Mathematical Methods (CAS) Unit 4 Revision Lecture Presented by Samuel Goh

Transcription

1 Mathematical Methods (CAS) Unit 4 Revision Lecture Presented by Samuel Goh

2 Introduction Multiple Choice Question Get familiar with the audience response software! What colour is this dress? a) Blue and Black. b) Gold and White. c) Pink and White. d) Something else. e) Nice dress! Respond:

3 Topics What are we covering? 1. Examination Information 2. Examination Advice and Tips 3. Area of Study Overview 4. Calculus Revision 5. Probability Revision 6. Statistics Revision

4 Examination Information Examination 1 (2 nd November): - Closed book - Short answer and extended answer questions - 15 minutes reading time, 1 hour writing time - 40 marks total - Worth 22% of study score Examination 2 (3 rd November): - Bound reference and CAS calculator allowed - Multiple choice and extended answer - 15 minutes reading time, 2 hours writing time - 80 marks total, 20 marks multiple choice and 60 marks extended answer - Worth 44% of study score

5 Examination Advice and Tips Practice Exams Take notes! - Keep track of the exams you complete! - Excel? Notepad? Pen and paper? How many? - The proficiency you have after completing x practice exams can be modelled by the function: P = 100k x - Some people might have a k=0.5, k=1 or even a k=5. It all depends on how you study and learn! -

6 Examination Advice and Tips Bound Reference Use the study design! - (Or do a Google search!) - Relevant information starts page 71. Do not go overboard!

7 Examination Advice and Tips CAS Calculator Use problems to keep organized! - Separate the problems based on exam question or exam section. Define everything! - Use functions from graphs in calculator, and functions from calculator pages in graphs. Take advantage of built in functions and navigation. - Tangent lines? Determinants? Normal Lines? Factorise? Solve System of Equations? Use the notes page for repetitive calculations.

8 Examination Advice and Tips General Exam Tips Exam 1 - Use your reading time wisely! - Show more working out than less. VCAA is generous with space. - If you have time, substitute back into your equations. - Extremely large numbers? Or large fractions? Take a step back. VCAA usually set simple answers. - Use the correct notation. If a question gives V= r, make sure you differentiate dv/dr, not dy/dx. Exam 2 - Use your reading time wisely! - Do not spend too long on Multiple Choice. - Do not use your calculator syntax as working out. - Make sure your calculator is in the correct mode! (Angles) - Beware of show that or using calculus, you must show extra working out to prove to the examiner you can reach the answer.

9 Skills Checklist Area of Study 3: Calculus Differential Calculus - Average and instantaneous rate of change - Deducing the graph of a derivative or antiderivative - Derivatives of x n, e x, log e (x), sin x, cos(x) and tan(x) - Product rule, quotient rule and chain rule - Applications involving identification of key features, maximum rate of increase or decrease, local maximum/minimum values

10 Skills Checklist Area of Study 3: Calculus Integral Calculus - Antiderivatives of polynomials, e x, sin x, cos x - Anti-differentiation by recognition that F x = f(x) implies f x dx = F x + C - Informal treatment of fundamental theorem of calculus a b f x dx = F b F(a) - Properties of anti-derivatives and definite integrals. - Applications of integration: known rate of change, area under a curve, area between curves, distance travelled, average value of a function.

11 Skills Checklist Area of Study 4: Probability/Statistics Discrete Random Variables - Discrete probability distributions using graphs, tables and probability mass functions - Mean, variance and standard deviation of discrete random variable - The binomial distribution - Calculations of probability for specific values and intervals

12 Skills Checklist Area of Study 4: Probability/Statistics Continuous Random Variables - Probability density functions - Mean, median, variance and standard deviation of a continuous random variable - The normal distribution - Calculation for probabilities for intervals, including conditional probability

13 Skills Checklist Area of Study 4: Probability/Statistics Statistical Inference - Parameters and statistics - Sample proportion as a binomial variable - Normality of sample proportion - Approximate confidence intervals

14 Next Topic! Revision of... Calculus - Graphs of Derivatives and Antiderivatives - Tangents and Normals - Integration by Recognition - Complex Areas

15 Graphs of Derivatives How do we sketch a derivative given a function? f x f (x) What observations can we make?

16 Graphs of Derivatives Things to look out for: - Stationary point/s (result in x-intercept/s on derivative) - Regions with positive and negative gradient (results in positive or negative values on derivative) - Steepness of slope (changes magnitude of derivative)

17 Graphs of Derivatives Written Example Sketch the graph of f (x) given f(x) below. f x f (x)

18 Graphs of Derivatives Multiple Choice Question Which of the following could be the graph of the derivative function y = f x? The graph of a function f(x) is shown below. Respond:

19 Graphs of Antiderivatives How do we sketch an antiderivative given a function? f x f x dx What observations can we make?

20 Graphs of Antiderivatives Things to look out for: - x intercept/s (results in turning point/s on antiderivative) - Areas with positive or negative sign (results in positive or negative gradient on antiderivative) - Magnitude (changes the steepness of the antiderivative)

21 Graphs of Antiderivatives Multiple Choice Question The graph of an antiderivative of f could be The graph of a function f is shown below. Respond:

22 Tangents and Normals Tangent: Normal: A straight line that touches a curve at one point and shares the gradient with that curve at that point. A line that is perpendicular to the tangent at a specific point. m N m T = 1

23 Tangents and Normals Tangent: How to find the tangent of f(x) at x = a? 1. Find out the gradient of f(x) at x=a, or f (a). 2. Find the y-coordinate of f(x) at x=a, or f(a) 3. Use the equation of a linear line: y y 1 = m(x x 1 ) or y f a = f (a)(x a) A straight line that touches a curve at one point and shares the gradient with that curve at that point. To find the normal, find the perpendicular gradient.

24 Tangents and Normals Written Example A normal to the graph of y = x has equation y = 4x + a, where a is a real constant. Find the value of a.

25 Tangents and Normals Written Example Let f: R R, f x = e x + k, where k is a real number. The tangent to the graph of f at the point where x = a passes through the point (0,0). Find the value of k in terms of a.

26 Integration by Recognition What is integration by recognition? Differentiating a function and using the result to help integrate a tough function. Strategy 1. Differentiate the given function to find f x 2. Recognise the tough function you need to integrate inside of that derivative f x. It may not be exactly the same! 3. Write out an expression connecting f x dx and f x 4. Manipulate the integral to get the integral of the tough function

27 Integration by Recognition Written Example A continuous random variable, X, has a probability density function π πx cos if x [0,2] f x = ቐ4 4 0 otherwise Given that d dx πx xsin 4 = πx 4 cos πx 4 + sin( πx ), find E(X). 4

28 Integration by Recognition Written Example Part of the graph f: R + R, f x = xlog e (x) is shown below. Find the derivative of x 2 log e (x). Use your answer to find the area of the shaded region in the form of alog e b + c.

29 Complex Areas Revision of Finding Areas How do we find areas under curves? One method is to approximate it by adding up lots of rectangles. The area of the first rectangle, for example, is A 1 = height width = f x 1 x The total approximate area is A = A 1 + A 2 + A A n If we make x smaller (width of rectangles), our approximate area becomes more accurate! What if we could make x infinitely small?

30 Complex Areas Revision of Finding Areas How do we find areas under curves? Adding up infinitely small rectangles? This is where the integral comes in. The special notation for a sum of an infinite amount of rectangles is: b න f x dx a This is: - The net area between f(x), the x axis from a to b. - This is now a definite integral!

31 Complex Areas Revision of Finding Areas Net Areas and Physical Areas A physical area does not include negative regions. So we need to make negative regions positive. Method 1: a b f x dx Method 2: b a f x dx

32 Complex Areas Revision of Finding Areas Areas Between Two Curves The area between two curves can be found by the area under the upper curve, and subtracting the area under the lower curve. Area between curves = Area under top curve - Area under bottom curve Area = a b ftop x f bottom x dx

33 Complex Areas Exam Style Questions Finding areas can become more difficult: - Sometimes you may need to subtract the area under a curve from a rectangle. - Sometimes you may need to use the inverse function if the graph cannot be integrated.

34 Complex Areas Written Example Consider the region bounded by the x-axis, the y-axis, the line with equation y=3 and the curve with equation y = log e (x 1). What is the exact value of the area of this region? CAS Calculator Allowed

35 Complex Areas Written Example Consider the region bounded by the x-axis, the y-axis, the line with equation y=3 and the curve with equation y = log e (x 1). What is the exact value of the area of this region? No CAS Calculator

36 Complex Areas Multiple Choice Question The graph of y = x 1 for x [1,5] is shown below. Which one of the following definite integrals could be used to find the area of the shaded region? a) 1 5 x 1 dx b) 0 2 x 1 dx c) 0 5 (2 x 1 )dx d) 0 2 (x 2 + 1) dx e) 0 2 x 2 dx Respond:

37 Complex Areas Multiple Choice Question To find the area of the shaded region in the diagram shown, four different students proposed the following calculations. i. 0 1 e 2x dx ii. iii. iv. e e 2x dx 1 e 2 e 2y dy 1 e 2 log e (x) 2 dx Which of the following is correct: a) ii. only b) ii. and iii. only c) i., ii., iii. and iv d) ii. and iv. only e) i. and iv. only Respond:

38 Next Topic! Revision of... Probability - Discrete Probability - Binomial Distribution - Normal Distribution

39 Probability Approaching Questions Getting started with probability questions: - Identify the type of distribution being used - Note all key values - Define the distribution mathematically

40 Probability Difficult Areas Exam style questions can often become more difficult when: - Not all information is given in a distribution - Questions take results from one distribution and use it in another distribution - Conditional probability

41 Discrete Distribution Discrete Distribution Rules There are two rules that must be satisfied for a distribution. 1) The sum of all probabilities must be 1. Pr X = x = 1 2) All probabilities must be in [0,1] 0 Pr X = x 1

42 Discrete Distribution Discrete Distribution Measures Mean: The sum of each discrete variable value (X) multiplied by its respective probability: E X = x Pr(X = x) Median: The value of X where 50% of the data lies above and below. Often in discrete probability, one specific X value will cover that range. Mode: The most frequently occurring X value. Variance: E X 2 E X 2 Standard Deviation: Var X

43 Discrete Distribution Multiple Choice Question The discrete random variable X has the following probability distribution. If the mean of X is 0.3, then a) a=0.1, b=0.3 b) a=0.2, b=0.4 c) a=0.4, b=0.2 d) a=0.1, b=0.5 e) a=0.7, b=0.3 X Pr(X = x) a b 0.4 Respond:

44 Discrete Distribution Written Example The discrete random variable X has the probability distribution: Find the value of k:

45 Binomial Distribution What is a Bernoulli Trial? - The trials are independent and identical. - Each trial has a success and failure. - The probability of success is always p and the failure is 1 p for each trial. To use the binomial distribution, your trials must be Bernoulli trials!

46 Binomial Distribution Binomial Distribution Calculations Binomial distribution deals with the probability of an event succeeding (p) or failing (q), over multiple repeated trials (n). X~Bi(n, p) The probability that an event with success probability p occurs x times out of n times is: Pr X = x = n C x p x q n x Pr X = x = n C x p x 1 p n x

47 Binomial Distribution Using your calculator Your calculator should have built in functions to perform binomial distribution calculations! On the Ti-Nspire: Calculating probability of a specific number of success: binompdf(n, p, x) Calculating probability of a range of success: binomcdf(n, p, x low, x up )

48 Binomial Distribution Measures of a binomial distribution Expected Value: E X = np Variance: Var X = npq Standard Deviation = Var(X) = npq

49 Binomial Distribution Written Example Seventy per cent of all scheduled trains through a particular station arrive on time. If 10 trains go through the station every day, find correct to 4 decimal places: The probability that at least 8 trains are on time for 9 out of the next 10 days.

50 Binomial Distribution Written Example Find the minimum number or rolls required for which the probability of obtaining at least one six on a die is greater than 0.95.

51 Binomial Distribution Multiple Choice Question X is a binomial random variable with parameters n and p. If Pr X = 0 = k, then Pr X = 1 is equal to: a) b) c) d) e) n 1 p k 1 p npk npk 1 p p n n 1 p 2 k 2 1 p 2 pk 1 p Respond:

52 Normal Distribution What is it? A normal distribution has a bell curve shape, where larger amounts of probability are concentrated in the centre.

53 Normal Distribution To characterize a normal distribution, we use: - Mean: where is the centre? - Standard Deviation: how spread out is our bell curve? For a normal distribution, the mean, median and mode are exactly the same, right in the centre due to its symmetrical shape. The notation we use is: X~N(μ, σ 2 )

54 Normal Distribution Calculations To find the exact probability within a region in a continuous distribution, use the normal distribution function on your calculator. On a TI-Nspire CAS, the function is: normcdf(x low, x up, μ, σ) On a CASIO Classpad, the function is: normcdf(x low, x up, μ, σ)

55 Normal Distribution % Rule μ 3σ μ 2σ μ σ μ μ + σ μ + 2σ μ + 3σ 68% 95% 99.7%

56 Normal Distribution Inverse Normal Used to find the value on a bell curve which corresponds to a specified amount of probability. Most inverse normal functions measure the probability from left to right. On a TI-Nspire CAS, the function is: a invnorm(area, μ, σ)

57 Normal Distribution Written Example The lifetime of a specific light globe is normally distributed with a mean of 400 and a standard deviation of 50 hours. 80% of these light globes last more than a hours. Find the value of a.

58 Normal Distribution Multiple Choice Question Butterflies of a particular species die T days after hatching, where T is normally distributed random variable with a mean of 120 days and a standard deviation of σ days. If, from a population of 2000 newly hatched butterflies, 150 are expected to die in the first 90 days, then the value of σ is closest to a. 7 days b. 13 days c. 17 days d. 21 days e. 37 days Respond:

59 Normal Distribution Multiple Choice Question In an orchard of 2000 apple trees it is found that 1735 trees have a height greater than 2.8 metres. The heights are distributed normally with a mean μ and a standard deviation of 0.2 metres. The value of μ is closest to: a b c d e Respond:

60 Normal Distribution Written Example Let the random variable X be normally distributed with mean 1.5 and standard deviation 0.4. Let Z be the standard normal variable, such that Z~N(0,1). Find a such that Pr X < a = Pr(Z > a 3 )

61 Next Topic! Revision of... Statistics - Key Terms - Proportions and Normality - Confidence Intervals

62 Statistics Key Terms Population: A collection of items in a set. The entire group as a whole to be investigated. Population Proportion: The proportion of items in a population which satisfy a success criteria. Sample: A subset of a larger population that can be collected and analysed to make inferences. Sample Proportion: The proportion of items in a sample which satisfy a success criteria.

63 Statistics Key Terms Statistical Inference Making judgements about the parameters of a population using the information gained from samples.

64 Statistics Sample Proportions

65 Statistics Sample Proportions Normality The mean and standard deviation of the sample proportion are: E P = p sd P = p(1 p) n

66 Statistics Confidence Intervals Motivation When we use a sample statistic to make predictions about a population parameter, there is uncertainty! This is because if we take a different sample, our statistics might change! We can use confidence intervals to capture how uncertain we are about a sample statistic.

67 Statistics Confidence Intervals Calculations The confidence interval for the population proportion is: ( pƹ z pƹ 1 pƹ n, p Ƹ + z pƹ 1 pƹ n ) What will change the confidence interval? How do we find z?

68 Statistics Confidence Intervals The Level of Confidence 95%

69 Statistics Confidence Intervals The Level of Error The level of error for a population proportion is: z pƹ 1 pƹ n The percentage travelled below and above the sample proportion (uncertainty) to create a particular confidence interval.

70 Statistics Multiple Choice Question An opinion pollster reported that for a random sample of 574 voters in a town, 76% indicated a preference for retaining the current council. An approximate 90% confidence interval for the proportion of the total voting population with a preference for retaining the current council can be found evaluating a) ( , ) 574 b) , c) , d) ( , ) e) ( , )

71 Statistics Written Example Confidence Intervals Fourteen of a random sample of 88 people said that they prefer to watch the news on a particular channel. Calculate a 95% confidence interval for the proportion of people in the population who prefer to watch the news on that channel.

72 Statistics Written Example Let us do our own investigation!

73 Questions Ask me anything!

74 Mathematical Methods (CAS) Unit 4 Revision Lecture Thanks for attending!