Increasing decreasing functions
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The document discusses increasing and decreasing functions. It defines an increasing function as one where the gradient (dy/dx) is always positive, and a decreasing function as one where the gradient is always negative. To determine if a function is increasing or decreasing, you find where the derivative (dy/dx) is equal to 0 to find critical points, and use a nature table to analyze the sign of the derivative on each interval. The derivative also needs to be analyzed to determine if a function is always increasing or decreasing.
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Increasing decreasing functions
- 2. What is to be learned? • What is meant by increasing/decreasing functions • How we work out when function is increasing/decreasing • How to show if a function is always increasing/decreasing
- 3. increasing increasing decreasing need to find SPs dy /dx = 0 Increasing → dy /dx is +ve Decreasing → dy /dx is -ve
- 4. Ex y = 4x3 – 3x2 + 10 Function Decreasing? For SPs dy /dx = 0 dy /dx = 12x2 – 6x 12x2 – 6x = 0 6x(2x – 1) = 0 6x = 0 or 2x – 1 = 0 x = 0 or x = ½
- 5. Nature Table y = 4x3 – 3x2 + 10 dy /dx = 12x2 – 6x = 6x(2x – 1) SPs at x = 0 and ½ x 0 dydy //dxdx = 6x(2x – 1)= 6x(2x – 1) 0 -1 ¼ = + = - Slope Max TP at x = 0 ½ 1 = + 0 Min TP at x = ½ - X - + X - + X + Decreasing 0 < x < ½
- 6. Function always increasing? • dy /dx always +ve (i.e > 0) Ex y =x3 + 7x dy /dx = 3x2 + 7 Increasing as dy /dx > 0 for all x.
- 7. Function always decreasing • dy /dx always -ve (i.e < 0) Ex y = -6x -x3 dy /dx = -6 - 3x2 Decreasing as dy /dx < 0 for all x.
- 8. Less obvious y = 1 /3x3 + 3x2 + 11x dy /dx= x2 + 6x + 11 completing square (x + 3)2 – 9 + 11 (x + 3)2 + 2 Increasing as dy /dx > 0 for all x.
- 9. Increasing/Decreasing Functions • Increasing → Gradient +ve (dy /dx > 0) • Decreasing → Gradient -ve (dy /dx < 0) • Find SPs (only need x values) • Completing square can be handy tactic
- 10. Ex y = 1 /3x3 + 4x2 + 17x dy /dx= x2 + 8x + 17 completing square (x + 4)2 – 16 + 17 (x + 4)2 + 1 Increasing as dy /dx > 0 for all x.