Application of Derivatives
Board: JEE Main |Exam:JEEMain
JEEMainstudynotesforApplicationofDerivativesbyAjayYadav.
KeyConcepts
RateofChangeandApproximation
dy/dtasrate.Approximation:f(x+Δx)&asympf(x)+f'(x)Δx.
TangentandNormal
Tangent:y-y₀=m(x-x₀).Normal:y-y₀=(-1/m)(x-x₀).Anglebetweencurves:tanθ=|(m1-m2)/(1+m1m2)|.
Increasing/Decreasing
f'(x)>0 ⇒ strictly increasing. f'(x)<0 ⇒ strictly decreasing.
Maxima and Minima
First derivative test. Second derivative test. Absolute max/min on closed interval.
Important Formulas
| Tangent | y-y₀=f'(x₀)(x-x₀) |
| Normal | y-y₀=-(x-x₀)/f'(x₀) |
| f”<0 | Local maximum |
| f”>0 | Local minimum |
Solved Examples
Example 1: Find approximate value of √25.2.
Solution: f(x)=√x, f'(x)=1/(2√x). f(25)=5, f'(25)=0.1. √25.2 ≈ 5+0.1(0.2)=5.02.
Practice Questions
- Find tangent to y=x³ at (1,1).
- Find maxima/minima: f(x)=x³-12x.
- A window is rectangular with semicircular top, fixed perimeter. Find dimensions for max light.
- Find intervals of increase: f(x)=x³-6x²+9x.
- Find angle between y=x² and y=x³ at (1,1).