# 11 x1 t10 01 first derivative (2012)

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<ol><li> 1. Geometrical Applications of DifferentiationThe First Derivative </li><li> 2. Geometrical Applicationsof Differentiation ddyThe First Derivative y, f x , f x , dx dx </li><li> 3. Geometrical Applicationsof Differentiation ddyThe First Derivative y, f x , f x , dx dxdy measures the slope of the tangent to a curvedx </li><li> 4. Geometrical Applicationsof Differentiation d dyThe First Derivative y, f x , f x ,dx dxdy measures the slope of the tangent to a curvedx If f x 0, the curve is increasing </li><li> 5. Geometrical Applicationsof Differentiation d dyThe First Derivative y, f x , f x ,dx dxdy measures the slope of the tangent to a curvedx If f x 0, the curve is increasing If f x 0, the curve is decreasing </li><li> 6. Geometrical Applicationsof Differentiation d dyThe First Derivative y, f x , f x ,dx dxdy measures the slope of the tangent to a curvedx If f x 0, the curve is increasing If f x 0, the curve is decreasing If f x 0, the curve is stationary </li><li> 7. Geometrical Applicationsof Differentiation d dyThe First Derivative y, f x , f x ,dx dxdy measures the slope of the tangent to a curvedx If f x 0, the curve is increasing If f x 0, the curve is decreasing If f x 0, the curve is stationary e.g. For the curve y 3x 2 x 3 , find all of the stationary pointsand determine their nature.Hence sketch the curve </li><li> 8. Geometrical Applicationsof Differentiation d dyThe First Derivative y, f x , f x ,dx dxdy measures the slope of the tangent to a curvedx If f x 0, the curve is increasing If f x 0, the curve is decreasing If f x 0, the curve is stationary e.g. For the curve y 3x 2 x 3 , find all of the stationary pointsand determine their nature.Hence sketch the curvedy 6 x 3x 2dx </li><li> 9. dyStationary points occur when0 dx </li><li> 10. dy Stationary points occur when0dxi.e. 6 x 3x 2 0 </li><li> 11. dy Stationary points occur when0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 </li><li> 12. dy Stationary points occur when0dxi.e. 6 x 3x 2 0 3x2 x 0 x 0 or x 2 stationary points occur at 0,0 and 2,4 </li><li> 13. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 </li><li> 14. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 xdydx </li><li> 15. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x0dydx </li><li> 16. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x0dy0dx </li><li> 17. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0 0dy0dx </li><li> 18. dyStationary points occur when0 dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0 0 0dy0dx </li><li> 19. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0dy 0dx </li><li> 20. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0dy(-9) 0dx </li><li> 21. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0dx </li><li> 22. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx </li><li> 23. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point </li><li> 24. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 </li><li> 25. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 xdydx </li><li> 26. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2dydx </li><li> 27. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2dy 0dx </li><li> 28. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 22dy 0dx </li><li> 29. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 22 2dy 0dx </li><li> 30. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2(1)2 2dy 0dx </li><li> 31. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2(1)2 2dy(3) 0dx </li><li> 32. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2(1)2 2 (3)dy(3) 0dx </li><li> 33. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2(1)2 2 (3)dy(3) 0(-9)dx </li><li> 34. dyStationary points occur when 0dxi.e. 6 x 3x 2 0 3x2 x 0x 0 or x 2 stationary points occur at 0,0 and 2,40,0 x 0(-1) 0 0 (1)dy(-9) 0(3)dx 0,0 is a minimum turning point2,4 x 2(1)2 2 (3)dy(3) 0(-9)dx 2,4 is a maximum turning point </li><li> 35. yx </li><li> 36. yx </li><li> 37. y2,4x </li><li> 38. y2,4xy 3x 2 x 3 </li><li> 39. Exercise 10A; 1, 2ace, 4, 5, 6ac, 7, 8, 9ace etc, 11, 12, 13, 15ace, 16, 17Exercise 10B; 1ad, 2ac, 3ac, 5, 6, 7ac, 8, 10, 12</li></ol>