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AP Calculus Theorems

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Section 1

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When does the limit not exist?

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Calculus Math

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Mar 14, 2020

Cards (17)

Section 1

(17 cards)

When does the limit not exist?

Front

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

Back

Rolle's Theorem

Front

Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c)= 0

Back

Chain Rule

Front

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

Back

Second Fundamental Theorem of Calculus

Front

If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x)

Back

Intermediate Value Theorem

Front

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b) then there is at least one number c in [a, b] such that f(c) = k

Back

Fundamental Theorem of Calculus

Front

The integral on (a, b) of f(x) dx = F(b) - F(a)

Back

Mean Value Theorem

Front

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

Back

Definition of Continuity

Front

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

Back

Mean Value Theorem (Integrals, Rectangle Format)

Front

The integral on (a, b) of f(x) dx = f(c) (b - a)

Back

The first derivative gives what?

Front

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

Back

Quotient Rule

Front

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

Back

Average Value Theorem/MVT for Integrals

Front

1/ (b-a) times the integral on (a, b) of f(x) dx

Back

Derivative of an Inverse Function

Front

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

Back

The second derivative gives what?

Front

1. points of inflection 2. concavity

Back

Product Rule

Front

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

Back

Extrema Value Theorem

Front

If f is continuous on the closed interval [a, b], then f has both a maximum and a minimum on the interval.

Back

Definition of a Derivative

Front

lim h→0 (f(x+h) - f(x)) / h

Back