Definitions
If f'(x)=0
there is a POSSIBLE max or min for f(x)
If f''(x)>0
f(x) is concave up
Average Rate of Change
slope os secant line
f' is increasing
f'' > 0 and f is concave up
possible inflection point
f'' = 0 or undefined
If f'(x)>0
f(x) is increasing
If f''(x)<0
f(x) is concave down
Instantaneous Rate of Change
slope of tangent line
f' is decreasing
f'' <0 and f is concave down
relative max
f' = 0 or undefined and f' changes from + to -
If f'(x)<0
f(x) is decreasing
f is concave down
f'' <0
What's required for continuity at a point?
Limit = function value
f is decreasing
f' is negative
relative min
f' = 0 or undefined and f' changes from - to +
If x=a is a critical value
f'(a)=0 or f'(a)=DNE
f is concave up
f '' >0
Differentiability
means derivative exists
differentiability implies continuity
not continuous Implies not differentiable
f is increasing
f' is positive
f''> 0
relative min
What is required for a limit to exist?
Left limit = right limit
inflection point
f'' = 0 or undefined AND f'' changes signs
Examples of Not Differentiable at a Point
vertical tangent
sharp turn
cusp
not continuous
Critical Point
f(x)= inc/dec or dec/inc
f' = 0 or und.
f''<0
relative max