AP Calculus Chapter 3
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Critical Numbers
Values of x in the domain of f (x) where f 'x= 0 or f 'x is undefined

How do you find absolute minimums and maximums on a closed interval?
 1. Find critical numbers
 2. Find values for the function at each endpoint and critical number
 3. Determine absolute max and absolute min for the interval by your results from step 2.

Rolle’s Theorem
 If f is differentiable on the interval (a , b) and f(a) = f(b) then there is at least one place in the interval
 where f 'x= 0

Mean Value Theorem (MVT)
 If f (x) is continuous and differentiable on the interval [a , b], there is someplace in the interval where f ' x
 equals the slope of the line going through the endpoints of the interval.

First Derivative ITSC
1) Use critical numbers to determine intervals.
 Interval
 Test Value
 Sign of f ' x
 Conclusion

If sign of f ' x is positive what is happening to f(x)?
f(x) is increasing

If sign of f ' x is negative then f(x) is _______.
f(x) is decreasing

First Derivative Test
Do a first derivative ITSC,
 If f ' x is changing from pos to neg, then relative max
 If f ' x is changing from neg to pos, then relative min

Concave Up
f'(x) is increasing, f''(x)is positive, f(x) will hold water