AP Calculus AB Formula List Review Flashcards

Flashcards for reviewing the 2020 AP Calculus AB formula list, covering definitions, theorems, and applications like particle motion and rate in/rate out problems.

Definition of the derivative

f'(x) = lim (h->0) [f(x+h) - f(x)] / h (Alternative form) f'(a) = lim (x->a) [f(x) - f(a)] / (x-a)

Definition of continuity

f is continuous at c if and only if: 1) f(c) is defined; 2) lim (x->c) f(x) exists; 3) lim (x->c) f(x) = f(c)

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c on (a, b) such that f'(c) = [f(b) - f(a)] / (b - a)

Intermediate Value Theorem

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

Definition of a definite integral

Integral from a to b of f(x) dx = lim (delta x -> 0) [sum from k=1 to n of f(x_k) * delta x]

Definition of a Critical Number

Let f be defined at c. If f'(c) = 0 or if f'(c) is undefined at c, then c is a critical number of f.

First Derivative Test

If f' changes from negative to positive at c, then (c, f(c)) is a relative minimum of f. If f' changes from positive to negative at c, then (c, f(c)) is a relative maximum of f.

Second Derivative Test

If f'(c) = 0 and f''(c) > 0, then (c, f(c)) is a relative minimum. If f'(c) = 0 and f''(c) < 0, then (c, f(c)) is a relative maximum

Definition of Concavity

The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

Test for Concavity

If f''(x) > 0 for all x in I, then the graph of f is concave upward in I. If f''(x) < 0 for all x in I, then the graph of f is concave downward in I.

Definition of an Inflection Point

A function f has an inflection point at (c, f(c)) if f''(c) = 0 or f''(c) does not exist and f''(x) changes sign from positive to negative or negative to positive at x=c OR if f'(x) changes from increasing to decreasing or decreasing to increasing at x = c.

First Fundamental Theorem of Calculus

Integral from a to b of f'(x) dx = f(b) - f(a)

Second Fundamental Theorem of Calculus

d/dx [Integral from a to x of f(t) dt] = f(x) Chain Rule Version: d/dx [Integral from a to g(x) of f(t) dt] = f(g(x)) * g'(x)

Average rate of change of f(x) on [a, b]

[f(b) - f(a)] / (b - a)

Average value of f(x) on [a, b]

AVE = (1 / (b - a)) * Integral from a to b of f(x) dx

Particle Motion: Velocity

v(t) = s'(t) where s(t) is the position function

Particle Motion: Speed

|v(t)|

Particle Motion: Acceleration

a(t) = v'(t) = s''(t)

Particle Motion: Displacement from x=a to x=b

Integral from a to b of v(t) dt

Particle Motion: Total Distance Traveled from x=a to x=b

Integral from a to b of |v(t)| dt

Particle Motion: At rest

v(t) = 0

Particle Motion: Moving left (down)

v(t) < 0

Particle Motion: Moving right (up)

v(t) > 0

Particle Motion: Changes direction

When velocity changes signs.

Particle Motion: Speeding up

Velocity and acceleration have the same sign.

Particle Motion: Slowing down

Velocity and acceleration have different signs.

Rate In/Rate Out: Amount at time t

A(t) = initial amount + Integral from 0 to t of [rate in - rate out] dt

L’Hospital’s Rule

If lim (x->a) f(x) = 0 and lim (x->a) g(x) = 0 OR lim (x->a) f(x) = +/- infinity and lim (x->a) g(x) = +/- infinity, then lim (x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x)