Calc Chp 5 BIG QUIZ

Intermediate Value Theorem

A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b); if f(c) is between f(a) and f(b), then c is in [a,b].

Mean Value Theorem

If f(x) is continuous @ [a,b] and differentiable at every point ∈ (a,b), then there is at least one point c in (a,b) where:

Extreme Value Theorem

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

First Derivative Test

• If f’(x) > 0 for x < c and f’(x) < 0 for x > c, then f has a local maximum at c.

• If f’(x) < 0 for x < c and f’(x) > 0 for x > c, then f has a local minimum at c.

• If f’ does not change sign at x = c, then f does not have a local extremum at c.

Second Derivative Test

• If f’(c) = 0 and f”(c) > 0, then x = c is a local minimum.

• If f’(c) = 0 and f”(c) < 0, then x = c is a local maximum.

Antiderivative

F(x) is an antiderivative of a function f(x) if F’(x) = f(x) for all x in the domain of f. The process of finding an antiderivative is antidifferentiation

Critical Point

A point at which f’ = 0 or f’ does not exist (DNE)

Stationary Point

A point at which f’ = 0