AP Calculus AB Formula and Concept Sheet

Limit Laws

lim [f(x) ± g(x)] = lim f(x) ± lim g(x), lim [f(x) · g(x)] = lim f(x) · lim g(x), lim [f(x)/g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0

Continuity at x = a

lim x→a f(x) = f(a)

Point-slope form

y - f(a) = f'(a)(x - a)

Power Rule

d/dx [x^n] = n·x^(n-1)

Constant Rule

d/dx [c] = 0

Constant Multiple

d/dx [c·f(x)] = c·f'(x)

Sum/Diff Rule

d/dx [f ± g] = f' ± g'

Product Rule

(fg)' = f'·g + f·g'

Quotient Rule

(f/g)' = (f'·g - f·g') / g²

Chain Rule

d/dx [f(g(x))] = f'(g(x))·g'(x)

Trig Derivatives

d/dx [sin x] = cos x, d/dx [cos x] = -sin x, d/dx [tan x] = sec² x, d/dx [sec x] = sec x·tan x, d/dx [csc x] = -csc x·cot x, d/dx [cot x] = -csc² x

Implicit Differentiation Example

d/dx [y²] = 2y·dy/dx

Indefinite Integrals

∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ e^x dx = e^x + C, ∫ a^x dx = a^x / ln(a) + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C, ∫ csc² x dx = -cot x + C, ∫ sec x·tan x dx = sec x + C, ∫ csc x·cot x dx = -csc x + C

Common Antiderivatives

∫ dx = x + C, ∫ k dx = kx + C, ∫ x^n dx = x^(n+1)/(n+1) + C, ∫ 1/x dx = ln|x| + C, ∫ e^x dx = e^x + C, ∫ a^x dx = a^x / ln(a) + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec x dx = ln|sec x + tan x| + C, ∫ csc x dx = -ln|csc x + cot x| + C

FTC Part 2

∫ from a to b of f(x) dx = F(b) - F(a)

FTC Part 1

d/dx [∫ from a to x of f(t) dt] = f(x)

Average Value of a Function

(1 / (b - a)) ∫ from a to b of f(x) dx

Critical Points

f'(x) = 0 or undefined

Increasing Function

f'(x) > 0

Decreasing Function

f'(x) < 0

Concave Up

f''(x) > 0

Concave Down

f''(x) < 0

Inflection Point

f''(x) = 0 AND sign changes

Mean Value Theorem

f'(c) = [f(b) - f(a)] / (b - a), f(x) must be continuous and differentiable

Rolle's Theorem

f(a) = f(b) → f'(c) = 0 somewhere between a and b

Velocity

v(t) = x’(t)

Acceleration

a(t) = v'(t) = x''(t)

Displacement

∫ from a to b of v(t) dt

Total Distance Traveled

∫ from a to b of |v(t)| dt

Area Between Curves

∫ from a to b of [Top - Bottom] dx (or Right - Left if in terms of y)

Volume by Disk Method

V = π ∫ from a to b of [R(x)]² dx, Use when rotating around axis with no hole

Volume by Washer Method

V = π ∫ from a to b of ([R(x)]² - [r(x)]²) dx, Use when rotating around axis and there's a hole

Volume by Cross Sections

V = ∫ from a to b of A(x) dx, Use when base is bounded and cross sections are known shapes

Separable DEs

dy/dx = f(x)·g(y) → (1/g(y)) dy = f(x) dx, Integrate both sides

Slope Fields

Small line segments showing slope dy/dx at (x, y), Estimate solution curves using the slope pattern

Unit Circle Values

θ (Radians) | sin θ | cos θ, 0 | 0 | 1, π/6 | 1/2 | √3/2, π/4 | √2/2 | √2/2, π/3 | √3/2 | 1/2, π/2 | 1 | 0, 2π/3 | √3/2 | -1/2, 3π/4 | √2/2 | -√2/2, 5π/6 | 1/2 | -√3/2, π | 0 | -1, 7π/6 | -1/2 | -√3/2, 5π/4 | -√2/2 | -√2/2, 4π/3 | -√3/2 | -1/2, 3π/2 | -1 | 0, 5π/3 | -√3/2 | 1/2, 7π/4 | -√2/2 | √2/2, 11π/6 | -1/2 | √3/2, 2π | 0 | 1