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

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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

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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 EXAMINATION 2

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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.

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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.

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

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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.

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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+)

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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.

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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.

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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.

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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.

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Revision Quiz

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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

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

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Bonus Prize!!

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

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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

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

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Question 4 a) b)

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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!)

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

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

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Consider the function So the maximal domain is:

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

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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.

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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. b)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.

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

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Question 6 Determine the graph of

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Reflection about the y-axis ANSWER: A

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

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Graphs of Power Functions

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

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

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

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

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Absolute Value Functions Question 14 ANSWER: D

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

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

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Composite Functions For the composite functionto be definedWhen the composite functionis defined

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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: Investigating Composite Functions

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Question 16

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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

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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.

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ANSWER: A Question 18

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ANSWER: C Question 19 Graph of the inverse function

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Question 21

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

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b) ii)

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

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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.

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Question 26 ANSWER: C

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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. Solving logarithmic equations

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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.

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ANSWER: A Question 28

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Change of Base Rule for Logarithms Question 29 ANSWER: D

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

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ANSWER: C Question 30

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

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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.

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ANSWER: E Question 34

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Question 36 Analysis Question a)

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b)

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c)

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d)

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e) i)

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e) ii)

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Revision Quiz

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Question 1 The derivative ofis equal to a)b)c) d) e) A

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Bonus Prize!!

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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 Question 3 a) d) b) e) c) D

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

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Which one of the following sets of statements is true? a) 2121 and b) 2121 and c) 2121 and d) 2121 and e) 2121 and A Question 5

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DIFFERENTIAL CALCULUS Chain Rule: Product Rule: Quotient Rule:

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Further Rules of Differentiation Square Root Functions

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Further Rules of Differentiation Trigonometric Functions

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Further Rules of Differentiation Logarithmic Functions

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Further Rules of Differentiation Exponential Functions

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ANSWER: D Question 37

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ANSWER: B Question 38

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Graphs of Derived Functions ANSWER: C Question 40

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Question 42 a)

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b)

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Question 43

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b)

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c)

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Approximations ANSWER: D Question 45

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Related Rates Question 46

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Question 47 Analysis Question a)

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

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c)i)

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ii)

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

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

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

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d)ii)

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Antidifferentiation and Integral Calculus

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ANSWER: B Question 48

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Trigonometric Functions Rules of Antidifferentiation

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Exponential Functions

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Rules of Antidifferentiation Logarithmic Functions

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Example

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Definite Integrals

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Properties of Definite Integrals ANSWER: E Question 50

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Integration by recognition

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

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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.

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

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Question 54 a)

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

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Area between curves

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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. Method

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Question 55 Find the solution to the equation

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

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Numerical techniques for finding area ANSWER: D Question 56

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

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b) i)

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

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c) i)

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c) ii)

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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.

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Expected value and expectation theorems

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Variance and Standard Deviation

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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: Question 59 ANSWER: D

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ANSWER: B x Question 60

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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.

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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?

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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

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The Binomial Distribution

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

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ANSWER: A Question 65

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

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Question 66 a)

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b)

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

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Question 67 ANSWER: E

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

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Using symmetry properties

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

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b)

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c)

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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: Solving normal distribution problems

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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 Question 72

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ANSWER: E

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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.

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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) 243b) 247c) 250d) 254e) 257

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ANSWER: E

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a) Question 77 Analysis Question

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b)

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c) d) i)

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ii) iii)

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iv) Conditional probability

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THE FINAL RESULT

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