AP calculus BC formulas

Definition of Derivative

f'(x) = limₕ→0 [f(x+h) – f(x)] / h

Power Rule

d/dx [x^n] = n·x^(n–1)

Product Rule

d/dx [u·v] = u'·v + u·v'

Quotient Rule

d/dx [u/v] = [v·u' – u·v'] / v²

Chain Rule

d/dx f(g(x)) = f'(g(x)) · g'(x)

Derivative of sin x

cos x

Derivative of cos x

–sin x

Derivative of tan x

sec²x

Derivative of ln x

1/x

Derivative of e^x

e^x

Mean Value Theorem

f'(c) = [f(b)–f(a)] / (b–a)

Linear Approximation

L(x) = f(a) + f'(a)(x–a)

Differential

dy ≈ f'(x)·dx

Fundamental Theorem of Calculus Part 1

d/dx ∫ₐˣ f(t) dt = f(x)

Fundamental Theorem of Calculus Part 2

∫ₐᵇ f(x) dx = F(b) – F(a)

Average Value of a Function

(1/(b–a))·∫ₐᵇ f(x) dx

Disk Method Volume

π·∫ₐᵇ [R(x)]² dx

Washer Method Volume

π·∫ₐᵇ ([R(x)]² – [r(x)]²) dx

Shell Method Volume

2π·∫ₐᵇ [radius·height] dx

Arc Length Parametric

∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt

Arc Length Function y(x)

∫ₐᵇ √[1 + (dy/dx)²] dx

Surface Area of Revolution

2π·∫ₐᵇ y·√[1 + (dy/dx)²] dx

Parametric Derivative

dy/dx = (dy/dt)/(dx/dt)

Second Derivative Parametric

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

Area in Polar Coordinates

(1/2)·∫ₐᵇ [r(θ)]² dθ

Arc Length Polar

∫ₐᵇ √[r² + (dr/dθ)²] dθ

Exponential Growth Solution

y = Ce^(kt)

Taylor Series

Σ [f⁽ⁿ⁾(a)/n!]*(x–a)^n

Maclaurin Series for e^x

Σ x^n / n!

Maclaurin Series for sin x

Σ (–1)^n · x^(2n+1)/(2n+1)!

Maclaurin Series for cos x

Σ (–1)^n · x^(2n)/(2n)!

Geometric Series Sum

a/(1–r) if |r|<1

Ratio Test

limₙ→∞ |aₙ₊₁ / aₙ|

nth Term Test for Divergence

If limₙ→∞ aₙ ≠ 0, the series diverges

p-Series Test

Σ1/n^p converges if p>1, diverges if p≤1

Integral Test

Convergence determined by ∫₁^∞ f(x) dx

Alternating Series Error Bound

Error ≤ |next term|