11X1 T09 04 concavity

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  • 1. Concavity

2. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavity 3. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavity If f x 0, the curve is concave up 4. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavity If f x 0, the curve is concave up If f x 0, the curve is concave down 5. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavity If f x 0, the curve is concave up If f x 0, the curve is concave down If f x 0, possible point of inflection 6. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2 7. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dy 3 x 2 10 x 3dx 8. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dy 3 x 2 10 x 3dx d2y2 6 x 10 dx 9. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dy 3 x 2 10 x 3dx d2y2 6 x 10 dxd2y Curve is concave up when 2 0 dx 10. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dy 3 x 2 10 x 3dx d2y 2 6 x 10 dxd2y Curve is concave up when 2 0 dx i.e. 6 x 10 0 5 x 3 11. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dyy 3 x 2 10 x 3dx d2y 2 6 x 10 dxd2y Curve is concave up when 2 0 5x dx i.e. 6 x 10 03 5x 3 12. Concavity The second deriviative measures the change in slope with respect to x, this is known as concavityIf f x 0, the curve is concave upIf f x 0, the curve is concave downIf f x 0, possible point of inflectione.g. By looking at the second derivative sketch y x 3 5 x 2 3 x 2dyy 3 x 2 10 x 3dx d2y 2 6 x 10 dxd2y Curve is concave up when 2 0 5x dx i.e. 6 x 10 03 5x 3 13. Turning Points 14. Turning Points All turning points are stationary points. 15. Turning Points All turning points are stationary points. If f x 0, minimum turning point 16. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning point 17. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 18. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx 19. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx d2y 2 6x 2 dx 20. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx d2y 2 6x 2 dxdyStationary points occur when 0dx 21. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx d2y 2 6x 2 dxdyStationary points occur when 0dxi.e. 3 x 2 2 x 1 0 22. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx d2y 2 6x 2 dxdyStationary points occur when 0dxi.e. 3 x 2 2 x 1 0 3x 1 x 1 0 23. Turning Points All turning points are stationary points. If f x 0, minimum turning pointIf f x 0, maximum turning pointe.g. Find the turning points of y x 3 x 2 x 1 dy 3x 2 2 x 1 dx d2y 2 6x 2 dxdyStationary points occur when 0dxi.e. 3 x 2 2 x 1 0 3x 1 x 1 01 x or x 13 24. d2y when x 1, 2 6 1 2 dx 4 0 25. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 26. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y 1 when x , 2 6 2 3 dx 3 40 27. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 28. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 Inflection Points 29. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 Inflection Points A point of inflection is where there is a change in concavity, to see if there is a change, check either side of the point. 30. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 Inflection Points A point of inflection is where there is a change in concavity, to see if there is a change, check either side of the point. e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2 31. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 Inflection Points A point of inflection is where there is a change in concavity, to see if there is a change, check either side of the point. e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2 dy 12 x 2 12 x dx 32. d2y when x 1, 2 6 1 2 dx 4 0 - 1,2 is a maximum turning point 1 d2y1 when x , 2 6 2 3 dx 340 1 , 22 is a minimum turning point 3 27 Inflection Points A point of inflection is where there is a change in concavity, to see if there is a change, check either side of the point. e.g. Find the inflection point(s) of y 4 x 3 6 x 2 2 dyd2y 12 x 2 12 x2 24 x 12 dx dx 33. d2y Possible points of inflection occur when 2 0 dx 34. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 35. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 1 x 2 36. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 1 x x 2d2 y 2dx 37. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 x x 22d2 y 2dx 38. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 x x 22d2 y 0 2dx 39. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 11 x x 22 2 2 d2 y02 dx 40. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 1 1 x x 22 (-1) 2 2 d2 y 02 dx 41. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 11 x x 22 (-1) 22 d2 y (-12)02 dx 42. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 11 x x 22 (-1) 22 (0) d2 y (-12)02 dx 43. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 11 x x 22 (-1) 22 (0) d2 y (-12)0(12) 2 dx 44. d2y Possible points of inflection occur when 2 0 dxi.e. 24 x 12 0 11 1 1 x x 22 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12)2dx 45. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0 (12) 2 ,3 is a point of inflection1 dx 2 46. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondx 47. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 48. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx1 d3ywhen x , 3 24 02 dx 49. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0 2 dx ,3 is a point of inflection 1 2 50. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0 2 dx ,3 is a point of inflection 1 2 Horizontal Point of Inflection; 51. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0 2 dx ,3 is a point of inflection 1 2 dyHorizontal Point of Inflection; 0 dx 52. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0 2 dx ,3 is a point of inflection 1 2 dyd2yHorizontal Point of Inflection; 020 dxdx 53. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 1 1 x x 2 2 (-1) 2 2 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0 2 dx ,3 is a point of inflection 1 2 dyd2y d3yHorizontal Point of Inflection; 02030 dxdxdx 54. d2y Possible points of inflection occur when 2 0dxi.e. 24 x 12 0 1 1 11 x x 2 2 (-1) 22 (0) there is a change in concavity d 2 y (-12) 0(12) 2 ,3 is a point of inflection1 dx 2 d3yIf 3 0, then it is a point of inflectiondxd3y e.g . 3 24dx 1 d3ywhen x , 3 24 0Exercise 10E; 1, 2bc, 3, 6ac, 2 dx ,3 is a point of inflection 1 7bd, 8, 10, 12, 14, 16, 18 2 dyd2y d3yHorizontal Point of Inflection; 02030 dxdxdx