Applications of Calculus: JEE Main Complete Guide

Applications of derivatives is the calculus chapter where pure technique meets problem-solving creativity. JEE Main tests tangents and normals, rate of change, monotonicity, maxima and minima, and the Mean Value Theorems — together contributing three to four questions per session. This guide covers every application type with the systematic approach that JEE Main rewards.

Tangents and Normals

Slope of tangent to y = f(x) at point (x₀, y₀): m_t = f'(x₀). Equation of tangent: y − y₀ = m_t(x − x₀). Slope of normal: m_n = −1/m_t (perpendicular to tangent). Equation of normal: y − y₀ = m_n(x − x₀). JEE Main questions: (1) Find the equation of the tangent/normal to a curve at a given point; (2) Find points where the tangent is horizontal (f'(x) = 0) or vertical (f'(x) undefined); (3) Find the tangent that passes through a given external point (solve simultaneously).

Length of tangent: |y₀|√(1 + 1/m_t²). Length of normal: |y₀|√(1 + m_t²). Length of subtangent: |y₀/m_t|. Length of subnormal: |y₀m_t|. JEE Main tests subtangent and subnormal lengths — these are the x-intercept distances from the foot of the perpendicular from the point to the tangent and normal respectively. For the differentiation foundation, see our Continuity & Differentiability Guide. Take a free mock test on applications of derivatives to practise tangent and maxima-minima problems.

Monotonicity and Maxima-Minima

f is increasing on an interval if f'(x) > 0 on that interval; decreasing if f'(x) < 0. Critical points: f'(x) = 0 or f'(x) does not exist. Local maximum at x = c: f'(c) = 0 and f''(c) < 0 (concave down). Local minimum: f'(c) = 0 and f''(c) > 0 (concave up). If f''(c) = 0: use first derivative test (sign change of f' determines max/min/neither).

Absolute (global) maximum and minimum on [a, b]: evaluate f at all critical points in (a, b) plus the endpoints f(a) and f(b). The largest value is the absolute maximum; smallest is the absolute minimum. JEE Main optimization problems: express the quantity to be maximised/minimised as a function of one variable, differentiate, set to zero, verify it is max or min. Common problems: maximum area of rectangle inscribed in a circle, minimum surface area of a box with fixed volume, shortest path problems.

Rolle's Theorem and Lagrange's Mean Value Theorem

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c ∈ (a,b) such that f'(c) = 0. Geometrically: the tangent is horizontal at some interior point. Lagrange's Mean Value Theorem (LMVT): if f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) such that f'(c) = [f(b) − f(a)]/(b − a). Geometrically: the tangent at c is parallel to the secant through (a, f(a)) and (b, f(b)).

JEE Main applications of MVT: (1) verifying the conditions and finding c for a specific function; (2) using LMVT to prove inequalities (if f'(x) < M on (a,b), then |f(b) − f(a)| < M(b−a)); (3) identifying which function satisfies Rolle's condition from multiple choices.

Rate of Change Problems

dy/dx represents the instantaneous rate of change of y with respect to x. Related rate problems: two quantities changing with time — differentiate both with respect to t. Example: a spherical balloon being inflated at 10 cm³/s — find the rate of change of radius when r = 5 cm. V = (4/3)πr³ → dV/dt = 4πr² × dr/dt → 10 = 4π(25) × dr/dt → dr/dt = 10/(100π) = 1/(10π) cm/s. JEE Main related-rate problems always have the form: given dA/dt, find dB/dt at a specific instant.

Approximations Using Derivatives

Linear approximation: f(x+Δx) ≈ f(x) + f'(x)·Δx. JEE Main uses this to evaluate approximate values: √(26) ≈ √25 + (1/(2√25)) × 1 = 5 + 0.1 = 5.1. Differentials: dy = f'(x)dx. For the integration applications that follow this chapter, see our Area Under Curves Guide.

Exam Strategy

Applications of derivatives questions are problem-set problems — each requires recognising the application type, applying the appropriate technique, and completing the algebra accurately. The five types (tangents/normals, monotonicity, maxima-minima, MVT, related rates) cover 95% of JEE Main questions. Practise 5 problems of each type under time pressure. For the complete calculus preparation plan, see our Calculus Complete Guide. Upgrade for ₹149/month for 250+ applications of derivatives problems at all difficulty levels.