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Stationary Points

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Stationary Points

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  1. Stationary Points

  2. What is to be learned • How to use differentiation to identify SPs • What SPs are!!!!!! • What the difference between a stationary point and a stationary value • What SVs are • How to find SP/SVs • How to finf the nature of SP/SVs

  3. m ~ 3 m ~ ½

  4. m ~ ½ m ~ 3 m ~ ½

  5. m ~ ½ m ~ 3 m ~ ½

  6. m = 0 m ~ ½ m ~ 3 m ~ ½

  7. m = 0 m ~ ½ m ~ 3 m ~ ½

  8. At Turning Points the gradient = 0 Turning points are known as stationary points (or values) (SPs or SVs) So for SPs the derivative = 0

  9. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0

  10. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0

  11. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y?

  12. Ex y = x2 -8x + 10 SPs? For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y? at x = 4 y = 42 – 8(4) + 10 = - 6 SP is (4 , -6)

  13. (0 , 10) and (2 , 2) Ex y = 2x3 -6x2 + 10 SPs? For SPs dy/dx = 0 dy/dx = 6x2 – 12x 6x2 – 12x = 0 6x(x – 2) = 0 6x = 0 or x – 2 = 0 x = 0 or x = 2 x = 0 y = 10 x = 2 y =2(2)3 – 6(2)2 + 10 = 2 Quadratic Equation Factorise

  14. Stationary Points • Max TPs, Min TPs or……. • For SPs gradient = 0 • For SPs dy/dx = 0 Tactics Find derivative Solve equation → x y → sub x into original equation = 0

  15. Ex y = 1/3x3 - 2x2 - 12x SPs? For SPs dy/dx = 0 dy/dx = x2 – 4x – 12 x2 – 4x – 12 = 0 (x – 6)(x + 2) = 0 x–6= 0 or x+2 = 0 x = 6 or x = -2 Quadratic Equation Factorise

  16. x = 6 y = 1/3(6)3 – 2(6)2 – 12(6) = - 72 Ex y = 1/3x3 - 2x2 - 12x SPs? For SPs dy/dx = 0 dy/dx = x2 – 4x – 12 x2 – 4x – 12 = 0 (x – 6)(x + 2) = 8 x–6= 0 or x+2 = 0 x = 6 or x = -2 x = -2 y = 1/3(-2)3 – 2(-2)2 – 12(-2) = 132/3 SPs are (-2 , 132/3) and (6 , -72)

  17. Finding the nature

  18. m negative m positive m = 0 for nature need to know gradient just before and after SP

  19. Making a nature table

  20. Making a Nature Table y = x2 -8x + 10 For SPs dy/dx = 0 dy/dx = 2x – 8 2x – 8 = 0 2x = 8 x = 4 y? at x = 4 y = 42 – 8(4) + 10 = - 6 SP is (4 , -6)

  21. SP is (4 , -6) Making a Nature Table y = x2 -8x + 10 dy/dx = 2x – 8 x 5 3 4 - 0 + dy/dx = 2x - 8 Slope Min TP at (4 , -6)

  22. SPs (0 , 10) and (2 , 2) Making a Nature Table y = 2x3 -6x2 + 10 dy/dx = 6x2 – 12x x 1 3 -1 2 0 + 0 + dy/dx = 6x2 – 12x 0 - Slope Min TP Max TP at (2 , 2) at (0 , 10)

  23. Nature Table • Used to find nature of SPs • Show gradient just before and after SPs • Use derivative

  24. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 + 0 + dy/dx = x2 – 4x - 12 0 - Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  25. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 + 0 + dy/dx = x2 – 4x - 12 0 - Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  26. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 - X - 0 dy/dx = (x - 6)(x + 2) 0 - X + + X + = + = - = + Slope Min TP Max TP at (6 , -72) at (-2, 132/3)

  27. SPs (-2, 132/3) and (6 , -72) Making a Nature Table y = 1/3x3 - 2x2 - 12x dy/dx = x2 – 4x - 12 x 0 7 -3 6 -2 * + 0 + dy/dx = x2 – 4x - 12 0 - Slope *could use factorised derivative Min TP Max TP at (6 , -72) at (-2, 132/3)