave up - does hold water
B. Problem:
A coin is in a "cup" formed by 4 matchsticks.
Try to get the coin out of the cup by moving only 2
matchsticks to new positions to form a "congruent
cup" in a new position.
C. What is happening to the tangent lines in a graph as you
go from left to right?
concave up - they are increasing
concave down - they are decreasing
The second derivative is a rate of change of the
first derivative.
if f''(x) > 0, then concave up.
if f''(x) < 0, then concave down.
Use Mr. Smiley and Mr. Frowny to help you remember this:
3. Points of Inflection
Points of inflection occur when the concavity changes.
Test: If there is a point of inflection, the second
derivative is zero.
BUT just because the second derivative is zero
doesn't guarantee a point of inflection.
Draw diagrams on board to illustrate.
Example where 2nd derivative is zero, but Not a point of
inflection:
The Normal Distribution Curve
The point of inflection on the normal
curve is where the first standard
deviation occurs.
4. To test for points of inflection and concavity:
(1) Take the second derivative of the function.
(2) Set it equal to zero and solve for x.
These values of x are possible points of inflection.
(3) Test on either side of these points to check
concavity. If the concavity changes from positive to
negative or from negative to positive, it is a point
of inflection.
5. Example:
Test the following function for intervals where it
increases and decreases, for intervals where it is
concave up and concave down, and for points of
inflection.
6. Discuss the S-Curve (Handout)
Click here for S-Curve
7. Assignment:
p. 276 (9, 11, 13, 15, 16, 29, 35, 36, 37)
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