Calculus
Calculus in NEB Class 12 spans limits, continuity, differentiation, applications of derivatives, integration and areas. Marks are lost less often in the calculus idea than in algebraic setup and endpoint checks.
Core rules
- A limit exists only if left-hand and right-hand limits agree. Standard limits: lim(x→0) sin x/x = 1, lim(x→0) (1 - cos x)/x² = 1/2, lim(x→0) (e^x - 1)/x = 1.
- Product, quotient and chain rules; for parametric x = x(t), y = y(t), dy/dx = (dy/dt)/(dx/dt).
- Tangent slope is f′(x₁); normal slope is -1/f′(x₁) when f′(x₁) ≠ 0.
- Stationary points: solve f′(x) = 0; use f″(x) (positive minimum, negative maximum) or the first-derivative sign change.
- Integration: substitution reverses the chain rule; by parts ∫u dv = uv - ∫v du; definite integrals need an antiderivative and evaluation at both limits.
Worked example
Differentiate y = x² sin x and find the tangent at x = π/2. By the product rule, y′ = 2x sin x + x² cos x. At x = π/2, y = π²/4 and y′ = π. Hence the tangent is y - π²/4 = π(x - π/2).
Traps
- Differentiating sin x while the calculator is in degree mode; calculus formulas assume radians.
- Omitting +C in an indefinite integral.
- Applying the chain rule only to the outer function.
- Finding stationary points but not comparing them with endpoints for a closed-interval maximum or minimum.
- Treating area below the x-axis as positive without splitting the integral.