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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
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑓 ′ (𝑥) = lim
ℎ→0
ℎ
Derivative of polynomial functions and power
functions
Displacement and velocity
Applications to modelling
Local maxima and minima
Subtopic 1.2: Differentiation Rules
Derivative of 𝒙𝒏
Simple rules of differentiation
If 𝑦 = 𝑥 𝑛
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
𝑛
𝑦 = 𝑓(𝑥)𝑛
Derivative of √𝑓(𝑥)
𝑑𝑦
Review indices laws including
= 𝑛 ∙ 𝑓(𝑥)𝑛−1 ∙ 𝑓 ′ (𝑥)
𝑚
𝑑𝑥
𝑛
𝑚
𝑥 𝑛 = √𝑥
Problems involving
1
−𝑛
1
𝑥 = 𝑛
𝑦=
= 𝑓(𝑥)−𝑛
𝑥
𝑓(𝑥)𝑛
Then,
𝑑𝑦
𝑑𝑥
= 𝑛𝑥 𝑛−1
The Product and Quotient Rule
Product rule
𝑦 = 𝑢𝑣
′
𝑦 = 𝑢′ 𝑣 + 𝑢𝑣′
Quotient rule
𝑢
𝑦=
𝑣
𝑢′ 𝑣 − 𝑢𝑣′
𝑦=
𝑣2
Week Single Lesson
Double Lesson
Single Lesson
Subtopic 1.5: The Second Derivative
The Second Derivative
If 𝑓(𝑥) = 𝑦
𝑑𝑦
𝑓 ′ (𝑥) = 𝑦 ′ =
𝑑𝑥
𝑑2 𝑦
𝑓 ′ ′(𝑥) = 𝑦 ′′ = 2
𝑑𝑥
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 𝑠(𝑡) = displacement,
𝑠 ′ (𝑡) = 𝑣(𝑡),
and 𝑠"(𝑡) = 𝑎(𝑡)
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 𝐲 = 𝐞𝐱 and y= 𝐞𝐟(𝐱)
If 𝑦 = 𝑒 𝑓(𝑥)
then, 𝑦 ′ = 𝑓′(𝑥) ∙ 𝑒 𝑓(𝑥)
Exponential, Surge and Logistic Modelling
Growth and decay functions
𝑦 = 𝑎𝑒 𝑏𝑡
Surge functions
𝑦 = 𝐴𝑡𝑒 −𝑏𝑡
Logistic functions
𝐶
𝑦=
1 + 𝐴𝑒 −𝑏𝑡
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 𝑎 𝑥 = 𝑏 then 𝑥 = 𝑙𝑜𝑔𝑎 𝑏 leading to
when 𝑦 = 𝑒 𝑥 then 𝑥 = 𝑙𝑜𝑔𝑒 𝑦 = 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 𝑦 = ln𝑥 and its properties
The graph of functions in the form 𝑦 = 𝑘ln(𝑏𝑥 + 𝑐)
The relationship between the graphs of 𝑦 = ln𝑥 and
𝑦 = 𝑒𝑥
Subtopic 4.3: Calculus of Logarithmic Functions
The derivatives of 𝐲 = 𝐥𝐧𝐱 and 𝐲 = 𝐥𝐧𝐟(𝐱)
If 𝑦 = ln(𝑓(𝑥))
𝑓′(𝑥)
then, 𝑦 ′ =
𝑓(𝑥)
1
∫ 𝑥 𝑑𝑥 = ln𝑥 + 𝑐 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 𝑐 = 180°
180°
𝜋𝑐
1𝑐 =
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
𝑦 = 𝑎 sin[𝑛(𝑥 − 𝑏)] + 𝑐
and y = 𝑎 cos[𝑛(𝑥 − 𝑏)] + 𝑐
to model periodic behaviour
Derivatives of 𝐬𝐢𝐧(𝒙) , 𝐜𝐨𝐬(𝒙) , 𝐭𝐚𝐧(𝒙)
Applying the chain rule to trigonometric functions
𝑦 = cos 𝑓(𝑥) and 𝑦 = sin 𝑓(𝑥)
Using the product and quotient rules with
trigonometric functions
Derivatives of functions such as:
𝑦 = 𝑥 sin(𝑥)
𝑒𝑥
𝑓(𝑥) =
cos(𝑥)
𝜋
1
e.g. sin(30°) = sin ( 6 ) = 2
Unit circle quadrants (ASTC)
Negative angles
Complementary angles
Special cases
11
Derivatives of 𝐬𝐢𝐧(𝒙) , 𝐜𝐨𝐬(𝒙) , 𝐭𝐚𝐧(𝒙)
If 𝑦 = sin(𝑥), then 𝑦 ′ = cos 𝑡
If 𝑦 = cos(𝑥), then 𝑦 ′ = −sin 𝑡
Use the product rule to find derivative of tan(𝑥)
sin(𝑥)
= tan(𝑥)
cos(𝑥)
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
∞
The mean from 𝐸(𝑥) = ∫−∞ 𝑥𝑓(𝑥)𝑑𝑥 and the
∞
standard deviation 𝜎 = √∫−∞[𝑥 − 𝐸(𝑥)]2 𝑓(𝑥)𝑑𝑥
The standard normal distribution with 𝜇 = 0 and 𝜎 = Subtopic 5.3: Sampling
1
Distribution of Sample Means
𝑋−𝜇
For a sample of a random variable 𝑋̅𝑛
Standardising a normal distribution using 𝑍 = 𝜎
mean 𝜇𝑋̅ = 𝜇 (same as 𝑋)
𝜎
standard deviation 𝜎𝑋̅ =
√𝑛
Sampling Distributions
𝑆𝑛 the outcome of adding n independent observations
of X
̅𝑋̅̅𝑛̅ the outcome of averaging n independent
observations of X
If 𝑋~𝑁(𝜇, 𝜎) then 𝑆𝑛 ~𝑁(𝑛𝜇, 𝜎√𝑛) and
̅𝑋̅̅𝑛̅~𝑁 (𝜇,
𝜎
)
√𝑛
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 𝑥−𝜇 2
1
− (
)
𝑓(𝑥) =
𝑒 2 𝜎
𝜎√2𝜋
Using electronic technology to calculate proportions,
probabilities, and the upper or lower limit of a certain
proportion
Simple random sample 𝑋̅
If 𝑋~𝑁(𝜇, 𝜎) then 𝑋̅~𝑁 (𝜇,
𝜎
)
√𝑛
for a sample size 𝑛
Central limit theorem
* using the notation 𝑁(𝜇, 𝜎) for a normal distribution with mean
𝜇 and standard deviation 𝜎
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
𝜎
normal for a sufficiently large sample. 𝑋̅~𝑁 (𝜇, 𝑛)
Confidence Intervals for Means
e.g. 95% confidence intervals
𝜎
𝜎
𝑥̅ − 1.96
< 𝜇 < 𝑥̅ + 1.96
√𝑛
√𝑛
Increasing sample size decreases interval width
Finding the sample mean 𝑥̅ and standard deviation 𝜎
based on a confidence interval
√
A confidence interval can be created around the
sample mean that may contain the population mean.
𝑠
If 𝑥̅ is the sample mean then the CI is 𝑥̅ − 𝑧 𝑛 ≤
𝑠
√
𝜇 ≤ 𝑥̅ + 𝑧
, where 𝑧 is determined by the
√𝑛
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 𝜇
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
𝜎 = √∑[𝑥 − 𝐸(𝑋)]2 𝑝(𝑥)
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 𝑛𝑝 and the standard deviation √𝑛𝑝(1 − 𝑝)
The Bernoulli distribution 𝑝(𝑥) = 𝑝𝑛 (1 − 𝑝)1−𝑛
Modelling scenarios using the binomial distribution.
The mean p and standard deviation √𝑝(1 − 𝑝).
Finding binomial probabilities using
Assigning Probabilities
𝑝(𝑋 = 𝑘) = 𝐶𝑘𝑛 𝑝𝑘 𝑝𝑛−𝑘 and electronic technology
i.e. using < , ≤ , > , ≥ , and = correctly in The shape of the binomial distribution for large
probability functions
values of 𝑛
The Binomial Probability Density Function and
the Binomial Cumulative Density Function
binomcdf (𝑛, 𝑝, 𝑥) and binompdf (𝑛, 𝑝, 𝑥)
Finding probabilities greater than using the binomcdf
function
8
Subtopic 6.2: Population Proportions
Population Proportions
Concept of a population proportion 𝑝.
Population Proportions
Continuation from previous lesson
Sample proportion 𝑝̂ =
𝑋
𝑛
mean 𝑝 and standard deviation √
𝑝(1−𝑝)
.
𝑛
As the sample size increases the distribution of 𝑝̂
becomes more like a normal distribution
Subtopic 6.3: Confidence Intervals for Population
Proportions
Confidence Intervals for Proportions
p̂ is the sample mean; the confidence interval is
𝑝̂ − 𝑧√
𝑝̂(1−𝑝̂)
𝑛
𝑝̂(1−𝑝̂)
𝑛
≤ 𝑝 ≤ 𝑝̂ + 𝑧√
,
where 𝑧 is
determined by the confidence that the interval will
contain the population mean.
e.g. 95% confidence intervals for proportions
𝑝̂ √1 − 𝑝̂
𝑝̂ √1 − 𝑝̂
𝑝̂ − 1.96√
< 𝑝 < 𝑝̂ + 1.96√
𝑛
𝑛
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 𝑓(𝑥) is 𝐹(𝑥)
The indefinite integral 𝐹 ′ (𝑥) = 𝑓(𝑥) ; and
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐
Integrals of 𝑥 𝑛 , 𝑒 𝑥 , 𝑠𝑖𝑛𝑥 𝑎𝑛𝑑 𝑐𝑜𝑠𝑥 including
consideration of functions of the form 𝑓(𝑎𝑥 + 𝑏)
These points are covered in more detail in the
program in the weeks below.
Anti-Differentiation
Revision and continuation from previous lesson
Using ∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
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
𝑏
Using the terminology ∫𝑎 𝑓(𝑥)𝑑𝑥 for the exact area
𝑏
for a positive continuous function, −∫𝑎 𝑓(𝑥)𝑑𝑥 for a
𝑏
negative continuous function, and ∫𝑎 [𝑓(𝑥) −
𝑔(𝑥)]𝑑𝑥 for the area between two curves where 𝑓(𝑥)
is above 𝑔(𝑥)
The observations
𝑎
∫𝑎 𝑓(𝑥)𝑑𝑥 = 0 and
𝑏
𝑐
𝑐
∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑏 𝑓(𝑥)𝑑𝑥 = ∫𝑎 𝑓(𝑥)𝑑𝑥
2
Subtopic 3.3: Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus
Know that
𝑏
Evaluating the exact area under a curve and the area
between two curves.
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑎
and,
𝑏
𝑐
𝑐
∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥
𝑎
𝑏
𝑎
𝑎
𝑏
∫ 𝑓(𝑥)𝑑𝑥 = − ∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑏
𝑎
𝑏
𝑏
∫ [𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥
𝑎
𝑎
The Fundamental Theorem of Calculus
Continuation from previous lesson
𝑎
The Indefinite Integral
" + 𝑐"
Discovering integrals by differentiation
Week Single Lesson
3
Integration
Simple Integrals
If 𝑓(𝑥) = 𝑘 (constant),
If 𝑓(𝑥) = 𝑥 𝑛 ,
1
If 𝑓(𝑥) = ,
𝑥
If 𝑓(𝑥) = 𝑒 𝑥 ,
4
Double Lesson
Single Lesson
Integration
Continuation from previous lesson
Integration
Integrating 𝑒 𝑎𝑥+𝑏 and (𝑎𝑥 + 𝑏)𝑛
1
∫ 𝑒 𝑎𝑥+𝑏 𝑑𝑥 = 𝑒 𝑎𝑥+𝑏 + 𝑐
𝑎
1 (𝑎𝑥 + 𝑏)𝑛+1
∫(𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 =
+𝑐
𝑎
𝑛+1
1
1
∫
𝑑𝑥 = ln|𝑎𝑥 + 𝑏| + 𝑐
𝑎𝑥 + 𝑏
𝑎
U-Substitution
Continuation of previous lesson
𝐹(𝑥) = 𝑘𝑥 + 𝑐
𝑥 𝑛+1
𝐹(𝑥) =
+𝑐
𝑛+1
𝐹(𝑥) = ln|𝑥| + 𝑐
𝐹(𝑥) = 𝑒 𝑥 + 𝑐
Integration
Continuation of previous lesson
U-Substitution
𝑑𝑢
Integrals in the form of ∫ 𝑓(𝑢) 𝑑𝑥 𝑑𝑥
𝑑𝑢
𝑑𝑥 = ∫ 𝑓(𝑢) 𝑑𝑢
𝑑𝑥
Integration of 𝐬𝐢𝐧(𝒙) and 𝐜𝐨𝐬(𝒙)
Mixed problems with sin(𝑥) and cos(𝑥)
∫ 𝑓(𝑢)
5
Integration of 𝐬𝐢𝐧(𝒙) and 𝐜𝐨𝐬(𝒙)
∫ sin(𝑥) 𝑑𝑥 = − cos(𝑥) + 𝑐
Linear Motion (Integration)
Total distance travelled vs. Displacement
The Absolute Value
∫ cos(𝑥) 𝑑𝑥 = sin(𝑥) + 𝑐
6
Linear Motion (Integration)
Continuation from the previous lesson
Definite Integrals
𝑏
Where ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
Definite Integrals
Continuation of previous lesson
𝑎
𝐹(𝑏) − 𝐹(𝑎) can be written as [𝐹(𝑥)]𝑏𝑎
7
Subtopic 3.4: Applications of Integration
Finding Areas
Area of region bounded by 𝑥-axis and 𝑥 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.
 the rate at which people enter a sports
Distance from velocity-time
venue
 water flow during a storm
 traffic flow
 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