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Stage 2 Mathematical Methods Program 2 This example program covers the following topics based on a four-lesson week format (two single lessons and one double lesson). There is additional time at the end of the program to allow for interruptions to occur or extra time spent on topics if required. Term 1 Term 2 Further Differentiation and Applications Logarithmic Functions Continuous Random Variables and the Normal Distribution Sampling and Confidence Intervals Discrete Random Variables Term 3 Integral Calculus Term 4 Exam Revision Many topics and subtopics specified in the program may be revisited or covered several times through the program. The topics and subtopics are not exclusive to those lessons or weeks as the placement of content and assessment tasks within this program is a guide only. Students demonstrate their learning through six Skills and Applications Tasks (50%) and one Mathematical Investigation (20%) for the school assessment, as specified in the Mathematical Methods subject outline for teaching in 2017. Week Single Lesson Double Lesson Single Lesson Term 1 Week 1 Limits and differentiation from first principles Slope of a tangent First principles π(π₯ + β) β π(π₯) π β² (π₯) = lim ββ0 β Derivative of polynomial functions and power functions Displacement and velocity Applications to modelling Local maxima and minima Subtopic 1.2: Differentiation Rules Derivative of ππ Simple rules of differentiation If π¦ = π₯ π 2 TOPIC 1: Further Differentiation and Applications Subtopic 1.1: Introductory Differential Calculus Rates of change Instantaneous rate of change Increasing and decreasing rates of change Derivatives of functions with negative and Composite Functions and the Chain Rule fractional indices Chain rule π π¦ = π(π₯)π Derivative of βπ(π₯) ππ¦ Review indices laws including = π β π(π₯)πβ1 β π β² (π₯) π ππ₯ π π π₯ π = βπ₯ Problems involving 1 βπ 1 π₯ = π π¦= = π(π₯)βπ π₯ π(π₯)π Then, ππ¦ ππ₯ = ππ₯ πβ1 The Product and Quotient Rule Product rule π¦ = π’π£ β² π¦ = π’β² π£ + π’π£β² Quotient rule π’ π¦= π£ π’β² π£ β π’π£β² π¦= π£2 Week Single Lesson Double Lesson Single Lesson Subtopic 1.5: The Second Derivative The Second Derivative If π(π₯) = π¦ ππ¦ π β² (π₯) = π¦ β² = ππ₯ π2 π¦ π β² β²(π₯) = π¦ β²β² = 2 ππ₯ The relationship between the function, its derivative, and the second derivative from graphical examples Curve Properties Increasing and decreasing functions Revision of intervals Increasing and decreasing intervals 3 The Product and Quotient Rule Continuation from previous lesson 4 Curve Properties Review Stationary points Maxima/Minima Inflection points Curve sketching including concavity and points of inflection SAT 1 Differential Calculus Test (No calculators or notes) 5 Motion in a Straight Line Displacement, acceleration, and velocity Where π (π‘) = displacement, π β² (π‘) = π£(π‘), and π "(π‘) = π(π‘) Motion in a Straight Line Average acceleration Instantaneous acceleration Speed (increasing / decreasing) Sign diagrams Motion diagrams Optimisation Examples of optimisation 6 Subtopic 1.3: Exponential Functions The derivative of π² = ππ± and y= ππ(π±) If π¦ = π π(π₯) then, π¦ β² = πβ²(π₯) β π π(π₯) Exponential, Surge and Logistic Modelling Growth and decay functions π¦ = ππ ππ‘ Surge functions π¦ = π΄π‘π βππ‘ Logistic functions πΆ π¦= 1 + π΄π βππ‘ Using exponential functions Slope of tangents to graphs of functions Local maxima and minima Increasing and decreasing functions Displacement and velocity Applications to model actual scenarios using exponential functions Week Single Lesson 7 TOPIC 4: Logarithmic Functions Subtopic 4.1: Using Logarithms for Solving Exponential Equations Solving Exponential Equations Given π π₯ = π then π₯ = ππππ π leading to when π¦ = π π₯ then π₯ = ππππ π¦ = lny Solving exponential equations and the revision of the log laws Double Lesson Single Lesson Subtopic 4.2: Logarithmic Functions and their Graphs Graphs Logarithmic scale The graph of π¦ = lnπ₯ and its properties The graph of functions in the form π¦ = πln(ππ₯ + π) The relationship between the graphs of π¦ = lnπ₯ and π¦ = ππ₯ Subtopic 4.3: Calculus of Logarithmic Functions The derivatives of π² = π₯π§π± and π² = π₯π§π(π±) If π¦ = ln(π(π₯)) πβ²(π₯) then, π¦ β² = π(π₯) 1 β« π₯ ππ₯ = lnπ₯ + π provided x is positive Problem solving using the derivatives of logarithmic functions Applications of logarithmic functions. 8 Applications of Exponential and Logarithmic Mathematical Investigation Functions Transportation Calculus Continuation of previous lessons Mathematical Investigation Transportation Calculus 9 SAT 2 Applications of Differential Calculus Test Mathematical Investigation Transportation Calculus Mathematical Investigation Transportation Calculus 10 Subtopic 1.4: Trigonometric Functions Review of Radians and the Unit Circle Since, π π = 180° 180° ππ 1π = and 1° = π 180 Symmetry and exact values Trigonometric equations and graphs General solutions of trigonometric equations Graphs of sine and cosine functions Dilation Reflection Translation Period Amplitude Trigonometric Modelling Using functions like π¦ = π sin[π(π₯ β π)] + π and y = π cos[π(π₯ β π)] + π to model periodic behaviour Derivatives of π¬π’π§(π) , ππ¨π¬(π) , πππ§(π) Applying the chain rule to trigonometric functions π¦ = cos π(π₯) and π¦ = sin π(π₯) Using the product and quotient rules with trigonometric functions Derivatives of functions such as: π¦ = π₯ sin(π₯) ππ₯ π(π₯) = cos(π₯) π 1 e.g. sin(30°) = sin ( 6 ) = 2 Unit circle quadrants (ASTC) Negative angles Complementary angles Special cases 11 Derivatives of π¬π’π§(π) , ππ¨π¬(π) , πππ§(π) If π¦ = sin(π₯), then π¦ β² = cos π‘ If π¦ = cos(π₯), then π¦ β² = βsin π‘ Use the product rule to find derivative of tan(π₯) sin(π₯) = tan(π₯) cos(π₯) Mathematical Investigation Submission Week Single Lesson Double Lesson Single Lesson Term 2 Week 1 Using derivatives of trigonometric functions Review Using trigonometric functions Slope of tangents to graphs of functions Local maxima and minima Increasing and decreasing functions Displacement and velocity Review SAT 3 Further Differentiation Test TOPIC 5: Continuous Random Variables and the Normal Distribution Subtopic 5.1: Continuous Random Variables Continuous Random Variables Comparing discrete and continuous random variables The probability of a specific range of values Probability density functions Probability density functions and their graphs 2 3 Modelling periodic scenarios such as tidal heights, temperature changes and AC voltages β The mean from πΈ(π₯) = β«ββ π₯π(π₯)ππ₯ and the β standard deviation π = ββ«ββ[π₯ β πΈ(π₯)]2 π(π₯)ππ₯ The standard normal distribution with π = 0 and π = Subtopic 5.3: Sampling 1 Distribution of Sample Means πβπ For a sample of a random variable πΜ π Standardising a normal distribution using π = π mean ππΜ = π (same as π) π standard deviation ππΜ = βπ Sampling Distributions ππ the outcome of adding n independent observations of X Μ πΜ Μ πΜ the outcome of averaging n independent observations of X If π~π(π, π) then ππ ~π(ππ, πβπ) and Μ πΜ Μ πΜ ~π (π, π ) βπ provided n is sufficiently large.* Subtopic 5.2: Normal Distributions The Normal Distribution Standard deviation Conditions for a normal random variable The key properties of normal distributions The probability density function 1 π₯βπ 2 1 β ( ) π(π₯) = π 2 π πβ2π Using electronic technology to calculate proportions, probabilities, and the upper or lower limit of a certain proportion Simple random sample πΜ If π~π(π, π) then πΜ ~π (π, π ) βπ for a sample size π Central limit theorem * using the notation π(π, π) for a normal distribution with mean π and standard deviation π Week Single Lesson 4 Sampling Continuation from previous lesson Double Lesson Single Lesson TOPIC 6: Sampling and Confidence Intervals Topic 6.1: Confidence Intervals for Population Mean Confidence Intervals for Means Sample means are continuous random variables Distribution of sample means will be approximately π normal for a sufficiently large sample. πΜ ~π (π, π) Confidence Intervals for Means e.g. 95% confidence intervals π π π₯Μ β 1.96 < π < π₯Μ + 1.96 βπ βπ Increasing sample size decreases interval width Finding the sample mean π₯Μ and standard deviation π based on a confidence interval β A confidence interval can be created around the sample mean that may contain the population mean. π If π₯Μ is the sample mean then the CI is π₯Μ β π§ π β€ π β π β€ π₯Μ + π§ , where π§ is determined by the βπ confidence level that the interval will contain the population mean. 5 Confidence Intervals for Means Using a confidence interval for a claim Assessing a claim about a population mean π Other applications of confidence intervals Confidence intervals other than 95% Confidence Intervals for Means Continuation from previous lesson Review 6 Review SAT 4 Normal Distribution Test TOPIC 2: Discrete Random Variables Subtopic 2.1: Discrete Random Variables Discrete Random Variables Discrete vs. continuous Probability distributions Uniform/non-uniform discrete random variables Expected value Standard deviation of a discrete random variable π = ββ[π₯ β πΈ(π)]2 π(π₯) Week Single Lesson Double Lesson Single Lesson 7 Discrete Random Variables Continuation from previous lesson Subtopic 2.2: The Bernoulli Distribution Subtopic 2.3: Repeated Bernoulli Trials and the Binomial Distribution The Bernoulli Distribution Bernoulli variables are discrete random variables The binomial random variable and the binomial distribution with only two outcomes; success or failure The mean ππ and the standard deviation βππ(1 β π) The Bernoulli distribution π(π₯) = ππ (1 β π)1βπ Modelling scenarios using the binomial distribution. The mean p and standard deviation βπ(1 β π). Finding binomial probabilities using Assigning Probabilities π(π = π) = πΆππ ππ ππβπ and electronic technology i.e. using < , β€ , > , β₯ , and = correctly in The shape of the binomial distribution for large probability functions values of π The Binomial Probability Density Function and the Binomial Cumulative Density Function binomcdf (π, π, π₯) and binompdf (π, π, π₯) Finding probabilities greater than using the binomcdf function 8 Subtopic 6.2: Population Proportions Population Proportions Concept of a population proportion π. Population Proportions Continuation from previous lesson Sample proportion πΜ = π π mean π and standard deviation β π(1βπ) . π As the sample size increases the distribution of πΜ becomes more like a normal distribution Subtopic 6.3: Confidence Intervals for Population Proportions Confidence Intervals for Proportions pΜ is the sample mean; the confidence interval is πΜ β π§β πΜ(1βπΜ) π πΜ(1βπΜ) π β€ π β€ πΜ + π§β , where π§ is determined by the confidence that the interval will contain the population mean. e.g. 95% confidence intervals for proportions πΜ β1 β πΜ πΜ β1 β πΜ πΜ β 1.96β < π < πΜ + 1.96β π π 9 Confidence Intervals for Proportions Continuation from previous lesson Assessing Claims with Confidence Intervals Assessing Claims with Confidence Intervals What does it mean if p falls within/outside the Continuation from previous lesson confidence interval? How does the size of the sample affect the confidence interval width? Week Single Lesson 10 Term 3 Week 1 Double Lesson Single Lesson Review SAT 5 Discrete Random Variables and Proportions Test TOPIC 3: Integral Calculus Subtopic 3.1: Anti-differentiation Anti-Differentiation Changing a derivative to the original function The antiderivative of π(π₯) is πΉ(π₯) The indefinite integral πΉ β² (π₯) = π(π₯) ; and β« π(π₯)ππ₯ = πΉ(π₯) + π Integrals of π₯ π , π π₯ , π πππ₯ πππ πππ π₯ including consideration of functions of the form π(ππ₯ + π) These points are covered in more detail in the program in the weeks below. Anti-Differentiation Revision and continuation from previous lesson Using β«[π(π₯) + π(π₯)]ππ₯ = β« π(π₯)ππ₯ + β« π(π₯)ππ₯ Determining the specific constant of integration. Subtopic 3.2: The Area under Curves Definite Integrals Estimating with lower and upper sum Strategies to improve the estimate. Use of electronic technology to find the area. Properties of definite integrals Definite Integrals π Using the terminology β«π π(π₯)ππ₯ for the exact area π for a positive continuous function, ββ«π π(π₯)ππ₯ for a π negative continuous function, and β«π [π(π₯) β π(π₯)]ππ₯ for the area between two curves where π(π₯) is above π(π₯) The observations π β«π π(π₯)ππ₯ = 0 and π π π β«π π(π₯)ππ₯ + β«π π(π₯)ππ₯ = β«π π(π₯)ππ₯ 2 Subtopic 3.3: Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Know that π Evaluating the exact area under a curve and the area between two curves. β« π(π₯)ππ₯ = πΉ(π) β πΉ(π) π and, π π π β« π(π₯)ππ₯ + β« π(π₯)ππ₯ = β« π(π₯)ππ₯ π π π π π β« π(π₯)ππ₯ = β β« π(π₯)ππ₯ π π π π π β« [π(π₯) ± π(π₯)] ππ₯ = β« π(π₯)ππ₯ ± β« π(π₯)ππ₯ π π The Fundamental Theorem of Calculus Continuation from previous lesson π The Indefinite Integral " + π" Discovering integrals by differentiation Week Single Lesson 3 Integration Simple Integrals If π(π₯) = π (constant), If π(π₯) = π₯ π , 1 If π(π₯) = , π₯ If π(π₯) = π π₯ , 4 Double Lesson Single Lesson Integration Continuation from previous lesson Integration Integrating π ππ₯+π and (ππ₯ + π)π 1 β« π ππ₯+π ππ₯ = π ππ₯+π + π π 1 (ππ₯ + π)π+1 β«(ππ₯ + π)π ππ₯ = +π π π+1 1 1 β« ππ₯ = ln|ππ₯ + π| + π ππ₯ + π π U-Substitution Continuation of previous lesson πΉ(π₯) = ππ₯ + π π₯ π+1 πΉ(π₯) = +π π+1 πΉ(π₯) = ln|π₯| + π πΉ(π₯) = π π₯ + π Integration Continuation of previous lesson U-Substitution ππ’ Integrals in the form of β« π(π’) ππ₯ ππ₯ ππ’ ππ₯ = β« π(π’) ππ’ ππ₯ Integration of π¬π’π§(π) and ππ¨π¬(π) Mixed problems with sin(π₯) and cos(π₯) β« π(π’) 5 Integration of π¬π’π§(π) and ππ¨π¬(π) β« sin(π₯) ππ₯ = β cos(π₯) + π Linear Motion (Integration) Total distance travelled vs. Displacement The Absolute Value β« cos(π₯) ππ₯ = sin(π₯) + π 6 Linear Motion (Integration) Continuation from the previous lesson Definite Integrals π Where β« π(π₯) ππ₯ = πΉ(π) β πΉ(π) Definite Integrals Continuation of previous lesson π πΉ(π) β πΉ(π) can be written as [πΉ(π₯)]ππ 7 Subtopic 3.4: Applications of Integration Finding Areas Area of region bounded by π₯-axis and π₯ value Area between two functions The area of cross-sections 8 Review Applications of Definite Integration Further Applications The total change in a quantity given the rate of For example: change over a time period. ο· the rate at which people enter a sports Distance from velocity-time venue ο· water flow during a storm ο· traffic flow ο· electricity consumption Review SAT 6 Integral Calculus Test 9/10 Exam Revision / Time for adjustment to program due to interruptions, excursions, etc. Term 4 Revision / SWOT VAC / Examination