source: ncert
topic: calculus
class_level: class_12
chapter: ch6
difficulty: advanced
Class 12 | Ch6: Applications of Derivatives RATE OF CHANGE: dy/dx represents rate of change of y w.r.t. x. If x and y both functions of t: use chain rule. dy/dt=(dy/dx)×(dx/dt). TANGENT AND NORMAL: Slope of tangent at (x1,y1): m=dy/dx|(x1,y1). Equation of tangent: y-y1=m(x-x1). Slope of normal=-1/m. Equation of normal: y-y1=(-1/m)(x-x1). INCREASING AND DECREASING FUNCTIONS: f increasing on (a,b) if f'(x)>0 for all x in (a,b). f decreasing on (a,b) if f'(x)<0 for all x in (a,b). MAXIMA AND MINIMA: Critical points: where f'(x)=0 or f'(x) undefined. First Derivative Test: f'changes + to -: local maximum. f' changes - to +: local minimum. No change: neither. Second Derivative Test: f''(c)<0: local maximum. f''(c)>0: local minimum. f''(c)=0: test fails. ABSOLUTE MAXIMA/MINIMA on [a,b]: Evaluate f at all critical points AND endpoints. Largest=absolute max, smallest=absolute min. APPROXIMATIONS: f(x+deltax)≈f(x)+f'(x)×deltax. SOLVED EXAMPLES: Example 1: Find intervals where f(x)=2x^3-9x^2+12x-5 is increasing/decreasing. f'(x)=6x^2-18x+12=6(x^2-3x+2)=6(x-1)(x-2). f'(x)>0 when x<1 or x>2: increasing on (-inf,1) and (2,inf). f'(x)<0 when 1<x<2: decreasing on (1,2). Example 2: Local maxima/minima of f above. f'(1)=0 and f' changes + to -: local max at x=1. f(1)=2-9+12-5=0. f'(2)=0 and f' changes - to +: local min at x=2. f(2)=16-36+24-5=-1. Example 3: Two numbers with sum 24 and maximum product. Let numbers be x and 24-x. P=x(24-x)=24x-x^2. P'=24-2x=0 → x=12. P''=-2<0: maximum. Numbers: 12 and 12. Example 4: Rate: balloon volume V=(4/3)pir^3. Find dV/dt when r=5, dr/dt=2. dV/dt=4pir^2*(dr/dt)=4pi×25×2=200pi cm^3/s. COMMON MISTAKES: Critical points where f'=0 need NOT be maxima or minima (inflection points possible). Second derivative test FAILS when f''=0 (use first derivative test instead). For max product/area problems: always verify using second derivative test.