AP Calculus AB Theorems and Concepts

Mean Value Theorem

If f(x) is continuous and differentiable on the closed interval [a,b], then there exists some c in the open interval (a,b) such that f’(c) = f(b) - f(a) / b - a.

Or… The tangent line and secant line are parallel

Rolle’s Theorem

If f(x) is continuous and differentiable on the closed interval [a,b] and f(a) = f(b), then there exists some c in the open interval (a,b) such that f’(c) = 0.

Extreme Value Theorem

If f(x) is continuous on the closed interval from [a,b], then f(x) has a guaranteed absolute maximum and minimum.

1. lim x→a f(x)-f(a)/x-a (AROC equation)

2. lim h→0 f(a+h)-f(a)/h (lets you solve for any point)

What are the two limit definitions?

Limit Definition Justification of a Vertical Asymptote

If x=c is a vertical assymptote if and only iff the limit as x approaches c from the left or right is equal to positive or negative infinity.

Limit Definition Justification of a Horizontal Asymptote

y=c is a horizontal asymptote if and only if the limit of f(x) as x approaches infinity is equal to some constant where the constant is a real number.

Intermediate Value Theorem

If f(x) is continuous on the closed interval [a,b] and N is a number between f(a) and f(b) where f(a) does not equal f(b), then there exists a number c in the (a,b) such that f(c)=N.

1. Must not have a fraction in the function

2. Must be continous

3. Must move to the same value from both sides

What are the three criteria for a general limit to exist?

lim x→c [b⋅f(x)] = b ⋅lim x→c f(x)

What is the coefficient rule of limits?

The mathematical function applied to f(x) on the inside of the limit is equal to the function applied to the limit itself.

Ex: The sum of the limits is the limits of the sums.

How do limit rules work?

When the x value is in the domain of a continuous function defined by a limit.

When can you use the direct substitution property when finding limits?

Squeeze Theorem

If f(x) is less than or equal to g(x) is less than or equal to h(x) when x is near a, then the limit of g(x) is equal to the limit of f(x) and h(x).

Ex: lim x→1 f(x) = 9 < lim x→ 1 g(x) < lim x→ 1 h(x) = 9 then lim x→1 g(x) = 9

1. Defined at f(a)

2. Has a general limit at f(a)

3. The general limit is equal to the point itself.

   1. lim x→a f(x)=f(a)

What are the three criteria for a function to be continuous at a point?