Calculus (Copy)
14 Calculus – Cheat Sheet
14.1 Derived Function
- Derivative = rate of change of y with respect to x.
- Informal idea: slope of tangent to curve at a point.
14.2 Notations
- f′(x), f″(x) = first and second derivatives
- dy/dx = derivative of y with respect to x
- d²y/dx² = second derivative
14.3 Standard Derivatives
- d/dx (xⁿ) = n xⁿ⁻¹ (n ∈ ℚ)
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x
- Chain rule: d/dx [f(g(x))] = f′(g(x)) × g′(x)
14.4 Product & Quotient Rules
- Product rule: (uv)′ = u′v + uv′
- Quotient rule: (u/v)′ = (u′v − uv′) / v²
14.5 Gradients, Tangents, Normals
- Gradient at x₀ = f′(x₀)
- Tangent: y − y₁ = m(x − x₁)
- Normal: slope = −1/m (perpendicular to tangent)
14.6 Stationary Points
- Stationary point: f′(x) = 0
- Maxima/Minima via tests (see 14.9)
14.7 Rates of Change & Approximations
- Connected rates: dy/dt = (dy/dx)(dx/dt)
- Small increments: Δy ≈ (dy/dx) × Δx
14.8 Maxima & Minima (Applications)
- Use f′(x) = 0 to find critical points.
- Apply to area, volume, cost, etc.
14.9 First & Second Derivative Tests
- 1st derivative: sign change of f′(x)
↑→↓ = max, ↓→↑ = min - 2nd derivative:
f″(x) < 0 ⇒ max, f″(x) > 0 ⇒ min
14.10 Integration Basics
- Integration = reverse of differentiation.
- ∫ f′(x) dx = f(x) + C
14.11 Basic Integration Rules
- ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ −1)
- ∫ 1/x dx = ln|x| + C
- ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + C
14.12 Special Integrals
- ∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + C
- ∫ sin(ax+b) dx = −(1/a) cos(ax+b) + C
- ∫ cos(ax+b) dx = (1/a) sin(ax+b) + C
- ∫ sec²(ax+b) dx = (1/a) tan(ax+b) + C
- ∫ eᵃˣ⁺ᵇ dx = (1/a) eᵃˣ⁺ᵇ + C
14.13 Definite Integrals & Areas
- ∫ₐᵇ f(x) dx = area under curve y = f(x) from x=a to x=b
- Between two curves: ∫ (upper − lower) dx
14.14 Kinematics with Calculus
- s = displacement, v = velocity, a = acceleration
- v = ds/dt, a = dv/dt = d²s/dt²
- If v is given: integrate to get s; differentiate to get a
14.15 Motion Graphs
- s–t graph: slope = velocity
- v–t graph: slope = acceleration; area = displacement
- a–t graph: area = change in velocity