2. www.mathsrevision.com Higher Outcome 2 Higher Unit 3 www.mathsrevision.com Further Differentiation Trig Functions Harder Type Questions Further Integration Integration Exam Type Questions Integrating Trig Functions Differentiation The Chain Rule Harder Type Questions Differentiation Exam Type Questions

3. www.mathsrevision.com Higher Outcome 2 Trig Function Differentiation The Derivatives of sin x & cos x

4. www.mathsrevision.com Higher Outcome 2 Example Trig Function Differentiation

5. www.mathsrevision.com Higher Outcome 2 Example Trig Function Differentiation Simplify expression - where possible Restore the original form of expression

6. www.mathsrevision.com Higher Outcome 2 The Chain Rule for Differentiating To differentiate composite functions (such as functions with brackets in them) use: Example

7. www.mathsrevision.com Higher Outcome 2 The Chain Rule for Differentiating Trig Functions Trig functions can be differentiated using the same chain rule method. Worked Example:

8. www.mathsrevision.com Higher Outcome 2 The Chain Rule for Differentiating Example

9. www.mathsrevision.com Higher Outcome 2 Example The Chain Rule for Differentiating

10. www.mathsrevision.com Higher Outcome 2 Example The Chain Rule for Differentiating

11. www.mathsrevision.com Higher Outcome 2 The Chain Rule for Differentiating Trig Functions Example

12. www.mathsrevision.com Higher Outcome 2 Example The Chain Rule for Differentiating Trig Functions

13. www.mathsrevision.com Higher Outcome 2 Example Re-arrange: The slope of the tangent is given by the derivative of the equation. Use the chain rule: Where x = 3: The Chain Rule for Differentiating Functions

14. www.mathsrevision.com Higher Outcome 2 Is the required equation Remember y - b = m(x – a) The Chain Rule for Differentiating Functions

15. www.mathsrevision.com Higher Outcome 2 Example In a small factory the cost, C, in pounds of assembling x components in a month is given by: Calculate the minimum cost of production in any month, and the corresponding number of components that are required to be assembled. Re-arrange The Chain Rule for Differentiating Functions

16. www.mathsrevision.com Higher Outcome 2 Using chain rule The Chain Rule for Differentiating Functions

17. www.mathsrevision.com Higher Outcome 2 For x < 5 we have (+ve)(+ve)(-ve) = (-ve) Therefore x = 5 is a minimum Is x = 5 a minimum in the (complicated) graph? Is this a minimum? For x > 5 we have (+ve)(+ve)(+ve) = (+ve) The Chain Rule for Differentiating Functions For x = 5 we have (+ve)(+ve)(0) = 0 x = 5

18. www.mathsrevision.com Higher Outcome 2 The cost of production: Expensive components? Aeroplane parts maybe ? The Chain Rule for Differentiating Functions

19. www.mathsrevision.com Higher Outcome 2 Integrating Composite Functions By “reversing” the Chain Rule we see that: Worked Example

20. www.mathsrevision.com Higher Outcome 2 Integrating Composite Functions Example : Re-arrange Compare with standard form

21. www.mathsrevision.com Higher Outcome 2 Example Compare with standard form Integrating Composite Functions

22. www.mathsrevision.com Higher Outcome 2 Example Integrating Composite Functions

23. www.mathsrevision.com Higher Outcome 2 Example Integrating Composite Functions

24. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions Integration is opposite of differentiation Worked Example

25. www.mathsrevision.com Higher Outcome 2 From “reversing” the chain rule we can see that: Worked Example Integrating Trig Functions

26. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions Example Apply the standard form (twice). Break up into two easier integrals

27. www.mathsrevision.com Higher Outcome 2 Example Re-arrange Apply the standard form Integrating Trig Functions

28. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions (Area) Example A The diagram shows the graphs of y = -sin x and y = cos x a)Find the coordinates of A b)Hence find the shaded area C A S T 0o0o 180 o 270 o 90 o

29. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions (Area)

30. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions (Area)

31. www.mathsrevision.com Higher Outcome 2 Example Integrating Trig Functions Remember cos(x + y) =

32. www.mathsrevision.com Higher Outcome 2 Integrating Trig Functions

33. www.mathsrevision.com Higher Outcome 2 Example Use the “reverse” chain rule So we have: Giving: Integrating Functions

34. Differentiation Higher Mathematics www.maths4scotland.co.uk Next

35. Calculus Revision Back Next Quit Differentiate

36. Calculus Revision Back Next Quit Differentiate Split up Straight line form Differentiate

37. Calculus Revision Back Next Quit Differentiate

38. Calculus Revision Back Next Quit Differentiate Straight line form Differentiate

39. Calculus Revision Back Next Quit Differentiate Multiply out Differentiate

40. Calculus Revision Back Next Quit Differentiate Straight line form Differentiate

41. Calculus Revision Back Next Quit Differentiate multiply out differentiate

42. Calculus Revision Back Next Quit Differentiate Straight line form Differentiate

43. Calculus Revision Back Next Quit Differentiate Straight line form multiply out Differentiate

44. Calculus Revision Back Next Quit Differentiate multiply out Simplify Differentiate Straight line form

45. Calculus Revision Back Next Quit Differentiate Chain rule Simplify

46. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

47. Calculus Revision Back Next Quit Differentiate Chain Rule

48. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

49. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

50. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

51. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

52. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

53. Calculus Revision Back Next Quit Differentiate

54. Calculus Revision Back Next Quit Differentiate

55. Calculus Revision Back Next Quit Differentiate

56. Calculus Revision Back Next Quit Differentiate

57. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

58. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

59. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

60. Calculus Revision Back Next Quit Differentiate

61. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify Straight line form

62. Calculus Revision Back Next Quit Differentiate Straight line form

63. Calculus Revision Back Next Quit Differentiate

64. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

65. Calculus Revision Back Next Quit Differentiate multiply out Differentiate

66. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

67. Calculus Revision Back Next Quit Differentiate Straight line form

68. Calculus Revision Back Next Quit Differentiate Straight line form Multiply out Differentiate

69. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

70. Calculus Revision Back Next Quit Differentiate

71. Calculus Revision Back Next Quit Differentiate Multiply out Differentiate

72. Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

73. Quit C P D www.maths4scotland.co.uk © CPD 2004

74. Integration Higher Mathematics www.maths4scotland.co.uk Next

75. Calculus Revision Back Next Quit Integrate Integrate term by term simplify

76. Calculus Revision Back Next Quit Find

77. Calculus Revision Back Next Quit Integrate Multiply out brackets Integrate term by term simplify

78. Calculus Revision Back Next Quit Find

79. Calculus Revision Back Next Quit Integrate Standard Integral (from Chain Rule)

80. Calculus Revision Back Next Quit Find p, given

81. Calculus Revision Back Next Quit Evaluate Straight line form

82. Calculus Revision Back Next Quit Find Use standard Integral (from chain rule)

83. Calculus Revision Back Next Quit Find Integrate term by term

84. Calculus Revision Back Next Quit Integrate Straight line form

85. Calculus Revision Back Next Quit Integrate Straight line form

86. Calculus Revision Back Next Quit Integrate Straight line form

87. Calculus Revision Back Next Quit Integrate Split into separate fractions

88. Calculus Revision Back Next Quit Find Use standard Integral (from chain rule)

89. Calculus Revision Back Next Quit Find

90. Calculus Revision Back Next Quit Find

91. Calculus Revision Back Next Quit Integrate Straight line form

92. Calculus Revision Back Next Quit Given the acceleration a is: If it starts at rest, find an expression for the velocity v where Starts at rest, so v = 0, when t = 0

93. Calculus Revision Back Next Quit A curve for which passes through the point Find y in terms of x. Use the point

94. Calculus Revision Back Next Quit Integrate Split into separate fractions Multiply out brackets

95. Calculus Revision Back Next Quit If passes through the point express y in terms of x. Use the point

96. Calculus Revision Back Next Quit Integrate Straight line form

97. Calculus Revision Back Next Quit The graph of passes through the point (1, 2). express y in terms of x. If simplify Use the point Evaluate c

98. Calculus Revision Back Next Quit Integrate Straight line form

99. Calculus Revision Back Next Quit A curve for which passes through the point (–1, 2). Express y in terms of x. Use the point

100. Calculus Revision Back Next Quit Evaluate Cannot use standard integral So multiply out

101. Calculus Revision Back Next Quit Evaluate Straight line form

102. Calculus Revision Back Next Quit Evaluate Use standard Integral (from chain rule)

103. Calculus Revision Back Next Quit The curve passes through the point Find f(x ) use the given point

104. Calculus Revision Back Next Quit Integrate Integrate term by term

105. Calculus Revision Back Next Quit Integrate Integrate term by term

106. Calculus Revision Back Next Quit Evaluate

107. Quit C P D www.maths4scotland.co.uk © CPD 2004