ap calc ab unit 3 quiz 1 vocab theorems

Critical Number

If f'(c)=0 or f(c) is undefined, then c is a critical number of f.

Critical Point

(c, f(c)) is a critical point on f.

Local Extreme Values

If a function f has a local maximum value or a local minimum value at an interior point, c, of its domain, then f'(c) = 0 or f'(c) is undefined.

Special note about critical points

They must be in the domain of f & they do not always indicate a maximum or minimum (sometimes just a point of inflection).

Local Maximum

The value, f(c) within some open interval containing c which is greater than or equal to all other f(x) for x's within that interval.

Local Minimum

The value, f(c) within some open interval containing c which is less than or equal to all other f(x) for x's within that interval.

Extreme Value Theorem

If f is continuous over a closed interval, then f has a maximum and minimum value in that interval.

Maximum Value

A max value of a function is NOT an x or an (x,y), it is ONLY the y.

Absolute Maximum

The value, f(c) within some closed interval containing c which is greater than or equal to all other f(x) for x's within that interval.

Absolute Minimum

The value, f(c) within some closed interval containing c which is less than or equal to all other f(x) for x's within that interval.

Mean Value Theorem

If f is continuous over [a,b] and differentiable over (a,b) then there is some point c between a and b such that 𝑓'(𝑐) = 𝑓(𝑏)−𝑓(𝑎) / 𝑏−𝑎.

Rolle's Theorem

If f is continuous over [a,b] and differentiable over (a,b) and f(a) = f(b), then there is at least one number c in (a,b) such that f'(c) = 0.