Studied by 21 peopletest thurs-fri jan 11-12
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
Evaluate f at each critical number in (a,b)
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
If f ‘ ‘( c) > 0, then f( c) is a relative minimum
If f ‘ ‘( c) < 0, then f( c) is a relative maximum
If f’ ‘( c) = 0, refer to First Derivative Test
Note
Flashcard