Calculus Notes & Formulas

1. Definition

Study of continuous change - limits, differentiation, integration, and their applications.

2. Limits & Continuity

Standard Limits (Very Important)

• lim(x→0) sin x / x = 1
• lim(x→0) (1 + x)^(1/x) = e
• lim(x→0) (eˣ − 1) / x = 1
• lim(x→0) ln(1 + x) / x = 1
• lim(x→∞) (1 + 1/x)ˣ = e

Continuity: f is continuous at x=a if lim(x→a) f(x) = f(a). Must check: limit exists, f(a) defined, both equal.

L'Hôpital's Rule: If limit gives 0/0 or ∞/∞ form → differentiate numerator and denominator separately.

3. Differentiation (Very Important)

Key Rules

• Product: (uv)' = u'v + uv'
• Quotient: (u/v)' = (u'v − uv') / v²
• Chain: d/dx[f(g(x))] = f'(g(x))·g'(x)

Standard Derivatives

• d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(eˣ) = eˣ
• d/dx(ln x) = 1/x
• d/dx(sin x) = cos x, d/dx(cos x) = −sin x
• d/dx(tan x) = sec²x
• d/dx(sin⁻¹x) = 1/√(1−x²)
• d/dx(tan⁻¹x) = 1/(1+x²)

4. Partial Derivatives

For f(x,y): ∂f/∂x treats y as constant. ∂²f/∂x∂y = ∂²f/∂y∂x (Clairaut's theorem, if continuous).

Total Derivative: df = (∂f/∂x)dx + (∂f/∂y)dy

Euler's Theorem (for homogeneous function of degree n):
x·(∂f/∂x) + y·(∂f/∂y) = n·f

5. Integration (Very Important)

Standard Integrals

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• ∫eˣ dx = eˣ + C
• ∫1/x dx = ln|x| + C
• ∫sin x dx = −cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C
• ∫1/(1+x²) dx = tan⁻¹x + C
• ∫1/√(1−x²) dx = sin⁻¹x + C

Integration by Parts: ∫u dv = uv − ∫v du
Choose u by ILATE: Inverse trig → Log → Algebraic → Trig → Exponential

6. Definite Integrals & Properties

• ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx
• ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx (King's property - Very Important)
• ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even; = 0 if f is odd

7. Maxima & Minima

• Critical point: f'(x) = 0
• f''(x) > 0 → local minimum
• f''(x) < 0 → local maximum
• f''(x) = 0 → use higher derivatives (inflection point check)

For f(x,y): Let r=fxx, s=fxy, t=fyy. D = rt − s²

• D > 0, r > 0 → minimum | D > 0, r < 0 → maximum | D < 0 → saddle point | D = 0 → inconclusive

8. Multivariable - Gradient, Divergence, Curl

• Gradient: ∇f = (∂f/∂x)î + (∂f/∂y)ĵ + (∂f/∂z)k̂ - points in direction of max increase
• Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z - scalar field
• Curl: ∇×F - measures rotation. Conservative field → curl = 0

9. GATE Trick

King's property solves most definite integral problems in 2 lines. For maxima/minima of 2-variable: always compute D = rt − s² first. L'Hôpital's can be applied repeatedly until indeterminate form is resolved.