| --- |
| 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)pi*r^3. Find dV/dt when r=5, dr/dt=2. |
| dV/dt=4pi*r^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. |
