Calculus MTH 250 Lecture 13 Previous Lectures Summary

Calculus (MTH 250) Lecture 13

Previous Lecture’s Summary • Critical points • Increasing & decreasing Functions • Concavity & inflection points • Strategy of Graphing

Today’s Lecture • Recalls: • Relative Extrema • Extreme Value Theorem • First Derivative Test • Second Derivative test • Rolle’s Theorem • Mean Value Theorem

Recalls

Recalls

Relative extrema

Relative extrema

Relative extrema

Extreme value theorem

Extreme value theorem

Extreme value theorem A function might have derivative equal to zero at a number c, and therefore the function will have a horizontal tangent line at c, but the function will not have a local maximum or minimum at that point.

Extreme value theorem

Extreme value theorem

Extreme value theorem

Extreme value theorem

First derivative test

First derivative test

First derivative test y 16 x 2+9 y 2=144 P(x, y) x

First derivative test + 2 2. 12 2. 2

Second derivative test

Second derivative test Example: Proof:

Second derivative test

Second derivative test

Rolle’s theorem

Rolle’s theorem

Rolle’s theorem

Rolle’s theorem The conditions of Rolle’s theorem are essential. If they fail at even one point, the graph may not have a horizontal tangent

Mean value theorem

Mean value theorem

Mean value theorem

Mean value theorem

Mean value theorem

Mean value theorem

Lecture Summary • Relative Extremum • Extreme Value Theorem • First Derivative Test • Second Derivative test • Rolle’s Theorem • Mean Value Theorem