Calculus

Limits

lim x→a f(x) = L—Limit definition

lim x→a [f±g] = L±M—Sum/difference rule

lim x→a [f·g] = L·M—Product rule

lim x→a [f/g] = L/M (M≠0)—Quotient rule

L'Hôpital: lim f/g = lim f'/g'—For 0/0 or ∞/∞

lim x→0 (sin x)/x = 1—

lim x→0 (1-cos x)/x = 0—

lim n→∞ (1+1/n)ⁿ = e—Definition of e

Derivative Rules

d/dx [c] = 0—Constant rule

d/dx [xⁿ] = nxⁿ⁻¹—Power rule

d/dx [cf] = cf'—Constant multiple

d/dx [f±g] = f'±g'—Sum/difference rule

d/dx [fg] = f'g + fg'—Product rule

d/dx [f/g] = (f'g-fg')/g²—Quotient rule

d/dx [f(g(x))] = f'(g(x))·g'(x)—Chain rule

Common Derivatives

d/dx [eˣ] = eˣ—

d/dx [aˣ] = aˣ ln a—

d/dx [ln x] = 1/x—

d/dx [log_a x] = 1/(x ln a)—

d/dx [sin x] = cos x—

d/dx [cos x] = -sin x—

d/dx [tan x] = sec²x—

d/dx [cot x] = -csc²x—

d/dx [sec x] = sec x tan x—

d/dx [csc x] = -csc x cot x—

d/dx [arcsin x] = 1/√(1-x²)—

d/dx [arctan x] = 1/(1+x²)—

Integral Rules

∫xⁿ dx = xⁿ⁺¹/(n+1)+C—n ≠ -1

∫1/x dx = ln|x|+C—

∫eˣ dx = eˣ+C—

∫aˣ dx = aˣ/ln a+C—

∫sin x dx = -cos x+C—

∫cos x dx = sin x+C—

∫sec²x dx = tan x+C—

∫csc²x dx = -cot x+C—

∫sec x tan x dx = sec x+C—

∫1/(1+x²) dx = arctan x+C—

∫1/√(1-x²) dx = arcsin x+C—

Integration Techniques

u-substitution: ∫f(g(x))g'(x)dx—Let u=g(x)

By parts: ∫u dv = uv - ∫v du—LIATE rule for choosing u

Partial fractions—Decompose rational functions

Trig substitution—√(a²-x²): x = a sinθ

√(a²+x²): x = a tanθ—

√(x²-a²): x = a secθ—

Applications of Derivatives

f'(c) = 0 or undefined—Critical points

f' > 0 → increasing—First derivative test

f' < 0 → decreasing—

f'' > 0 → concave up (minimum)—Second derivative test

f'' < 0 → concave down (maximum)—

f'' = 0 → possible inflection point—

Related rates: implicit differentiation with respect to t—

Fundamental Theorem & Applications

∫ₐᵇ f(x)dx = F(b)-F(a)—FTC Part 1

d/dx ∫ₐˣ f(t)dt = f(x)—FTC Part 2

Area = ∫ₐᵇ |f(x)-g(x)| dx—Between curves

V = π∫ₐᵇ [f(x)]² dx—Disk method

V = 2π∫ₐᵇ x·f(x) dx—Shell method

Arc length = ∫ₐᵇ √(1+[f'(x)]²) dx—

Average value = 1/(b-a) ∫ₐᵇ f(x) dx—

Series & Sequences

Taylor: Σ f⁽ⁿ⁾(a)(x-a)ⁿ/n!—Taylor series

Maclaurin: Taylor at a=0—

eˣ = Σ xⁿ/n!—

sin x = Σ (-1)ⁿx²ⁿ⁺¹/(2n+1)!—

cos x = Σ (-1)ⁿx²ⁿ/(2n)!—

1/(1-x) = Σ xⁿ, |x|<1—Geometric series

ln(1+x) = Σ (-1)ⁿ⁺¹xⁿ/n—|x| ≤ 1

Ratio test: lim |aₙ₊₁/aₙ| < 1 → converges—

Integral test: compare Σ with ∫—

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