Application of Derivatives
Board: CBSE | Class: Class 12
Comprehensive study notes for Application of Derivatives by Ajay Yadav.
Key Concepts
Rate of Change
dy/dx measures rate of change of y w.r.t. x. If y=f(t) and x=g(t): dy/dx = (dy/dt)/(dx/dt).
Tangents and Normals
Tangent: y-y₀=m(x-x₀), m=f'(x₀). Normal: y-y₀=(-1/m)(x-x₀). Slopes of tangents: m1m2=-1 for perpendicular.
Increasing/Decreasing
f'(x)>0 ⇒ increasing. f'(x)<0 ⇒ decreasing. Critical points: f'(x)=0 or undefined.
Maxima and Minima
First derivative test: sign change of f’ around critical point. Second derivative test: f”(x)<0 ⇒ max, f”(x)>0 ⇒ min.
Absolute Max/Min
On closed interval [a,b], evaluate f at critical points and endpoints. Largest = absolute max, smallest = absolute min.
Important Formulas
| Tangent slope | m = f'(x₀) |
| Normal slope | mₙ = -1/m |
| f'(x) > 0 | Increasing function |
| f'(x) < 0 | Decreasing function |
| Second derivative test | f''(x)<0 ⇒ local max, f''(x)>0 ⇒ local min |
Solved Examples
Example 1: Find the rate of change of area of circle w.r.t. radius when r=5.
Solution: A=πr². dA/dr=2πr. At r=5: dA/dr=10π.
Example 2: Find tangent to y=x² at (1,1).
Solution: dy/dx=2x, m=2. Tangent: y-1=2(x-1) ⇒ y=2x-1.
Example 3: Find intervals of increase/decrease for f(x)=x³-6x²+9x+1.
Solution: f'(x)=3x²-12x+9=3(x-1)(x-3). Increasing: (-∞,1)∪(3,∞). Decreasing: (1,3).
Practice Questions
- Find the rate of change of volume of sphere w.r.t. r at r=4.
- Find normal to y=x² at (2,4).
- Find local maxima/minima of f(x)=2x³-15x²+36x+10.
- Find absolute max of f(x)=x³-3x on [0,2].
- The radius of a circle increases at 2 cm/s. Find rate of area increase when r=10.
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