Application of Derivatives
Madhya Pradesh Board · Class 12 · Mathematics
Quick revision notes for Application of Derivatives — Madhya Pradesh Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Rate of Change of Quantities
- Derivative dy/dx represents the rate of change of y with respect to x
- At a specific point x₀, f'(x₀) gives the instantaneous rate of change
- When both x and y vary with time t, use chain rule: dy/dx = (dy/dt)/(dx/dt)
Increasing and Decreasing Functions
- Function f is increasing on interval I if x₁ < x₂ implies f(x₁) < f(x₂)
- Function f is decreasing on interval I if x₁ < x₂ implies f(x₁) > f(x₂)
- If f'(x) > 0 on interval I, then f is increasing on I
Maxima and Minima
- Local maximum: f(c) ≥ f(x) for all x in a neighborhood of c
- Local minimum: f(c) ≤ f(x) for all x in a neighborhood of c
- Absolute maximum: largest value of f over its entire domain
Optimization Problems
- Set up the problem by identifying the quantity to be optimized
- Express the objective function in terms of a single variable
- Find the domain of the objective function
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