Calculus 15

Calculus

Lesson 15 Concavity

Concave up if is increasing.

Concave down if is decreasing.

Concavity changes at points of inflection!!!!

Since rate of change of is ,

indicates that is concave up at .

indicates that is concave down at .

indicates a possible point of inflection at .

Note that it doesn always mean that there is a point of inflections.

Example:

We must use a sign chart to make sure that changes sign.

If does not change sign, then is not a point of inflection.

For each of f, g, and h, identify all critical points and all points of inflection.

Same Examples as last lesson:

Note: never equals zero, so there are no points of inflection!

Also:

Test to see if they are points of inflection!

If , then the test is inconclusive - you can always use the

first derivative test!

Example:

On to Lesson 16 - Limits at Infinity