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1 Flinders Christian Community College
MAV REVISION LECTURE MATHEMATICAL METHODS UNITS 3 AND 4 Presenter: MICHAEL SWANBOROUGH Flinders Christian Community College

2 EXAMINATION 1 Short-answer questions (40 marks)
Questions are to be answered without the use of technology and without the use of notes Time Limit: 15 minutes reading time 60 minutes writing time

3 EXAMINATION 2 Part I: Multiple-choice questions
22 questions (22 marks) Part II: Extended response questions: 58 marks Time limit: 15 minutes reading time 120 minutes writing time

4 Examination Advice General Advice
Answer questions to the required degree of accuracy. If a question asks for an exact answer then a decimal approximation is not acceptable. When an exact answer is required, appropriate working must be shown.

5 Examination Advice General Advice
When an instruction to use calculus is stated for a question, an appropriate derivative or antiderivative must be shown. Label graphs carefully – coordinates for intercepts and stationary points; equations for asymptotes. Pay attention to detail when sketching graphs.

6 Examination Advice General Advice
Marks will not be awarded for questions worth more than one mark if appropriate working is not shown.

7 Examination Advice Notes Pages
Well-prepared and organised into topic areas. Prepare general notes for each topic. Prepare specific notes for each section of Examination 2. Include process steps as well as specific examples of questions.

8 Examination Advice Notes Pages
Include key steps for using your graphic calculator for specific purposes. Be sure that you know the syntax to use with your calculator (CtlgHelp is a useful APP for the TI-84+)

9 Strategy - Examination 1
Examination Advice Strategy - Examination 1 Use the reading time to carefully plan an approach for the paper. Momentum can be built early in the exam by completing the questions for which you feel the most confident. Read each question carefully and look for key words and constraints.

10 Strategy - Examination 2
Examination Advice Strategy - Examination 2 Use the reading time to plan an approach for the paper. Make sure that you answer each question in the Multiple Choice section. There is no penalty for an incorrect answer. It may be sensible to obtain the “working marks” in the extended answer section before tackling the multiple choice questions.

11 Strategy - Examination 2
Examination Advice Strategy - Examination 2 Some questions require you to work through every multiple-choice option – when this happens don’t panic!! Eliminate responses that you think are incorrect and focus on the remaining ones. Multiple Choice questions generally require only one or two steps – however, you should still expect to do some calculations.

12 Strategy - Examination 2
Examination Advice Strategy - Examination 2 If you find you are spending too much time on a question, leave it and move on to the next. When a question says to “show” that a certain result is true, you can use this information to progress through to the next stage of the question.

13 Revision Quiz 1 2 3 4 5

14 Question 1 where a, b and c are three different positive real numbers. The equation has exactly a) 1 real solution b) 2 distinct real solutions c) 3 distinct real solutions d) 4 distinct real solutions e) 5 distinct real solutions B

15 B Question 2 The range of the function with graph as shown is a) b) c)
d) e) B

16 Bonus Prize!!

17 E Question 4 For the equation the sum of the solutions on the interval
is a) b) E c) d) e)

18 B Question 5 What does V.C.A.A. stand for?
a) Vice-Chancellors Assessment Authority b) Victorian Curriculum and Assessment Authority c) Victorian Combined Academic Authority d) Victorian Certificate of Academic Aptitude e) None of the above B

19 The linear factors of the polynomial
Question 1 The linear factors of the polynomial are ANSWER: B

20 Question 4 a) b)

21 Functions and Their Graphs
Vertical line test - to determine whether a relation is a function A represents the DOMAIN B represents the CODOMAIN (not the range!)

22 Interval Notation Square brackets [ ] – included
Round brackets ( ) – excluded

23 Maximal (or implied) Domain
The largest possible domain for which the function is defined A function is undefined when: a) The denominator is equal to zero The square root of a negative number is present. The expression in a logarithm results in a negative number.

24 Consider the function So the maximal domain is:

25 Using Transformations
When identifying the type of transformation that has been applied to a function it is essential to state each of the following: NATURE – Reflection, Dilation, Translation MAGNITUDE (or size) DIRECTION

26 1. Translations a) Parallel to the x-axis – horizontal translation. b) Parallel to the y-axis – vertical translation. To avoid mistakes, let the bracket containing x equal zero and then solve for x. If the solution for x is positive – move the graph x units to the RIGHT. If the solution for x is negative – move the graph x units to the LEFT.

27 2. Dilations a) Parallel to the y-axis – the dilation factor is the number outside the brackets. This can also be described as a dilation from the x-axis. Parallel to the x-axis – the dilation factor is the reciprocal of the coefficient of x. This can also be described as a dilation from the y-axis. Note: A dilation of a parallel to the y-axis is the same as a dilation of parallel to the x-axis.

28 3. Reflections a) Reflection about the x-axis b) Reflection about the y-axis c) Reflection about both axes d) Reflection about the line

29 Question 6 Determine the graph of

30 ANSWER: A Reflection about the y-axis

31 ANSWER: B Question 7 Graph of
Reflected in the x-axis, Translated 2 units to the right, Translated 1 unit down Reflection: Translation: Translation: ANSWER: B

32 Graphs of Power Functions

33 Square Root Functions The graph is:
translated 2 units in the positive x direction translated 1 unit in the positive y direction

34 The rule of the graph shown could be
Question 9 The rule of the graph shown could be ANSWER: D

35 Graphs of Rational Functions
Question 10 The equations of the horizontal and vertical asymptotes of the graph with equation Vertical: Horizontal: ANSWER: E

36 The graph shown could be that of the function f whose rule is
Question 12 The graph shown could be that of the function f whose rule is a) b) c) d) e) ANSWER: A

37 Absolute Value Functions
Question 14 ANSWER: D

38 Part of the graph of is shown below. a) Sketch the graph of
Question 15 Part of the graph of is shown below. a) Sketch the graph of

39 b) Find the set of values of x for which
From the graph, solve

40 Composite Functions For the composite function to be defined
When the composite function is defined

41 Investigating Composite Functions
Step 1: Complete a Function, Domain, Range (FDR) table. Step 2: Check that the range of g is contained in the domain of f . Step 3: Substitute the function g(x) into the function f (x). Step 4: Remember that:

42 Question 16

43

44 Inverse Functions Key features:
The original function must be one-to-one Reflection about the line y = x Domain and range are interchanged Intersections between the graph of the function and its inverse occur on the line y = x

45 To find the equation of an inverse function
Step 1: Complete a Function, Domain, Range (FDR) table. Step 2: Interchange x and y in the given equation. Step 3: Transpose this equation to make y the subject. Step 4: Express the answer clearly stating the rule and the domain.

46 Question 18 ANSWER: A

47 Question 19 Graph of the inverse function ANSWER: C

48 Question 21

49 a) exists because the function f is one-to-one b) i)

50 b) ii)

51 Algebra of Functions Sum and Difference of Functions Key features:
Points of intersection of the two functions. Points where either graph crosses the x-axis The dominant function in different parts of the domain.

52

53 Solving indicial equations
Step 1: Use appropriate index laws to reduce both sides of the equation to one term. Step 2: Manipulate the equation so that either the bases or the powers are the same. Step 3: Equate the bases or powers. If this is not possible then take logarithms of both sides to either base 10 or base e.

54 Question 26 ANSWER: C

55 Solving logarithmic equations
Step 1: Use the logarithmic laws to reduce the given equation to two terms – one on each side of the equality sign. Step 2: Convert the logarithmic equation to indicial form. Step 3: Manipulate the given equation so that either the bases or the powers are the same.

56 Step 4:. Equate the bases or powers
Step 4: Equate the bases or powers. If this is not possible then take logarithms of both sides to either base 10 or base e. Step 5: Check to make sure that the solution obtained does not cause the initial function to be undefined.

57 Question 28 ANSWER: A

58 Change of Base Rule for Logarithms
Question 29 ANSWER: D

59 Circular (Trigonometric) Functions
Amplitude: a Horizontal translation: c units in the negative x-direction Period: Vertical translation: d units in the positive y-direction

60 Question 30 ANSWER: C

61 ANSWER: C Question 32 Dilation by a factor of 2 from the x-axis
Reflection about the x-axis

62 Solving Trigonometric Equations
Put the expression in the form sin(ax) = B Check the domain – modify as necessary. Use the CAST diagram to mark the relevant quadrants. Solve the angle as a first quadrant angle. Use symmetry properties to find all solutions in the required domain. Simplify to get x by itself.

63 Question 34 ANSWER: E

64 Question 36 Analysis Question a)

65 b)

66 c)

67 d)

68 e) i)

69 e) ii)

70 Revision Quiz 1 2 3 4 5

71 Question 1 The derivative of is equal to a) b) c) d) e) A

72 Bonus Prize!!

73 Question 3 Angie notes that 2 out of 10 peaches on her peach tree are spoilt by birds pecking at them. If she randomly picks 30 peaches the probability that exactly 10 of them are spoilt is equal to a) d) b) e) c) D

74 D Question 4 The total area of the shaded region shown is given by a)
c) d) D e)

75 A Question 5 Which one of the following sets of statements is true?
2 1 and s m < b) > c) d) e) = A

76 DIFFERENTIAL CALCULUS
Chain Rule: Product Rule: Quotient Rule:

77 Further Rules of Differentiation
Square Root Functions

78 Further Rules of Differentiation
Trigonometric Functions

79 Further Rules of Differentiation
Logarithmic Functions

80 Further Rules of Differentiation
Exponential Functions

81 Question 37 ANSWER: D

82 Question 38 ANSWER: B

83 Graphs of Derived Functions
Question 40 ANSWER: C

84 Question 42 a)

85 b)

86 Question 43

87 b)

88 c)

89 Approximations Question 45 ANSWER: D

90 Related Rates Question 46

91

92 Question 47 Analysis Question a)

93 Stationary point of inflexion at:
b) Find the EXACT COORDINATES of the two stationary points and their nature. Stationary point of inflexion at: Local maximum at:

94 c) i)

95 ii)

96 iii) Select a point and find the equation of the tangent at this point.

97 This tangent will pass through the origin when x = 0
Therefore the only two tangents that pass through the origin are when

98 d) i) Use CALCULUS to find the exact values of the constants p, q and k.

99 d) ii)

100 Antidifferentiation and Integral Calculus

101 Question 48 ANSWER: B

102 Rules of Antidifferentiation
Trigonometric Functions

103 Rules of Antidifferentiation
Exponential Functions

104 Rules of Antidifferentiation
Logarithmic Functions

105 Example

106 Definite Integrals

107 Properties of Definite Integrals
Question 50 ANSWER: E

108 Integration by recognition

109 Question 52 On the interval (a, b) the gradient of g(x) is negative. ANSWER: A

110 Calculating Area Sketch a graph of the function, labelling all x-intercepts. Shade in the region required. Divide the area into parts above the x-axis and parts below the x-axis. Find the integral of each of the separate sections, using the x-intercepts as the terminals of integration. Subtract the negative areas from the positive areas to obtain the total area.

111 Question 53 The total area bounded by the curve and the x-axis on the interval [a, c] is given by: ANSWER: D

112 Question 54 a)

113 b) Hence, find the exact area of the shaded region

114 Area between curves

115 Sketch the curves, locating the points of intersection.
Method Sketch the curves, locating the points of intersection. Shade in the required region. If the terminals of integration are not given – use the points of intersection. Check to make sure that the upper curve remains as the upper curve throughout the required region. If this is not the case then the area must be divided into separate sections. Evaluate the area.

116 Find the solution to the equation
Question 55 Find the solution to the equation

117 b) Use CALCULUS to find the area of the shaded region.

118 Numerical techniques for finding area
Question 56 ANSWER: D

119 Analysis Question Question 57
Write down the equation in x, the solutions of which give the x-coordinates of the stationary points of the curve a)

120 b) i)

121 b) ii) A repeated root at x = -1 indicates that the normal is a tangent to the curve at this point.

122 c) i)

123 c) i)

124 c) ii)

125 Discrete Random Variables
A discrete random variable takes only distinct or discrete values and nothing in between. Discrete variables are treated using either discrete or binomial distributions. These values are usually obtained by counting. A continuous random variable can take any value within a given domain. These values are usually obtained through measurement of a quantity. Continuous variables are often treated using normal distributions.

126 Expected value and expectation theorems

127 Variance and Standard Deviation

128 Question 59 The number of hours each day, X, spent cycling has the following probability distribution. The proportion of days for at least two hours of cycling is: ANSWER: D

129 Question 60 x 1 2 3 4 ANSWER: B

130 Markov Chains A Markov chain is a chain of events for which the probabilities of outcomes or states depend on what has happened previously. Tree diagrams are a useful tool for solving problems.

131 Question 61 If it has snowed the day before the probability of snow is 0.6. If it has not snowed on the previous day then the probability of snow is 0.1. If it has snowed on Thursday, what is the probability that it will not snow on the following Saturday?

132 Question 62 A bag contains three white ball and seven yellow balls. Three balls are drawn without replacement. The probability that they are all yellow is: ANSWER: D

133 The Binomial Distribution

134 Question 63 A random sample of 20 tickets is taken. The probability that this sample contains exactly twelve Adult tickets is equal to: ANSWER: B

135 Question 65 ANSWER: A

136 Continuous Random Variables
Properties of probability density functions a) for all real numbers x b) c)

137 Question 66 a)

138 b)

139 Continuous Random Variables
Mean: Mode: the value for which is a maximum Median: the value m such that Variance:

140 Question 67 ANSWER: E

141 The Normal Distribution
The mean, mode and median are the same. The total area under the curve is one unit.

142 Using symmetry properties

143

144

145 Question 71 X is normally distributed with a mean of 72 and a standard deviation of 8. Use the result that to find: a)

146 b)

147 c)

148 Solving normal distribution problems
Draw a diagram, clearly labelling the mean. Shade the region required. Use either the appropriate symmetry properties or a calculator to find the required probability Remember that:

149 Question 72 Volunteers for a weight loss program have weights which are normally distributed with a mean of 100 kg and a standard deviation of 8 kg. One person is selected at random. The probability that this person’s weight is over 110 kg is approximately

150 ANSWER: E

151 Applications of the normal distribution
Draw a diagram, clearly shading the region that corresponds to the given probability. Use the symmetry properties of the curve to write down the appropriate z value. Use the inverse normal function to find the required probability and the corresponding z value. Use the relationship to calculate the required x value.

152 Question 73 Black Mountain coffee is sold in packets labeled as being of 250 grams weight. The packing process produces packets whose weight is normally distributed with a standard deviation of 3 grams. In order to guarantee that only 1% of packets are under the labeled weight, the actual mean weight (in grams) would be required to be closest to a) 243 b) 247 c) 250 d) 254 e) 257

153 ANSWER: E

154 Question 77 Analysis Question a)

155 b)

156 c) d) i)

157 ii) iii)

158 iv) Conditional probability

159 THE FINAL RESULT


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