AP Calculus Unit 5

test thurs-fri jan 11-12

Rolle’s Theorem

Rolle’s Theorem

If f is continuous + differentiable, and f(a) = f(b), then there is a point, c, where f’( c) = 0

Mean Value Theorem

If f is continuous + differentiable, then there is a point, c, where f’( c) = (f(b) - f(a)) / (b - a) (inst r.o.c = avg r.o.c)

Extreme Value Theorem

If f is continuous on [a,b], then f has both an extreme maximum and minimum on [a,b]

Critical Numbers

Where f’( c) = 0 or DNE

First Derivative Test

c is a critical number

If f’(x) goes from neg to pos at c, then f( c) is a relative minimum

If f’(x) goes from pos to neg at c, then f( c) is a relative maximum

Canditate Test

Used for finding absolute extrema

1. Evaluate f at each critical number in (a,b)

2. Evaluate f at each endpoint of [a,b]

Test for Concavity

If f’(x) > 0 for all x in (a,b), then f is concave up on (a,b)

If f’(x) < 0 for all x in (a,b), then f is concave down on (a,b)

Point of Inflection

Where a graph changes concavity; either f’ ’(x) = 0 or f’(x) DNE

Second Derivative Test

Used for finding relative max or mins. f’( c) = 0

1. If f ‘ ‘( c) > 0, then f( c) is a relative minimum

2. If f ‘ ‘( c) < 0, then f( c) is a relative maximum

3. If f’ ‘( c) = 0, refer to First Derivative Test