Warm-upWarm-up 1.Find all values of c on the interval that satisfy the mean value theorem. 2. Find where increasing and decreasing.

Table of Contents 26. Section 4.4 The Shape of a Graph

The Shape of a graph Essential Question – What is the 2 nd derivative test and what does it tell you about a function?

Concavity Concave up – would catch water Concave down – water would roll off curve Concave up – curve lies above tangents Concave down – curve lies below tangents

Concavity test Concave up where y’ is increasing (y” > 0) Concave down where y’ is decreasing (y” < 0)

Example Where is this concave up and where concave down?

Example Where is this concave up and where concave down?

Points of Inflection Points where concavity changes Y”=0 or is undefined at points of inflection A graph crosses its tangent at point of inflection

Example Find all points of inflection of 02 Plug in values in each interval to f” Points of inflection

Example Use the graph of f to estimate where f’ and f” are 0, positive and negative

Looking at a graph On intervals f is increasing On intervals f is concave up At local extremes of f Inflection points of f f’ is pos f’ is increasing, f’’ is pos f’ =0 f’’ = 0

Particle movement A particle is moving along x-axis Find velocity and acceleration and describe motion Going right until t=1, then left until t=3.7, then right Slowing down before t=2.3, then speeding up

Second derivative test for local extrema If f’(c)=0 and f’’(c)<0, then f has a local max at x=c If f’(c)=0 and f’’(c)>0, then f has a local min at x=c.

Example

Assignment Pg 243 #1-13 odd, 22, 23, 29, 37, 43, all