Chapter 2 · Class 11-12 & JEE Main/Adv
Application of Derivatives
34 topics in 10 parts
Key ideas
Analytical Comparison & Distinction Matrix
| Analytical Method | Standard Representation | Parametric / Invariant Framework |
|---|---|---|
| Algebraic Nature | Explicit equations relating spatial coordinates | Parameterized using scalar parameters (λ, μ) or vectors |
| Symmetry Usage | Relies on coordinate expansion | Exploits vector dot/cross products for rapid simplification |
| Computational Efficiency | Direct for intercept evaluation | Superior for angles, orthogonal projections, and distances |
| Common Mistake | Sign confusion in determinants | Omitting boundary condition checks |
Conceptual Foundation & Core Theory
In 3D Euclidean space, a Plane is a flat two-dimensional surface that extends infinitely in all directions. A plane is uniquely positioned in space if we know either (1) its perpendicular distance from the origin and its normal direction, or (2) a point lying on the plane and a normal vector perpendicular to it.
General Cartesian Equation of a Plane:
Every first-degree equation in x, y, z represents a plane:
Ax + By + Cz + D = 0
Crucially, the coefficients <A, B, C> represent the Direction Ratios of the Normal Vector perpendicular to the plane!
Intercept Form: If a plane cuts off intercepts a, b, c from the coordinate axes X, Y, Z respectively, its equation is:
x / a + y / b + z / c = 1
Standard Formulas & Governing Conditions
// Plane Formulations in 3D Space
1. General Cartesian Form:
Ax + By + Cz + D = 0 (Normal vector n = Aî + Bĵ + Ck̂)
2. Vector Form Passing through a with Normal n:
(r - a) · n = 0 ⟺ r · n = d
3. Normal Form (Unit Normal n̂ and Perpendicular Distance p from Origin):
r · n̂ = p ⟺ lx + my + nz = p
4. Intercept Form (Intercepts a, b, c on axes):
x/a + y/b + z/c = 1
5. Perpendicular Distance of Point (x1, y1, z1) from Plane Ax + By + Cz + D = 0:
d = | Ax1 + By1 + Cz1 + D | / √(A² + B² + C²)
6. Distance Between Two Parallel Planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0:
d = | D1 - D2 | / √(A² + B² + C²)Standard Forms of Plane Equations
| Plane Equation Form | Formula | Geometrical Information Conveyed |
|---|---|---|
| General Form | Ax + By + Cz + D = 0 | Normal direction ratios are <A, B, C> |
| Normal Form | lx + my + nz = p | p is exact perpendicular distance from origin; <l, m, n> are DCs |
| Intercept Form | x/a + y/b + z/c = 1 | Plane cuts X at (a,0,0), Y at (0,b,0), Z at (0,0,c) |
| Three-Point Form | | x-x1 y-y1 z-z1 ; x2-x1 y2-y1 z2-z1 ; x3-x1 y3-y1 z3-z1 | = 0 | Plane passing through three non-collinear points |
Step-by-Step Solved Problem (JEE Main Standard)
Problem: Find the perpendicular distance between the parallel planes 2x - 2y + z + 3 = 0 and 4x - 4y + 2z + 5 = 0.
Multiply Plane 1 by 2: 4x - 4y + 2z + 6 = 0.
Plane 2: 4x - 4y + 2z + 5 = 0.
Here A = 4, B = -4, C = 2, D₁ = 6, D₂ = 5.
d = |D₁ - D₂| / √(A² + B² + C²)
d = |6 - 5| / √(4² + (-4)² + 2²) = 1 / √(16 + 16 + 4) = 1 / √36 = 1/6 units.
Checkpoint
1. What are the direction ratios of the normal to the plane 3x - 4y + 5z = 12?
All 34 topics in Application of Derivatives
Rate of Change & Approximations
- Rate Measure
- Approximations
- Related Rates
Tangents & Normals
- Equation of Tangents & Normals
- Angle Between Curves
- Length of Tangent, Normal, Sub-Tangent & Sub-Normal
- Shortest Distance Between Two Curves
- Parametric Curves
Monotonicity
- In an Interval
- At a Point
- Separating Intervals
- Inequalities
Point of Inflection & Concavity
- Point of Inflection
- Concavity & Convexity
Rolle's & Lagrange's Mean Value Theorems
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Cauchy's Mean Value Theorem
- Inequalities Using LMVT
Tests for Local Maximum/Minimum
- Definition of Maxima & Minima
- First Derivative Test
- Second Derivative Test
- Higher Order Derivative Test
- Local Extrema for Non-Differentiable Functions
- Critical Points
Concept of Global Maximum/Minimum
- Greatest & Least Value of Function
- Absolute Extrema in an Interval
Application of Maxima/Minima
- Optimisation
- Plane Geometry
- Solid Geometry
- Coordinate Geometry
Nature of Roots of Cubic Polynomials
- Roots of Cubic Polynomials
- Number of Roots of Equations
Graph Plotting
- Graph of Rational Functions
- Curve Tracing & Features of Graphs