Calculus Concepts: 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(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 denominator ≠ 0)

Continuity at a Point

f(x) is continuous at x = a if: f(a) exists; limₓ→a f(x) exists; limₓ→a f(x) = f(a)

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 [cf(x)] = c f'(x)

Sum/Difference Rule

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Product Rule

d/dx [f(x)g(x)] = f(x)g'(x) + f'(x)*g(x)

Quotient Rule

d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x)) / [g(x)]²

Chain Rule

d/dx [f(g(x))] = f'(g(x)) * g'(x)

Derivatives of Trig Functions

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

Derivatives of Inverse Trig

d/dx [sin⁻¹x] = 1/√(1 - x²); d/dx [cos⁻¹x] = -1/√(1 - x²); d/dx [tan⁻¹x] = 1 / (1 + x²)

Critical Points

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

Mean Value Theorem (MVT)

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

Rolle's Theorem

If f(a) = f(b) and f is continuous and differentiable, then ∃ c ∈ (a, b) such that f'(c) = 0

First Derivative Test

If f′ changes sign at c, it's a local max/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; ∫ aˣ dx = aˣ / ln(a) + C (a > 0)

Trig Integrals

∫ 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

Substitution Rule

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

Fundamental Theorem of Calculus (Part 1)

If F is antiderivative of f, then: d/dx [∫ₐˣ f(t) dt] = f(x)

FTC (Part 2)

∫ₐᵇ f(x) dx = F(b) - F(a)

Net Area

∫ₐᵇ f(x) dx is net area between curve and x-axis.

Average Value of a Function

(1/(b - a)) ∫ₐᵇ f(x) dx

Position, Velocity, Acceleration

If s(t) is position: v(t) = s'(t) (velocity); a(t) = v'(t) (acceleration)

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