Key Concepts in Calculus: Limits, Derivatives, and Integrals

Definition of Limit

limₓ→a f(x) = L means f(x) gets arbitrarily close to L as x approaches a.

Limit Laws

lim(f ± g) = lim f ± lim g, lim(fg) = lim f × lim g, lim(f/g) = lim f ÷ lim g (if denominator ≠ 0)

Continuity at a Point

f(x) is continuous at x = a if f(a) exists, limₓ→a f(x) exists, and they're equal.

Definition of Derivative

f'(x) = limₕ→0 [f(x + h) − f(x)] / h

Power Rule

d/dx [xⁿ] = n × xⁿ⁻¹

Constant Rule

d/dx [c] = 0

Constant Multiple Rule

d/dx [c × f(x)] = c × f'(x)

Sum/Difference Rule

d/dx [f ± g] = f' ± g'

Product Rule

d/dx [fg] = f × g' + f' × g

Quotient Rule

d/dx [f/g] = (f'g − fg') / g²

Chain Rule

d/dx [f(g(x))] = f'(g(x)) × g'(x)

Derivatives of Trig Functions

sin→cos, cos→−sin, tan→sec², csc→−csc·cot, sec→sec·tan, cot→−csc²

Derivatives of Inverse Trig

sin⁻¹→1/√(1−x²), cos⁻¹→−1/√(1−x²), tan⁻¹→1/(1 + x²)

Critical Points

Where f'(x) = 0 or f'(x) is undefined

Mean Value Theorem

There is some number c between a and b where f'(c) = (f(b) − f(a)) / (b − a)

Rolle's Theorem

If f(a) = f(b), there is some number c between a and b where f'(c) = 0

First Derivative Test

If f' changes sign at c, there is a local max or min

Second Derivative Test

If f'(c) = 0 and f''(c) > 0 → min, if f''(c) < 0 → max

Inflection Point

Where f''(x) changes sign

Power Rule for Integrals

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ −1)

Basic Integrals

∫ dx = x + C, ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C

Trig Integrals

∫ sin = −cos, ∫ cos = sin, ∫ sec² = tan, ∫ csc² = −cot, etc.

Substitution Rule

If u = g(x), then ∫ f(g(x))g'(x) dx = ∫ f(u) du

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)

Net Area

∫ₐᵇ f(x) dx = net area between the curve and x-axis

Average Value of a Function

(1 / (b − a)) × ∫ₐᵇ f(x) dx

Position, Velocity, Acceleration

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

Total Distance Traveled

∫ₐᵇ |v(t)| dt

Volume - Disc Method

V = π × ∫ₐᵇ [r(x)]² dx

Volume - Washer Method

V = π × ∫ₐᵇ ([R(x)]² − [r(x)]²) dx

Volume - Shell Method

V = 2π × ∫ₐᵇ (radius)(height) dx