Chapter 6 Applications of Derivatives

Notes

Notes: Applications of Derivatives chapter mainly deals with: • Increasing and decreasing functions • Maxima and minima • Tangent and normal • Rate of change • Approximation using differentials Important Definitions Increasing Function: If f'(x) > 0 in an interval, then function is increasing. Decreasing Function: If f'(x) < 0 in an interval, then function is decreasing. Critical Point: Point where f'(x) = 0 or undefined. Local Maximum: If f''(x) < 0 at critical point. Local Minimum: If f''(x) > 0 at critical point. Important Formulas Slope of tangent: m = dy/dx Equation of tangent: y – y₁ = m(x – x₁) Slope of normal: m₁ × m₂ = –1 Equation of normal: y – y₁ = –1/m(x – x₁) Approximation Formula: f(x + dx) ≈ f(x) + dy where, dy = f'(x)dx Second Derivative Test • If f''(x) > 0 → minimum • If f''(x) < 0 → maximum Short Tricks • Tangent parallel to x-axis means dy/dx = 0 • Normal slope = negative reciprocal of tangent slope • For maxima/minima always check second derivative Common Mistakes • Forgetting second derivative test • Wrong sign while solving inequalities • Using tangent slope in normal equation • Arithmetic mistakes in maxima-minima values Solved Example Find maxima of f(x) = –x² + 6x f'(x) = –2x + 6 –2x + 6 = 0 x = 3 f''(x) = –2 < 0 Maximum occurs. Maximum value: f(3) = –9 + 18 = 9