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