1. §2.5: Solving Problems Involving Rate of Change
October 8, 2010

2. Types of Points stationary point critical point
a point on the graph where the graph of the function stops increasing or decreasing
critical point
a stationary point or a point on the graph where the IROC does not exist

3. Example
stationary point

4. Example
critical point

5. Types of Stationary Points
Extrema
minima
maxima
Non-extrema

6. Types of Critical Points
Extrema
Cusps
Endpoints of Domains

7. Characteristics of Maxima
There is a maxima at x = a if
IROC is positive and relatively close to zero for values of x slightly less than a
IROC is negative and relatively close to zero for values of x slightly greater than a

8. Characteristics of Minima
There is a minima at x = a if
IROC is negative and relatively close to zero for values of x slightly less than a
IROC is positive and relatively close to zero for values of x slightly greater than a

9. Classifying Extrema Global Extrema (Absolute Extrema)
global maximum
global minimum
Local Extrema (Relative Extrema)
local maximum
local minimum

10. Graphically..
The slope of the tangent line at a point on the graph where there is a minimum or maximum value is zero (horizontal tangent line)

12. Proving Existence of Extrema
Determine IROC at the point using a preceding-following interval method
If the sign of the IROC is different on both sides then it is an extrema
If the sign of the IROC is the same on both sides then it is just a stationary point