AP Calc Theorems

Definition of Continuity

1. lim x→c f(x) exists.
2. f(c) exists.
3. lim x→c f(x) = f(c)

When does the limit not exist?

1. f(x) approaches a different number from the right as it does from the left as x→c
2. f(x) increases or decreases without bound as x→c
3. f(x) oscillates between two fixed values as x→c

Intermediate Value Theorem

If f is continuous on the closed interval [a

Definition of a Derivative

lim h→0 (f(x+h) - f(x)) / h

Product Rule

d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)

Quotient Rule

d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)^2

Chain Rule

d/dx f(g(x)) = f'(g(x)) g'(x)

Extrema Value Theorem

If f is continuous on the closed interval [a

The first derivative gives what?

1. critical points
2. relative extrema
3. increasing and decreasing intervals

The second derivative gives what?

1. points of inflection
2. concavity

Rolle's Theorem

Let f be continuous on the closed interval [a

Mean Value Theorem

f'(c) = (f(b) - f(a))/ (b - a)

Fundamental Theorem of Calculus

The integral on (a

Mean Value Theorem (Integrals)

The integral on (a

Average Value Theorem

1/ (b-a) times the integral on (a

Second Fundamental Theorem of Calculus

If f is continuous on an open interval containing a

Derivative of an Inverse Function

g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)