Year 11 mathematics extension 1 notes summary PDF

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Year 11 mathematics extension 1 notes summary.pdf

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Year 11 mathematics extension 1 ME-C1 Rates of change Outcomes A student: • uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 • applies understanding of the concept of a derivative in the solution of problems, including rates of change, exponential growth and decay and related rates of change ME11-4 • uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 • communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7

Glossary of terms Term

Description

direct variation

Two variables are in direct variation if one is a constant multiple of the other. This can be represented by the equation 𝑦 = 𝑘𝑥 , where 𝑘 is the constant of variation (or proportion). Also known as direct proportion, it produces a linear graph through the origin.

exponential growth and decay

Exponential growth occurs when the rate of change of a mathematical function is positive and proportional to the function’s current value. Exponential decay occurs in the same way when the growth rate is negative.

instantaneous rate of change

The instantaneous rate of change is the rate of change at a particular moment. For a differentiable function, the instantaneous rate of change at a point is the same as the gradient of the tangent to the curve at that point. This is defined to be the value of the derivative at that particular point.

rate of change

𝛥𝑦

A rate of change of a function , 𝑦 = 𝑓(𝑥) is 𝛥𝑥 where ∆𝑥 is

the change in 𝑥 and ∆𝑦 is the corresponding change in 𝑦.

sketch

A sketch is an approximate representation of a graph, including labelled axes, intercepts and any other important relevant features. Compared to the corresponding graph, a sketch should be recognisably similar but does not need to be precise.

Content Students learn to:

C1.1 Rates of change with respect to time • describe the rate of change of a physical quantity with respect to time as a derivative o use appropriate language to describe rates of change, for example ‘at rest’, ‘initially’, ‘change of direction’ and ‘increasing at an increasing rate’ • describe the rate of change of a physical quantity with respect to time as a derivative • investigate examples where the rate of change of some aspect of a given object with respect to time can be modelled using derivatives AAM 𝑑𝑄

• Find and interpret the derivative 𝑑𝑡 , given a function in the form 𝑄 = 𝑓(𝑡), for the amount of a physical quantity present at time 𝑡 • describe the rate of change with respect to time of the displacement of a particle moving along 𝑑𝑥 the 𝑥 -axis as a derivative 𝑑𝑡 or 𝑥󰇗 o Describe the rate of change with respect to time of the velocity of a particle moving along the 𝑥 -axis as a derivative

𝑑 2𝑥

𝑑𝑡 2

or 𝑥󰇘 .

C1.2: Exponential growth and decay construct, analyse and manipulate an exponential model of the form 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM • construct, analyse and manipulate an exponential model of the form 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM o establish the simple growth model,

𝑑𝑁 𝑑𝑡

= 𝑘𝑁

• where 𝑁 is the size of the physical quantity, 𝑁 = 𝑁(𝑡) at time 𝑡 and 𝑘 is the growth constant

• construct, analyse and manipulate an exponential model of the form 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM verify (by substitution) that the function 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 satisfies the relationship being the initial value of 𝑁

𝑑𝑁

𝑑𝑡

= 𝑘𝑁, with 𝐴

• construct, analyse and manipulate an exponential model of the form 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM o sketch the curve 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 for positive and negative values of 𝑘 recognise that this model states that the rate of change of a quantity varies directly with the size of the quantity at any instant solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems AAM 𝑑𝑁

• establish the modified exponential model, 𝑑𝑡 = 𝑘(𝑁 − 𝑃) for dealing with problems such as ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’ AAM 𝑑𝑁

• establish the modified exponential model, 𝑑𝑡 = 𝑘(𝑁 − 𝑃) for dealing with problems such as ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’ AAM

Content Students learn to:

verify (by substitution) that a solution to the differential equation 𝑘𝑡

𝑑𝑁 𝑑𝑡

= 𝑘(𝑁 − 𝑃) is 𝑁(𝑡) = 𝑃 +

𝐴𝑒 , for an arbitrary constant 𝐴, and 𝑃 a fixed quantity, and that the solution is 𝑁 = 𝑃 in the case when 𝐴 = 0 𝑑𝑁

• establish the modified exponential model, 𝑑𝑡 = 𝑘(𝑁 − 𝑃) for dealing with problems such as ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’ AAM o sketch the curve 𝑁(𝑡) = 𝑃 + 𝐴𝑒 𝑘𝑡 for positive and negative values of 𝑘

o note that whenever 𝑘 < 0 the quantity 𝑁 tends to the limit 𝑃 as 𝑡 → ∞ irrespective of the initial conditions recognise that this model states that the rate of change of a quantity varies directly with the difference in the size of the quantity and a fixed quantity at any instant solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems AAM

C1.3: Related rates of change • solve problems involving related rates of change as instances of the chain rule (ACMSM129) AAM develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied

ME-A1 Working with combinatorics Outcomes A student: • uses concepts of permutations and combinations to solve problems involving counting or ordering ME11-5 • uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 • communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7

Glossary of terms Term Arranging n objects in an ordered list

combination

factorial

Description The number of ways to arrange 𝒏 different objects in an ordered list is given by 𝒏(𝒏 − 𝟏)(𝒏 − 𝟐) × . . .× 𝟑 × 𝟐 × 𝟏 = 𝒏! A combination is a selection of 𝒓 distinct objects from 𝒏 distinct objects, where order is not important. The number 𝒏 of such combinations is denoted by 𝒏 𝑪𝒓 or ( ), and is given 𝒓 𝒏! by: 𝒓!(𝒏−𝒓)! The product of the first 𝑛 positive integers is called the factorial of 𝑛 and is denoted by 𝑛!. 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) × … × 3 × 2 × 1

By definition: 𝟎! = 𝟏 fundamental counting principle

permutation

The fundamental counting principle states that if one event has 𝒎 possible outcomes and a second independent event has 𝒏 possible outcomes, then there are a total of 𝒎 × 𝒏 possible outcomes for the two combined events. A permutation is an arrangement of 𝑟 distinct objects taken from 𝑛 distinct objects where order is important. The number of such permutations is denoted by 𝑛 𝑃𝑟 and is 𝑛! equal to: 𝑛 𝑃𝑟 = 𝑛(𝑛 − 1). . . (𝑛 − 𝑟 + 1) = (𝑛−𝑟)!

The number of permutations of 𝒏 objects is 𝒏!.

Content Students learn to: A1.1: Permutations and combinations • list and count the number of ways an event can occur • use the fundamental counting principle (also known as the multiplication principle) • use factorial notation to describe and determine the number of ways 𝑛 different items can be arranged in a line or a circle • use factorial notation to describe and determine the number of ways n different items can be arranged in a line or a circle • use factorial notation to describe and determine the number of ways n different items can be arranged in a line or a circle • solve problems involving cases where some items are not distinct (excluding arrangements in a circle) • understand and use permutations to solve problems (ACMSM001) • understand and use the notation 𝑛 𝑃𝑟 and the formula 𝑛 𝑃𝑟 =

𝑛!

(𝑛−𝑟 )!

• solve problems involving permutations and restrictions with or without repeated objects (ACMSM004) • understand and use combinations to solve problems (ACMSM007) 𝑛 • understand and use the notations ( ) and 𝑛 𝐶𝑟 and the formula 𝑛 𝐶𝑟 = 𝑟 (ACMMM045, ACMSM008)

𝑛!

𝑟!(𝑛−𝑟)!

• solve practical problems involving permutations and combinations, including those involving simple probability situations AAM • solve simple problems and prove results using the pigeonhole principle (ACMSM006)

o understand that if there are 𝑛 pigeonholes and 𝑛 + 1 pigeons to go into them, then at least one pigeonhole must hold 2 or more pigeons o generalise to: If 𝑛 pigeons are sitting in 𝑘 pigeonholes, where 𝑛 > 𝑘, then there is at 𝑛 least one pigeonhole with at least pigeons in it 𝑘

• prove the pigeonhole principle A1.2: The binomial expansion and Pascal’s triangle • expand (𝑥 + 𝑦)𝑛 for small positive integers 𝑛 (ACMMM046)

note the pattern formed by the coefficients of 𝑥 in the expansion of (1 + 𝑥)𝑛 and recognise links to Pascal’s triangle

• expand (𝑥 + 𝑦)𝑛 for small positive integers 𝑛 (ACMMM046) 𝑛 recognise the numbers ( ) (also denoted 𝑛 𝐶𝑟 ) as binomial coefficients (ACMMM047) 𝑟 Investigating the expansion of binomial products • Expand (𝑥 + 𝑦) 𝑛 for general n by considering the number of ways of obtaining a term of the form 𝑥 𝑛−𝑟 𝑦𝑟 Example • What is the coefficient of 𝑥 2 𝑦 in the expansion of (𝑥 + 𝑦)3 ? (𝑥 + 𝑦)3 = (𝑥 + 𝑦 )(𝑥 + 𝑦)(𝑥 + 𝑦) = 𝑥𝑥𝑥 + 𝑥𝑥𝑦 + 𝑥𝑦𝑥 + 𝑥𝑦𝑦 + 𝑦𝑥𝑥 + 𝑦𝑥𝑦 + 𝑦𝑦𝑥 + 𝑦𝑦𝑦

• From the 3 brackets we choose one y which can be done in of 𝑥 2 𝑦 is 𝟑 𝑪𝟏 .

𝟑

𝑪𝟏 ways. So the coefficient

Content Students learn to: • Generalising this to find the coefficient of 𝑥 𝑛−𝑟 𝑦𝑟 in the expansion of (𝑥 + 𝑦)𝑛 , we could select 𝑥 from the first 𝑛 − 𝑟 brackets and 𝑦 from the next 𝑟 brackets. Now the number of 𝑛!(𝑛−𝑟)! . Hence the coefficient of ways of arranging these (because some are the same) is 𝑟! 𝑥 𝑛−𝑟 𝑦𝑟 is

𝒏

𝑪𝒓.

• The binomial theorem

𝑛

𝑛 𝑛 𝑛 𝑛 (𝑎 + 𝑏) = 𝑎 + ( ) 𝑎 𝑛−1 𝑏 + ( ) 𝑎 2 𝑏 2 + ⋯ + ( ) 𝑎 𝑛−𝑟 𝑏𝑟 + ⋯ + 𝑏 𝑛 = ∑ ( ) 𝑎 𝑛−𝑘 𝑏𝑘 𝑟 2 1 𝑘 𝑛

𝑛

𝑘=0

Example • Expand (3 + 2𝑥)10 Applications of the binomial theorem • The general term in the expansion of (𝑥 + 𝑦)𝑛 can be written as 𝑛 𝐶𝑟 𝑥 𝑛−𝑘 𝑦𝑘

• Use a general term of the expansion of (𝑥 + 𝑦)𝑛 to find a specific term in the expansion

• derive and use simple identities associated with Pascal’s triangle (ACMSM009) o establish combinatorial proofs of the Pascal’s triangle relations 𝑛 𝐶0 = 1, 𝑛

𝑛

and

𝑛

𝐶𝑛 = 1; 𝐶𝑟 = 𝑛−1 𝐶𝑟−1 +

𝑛−1

for 1 ≤ 𝑟 ≤ 𝑛 − 1;

𝐶𝑟 = 𝑛 𝐶𝑛−𝑟

𝐶𝑟

ME-F1.1 Graphical relationships Outcomes A student: • uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 • manipulates algebraic expressions and graphical functions to solve problems ME11-2 • uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 • communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7

Glossary of terms Term

Description

composite function

In a composite function, the output of one function becomes the input of a second function. More formally, the composite of 𝑓 and 𝑔, acting on 𝑥, can be written as ((𝑥)), with (𝑥) being performed first.

reciprocal

1

The reciprocal of a quantity 𝑥 is define formally as 𝑥. The

product of a quantity and its reciprocal is equal to 1.

Informally, a reciprocal of a quantity, expressed as a fraction, can be generated by reversing its numerator and 2 5 denominator, eg) the reciprocal of 5 is 2. inequality

An inequality is a statement that compares two quantities and describes how they are different.

quadratic

Quadratic describes any function in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎, 𝑏 and 𝑐 are constants.

domain

function

The domain of a function is the set of 𝑥 values of 𝑦 = 𝑓(𝑥) for which the function is defined. Also known as the ‘input’ of a function. A function 𝑓 is a rule that associates each element 𝑥 in a set 𝑆 with a unique element (𝑥) from a set 𝑇.

The set 𝑆 is called the domain of 𝑓 and the set 𝑇 is called the co-domain of 𝑓. The subset of 𝑇 consisting of those elements of 𝑇 which occur as values of the function is called the range of 𝑓. The functions most commonly encountered in elementary mathematics are real functions of a real variable, for which both the domain and co-domain are subsets of the real numbers.

If we write 𝑦=(𝑥), then we say that 𝑥 is the independent variable and 𝑦 is the dependent variable. range (of function)

The range of a function is the set of values of the dependent variable for which the function is defined. A relation is a rule that associates elements 𝑥 in a set 𝑆 to elements in a set 𝑇 , where some element of 𝑥 maps to more than one element in 𝑇 .

relation

Informally, relations do not conform to the vertical line test for functions. causation

Causation describes a cause and effect relationship between two quantities, where the relationship is linear.

correlation

Correlation describes a linear relationship between two variables which is not necessarily causal (see causation above).

parameter

A parameter is a quantity that defines certain characteristics of a function or system. For example 𝜃 is a parameter in 𝑦 = 𝑥 cos 𝜃 A parameter can be a characteristic value of a situation. For example the time taken for a machine to produce a certain product.

polynomial

A polynomial is defined by 𝑃(𝑥) or 𝑄(𝑥). It is a type of function in the form 𝑃(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 . A polynomial is a series of powers of 𝑥 .

The degree of a polynomial, 𝑑𝑒𝑔 𝑃(𝑥), is defined by the highest power of 𝑥 , ie) 𝑑𝑒𝑔 𝑃(𝑥) = 𝑛 in the definition above.

Generally, polynomials are written in descending powers of 𝑥 . The first term written in this form is called the leading term and it is the term with the highest power of 𝑥 .

The leading coefficient is the coefficient of the leading term, ie) 𝑎𝑛 in the definition above.

Multiplicity of a root

Given a polynomial 𝑃(𝑥), if 𝑃(𝑥) = (𝑥 − 𝑎)𝑟 𝑄(𝑥) , 𝑄(𝑎) ≠ 0 and 𝑟 is a positive integer, then the root 𝑥 = 𝑎 has multiplicity 𝑟.

Content Students learn to:

• examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graph of 𝑦 =

1 𝑓(𝑥)

and hence

sketch the graphs (ACMSM099) • examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graphs of 𝑦 2 = 𝑓(𝑥) and 𝑦 = √𝑓(𝑥) and hence sketch the graphs

Content Students learn to: • examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graphs of 𝑦 = |𝑓(𝑥)| and 𝑦 = 𝑓(|𝑥|) and hence sketch the graphs (ACMSM099) • examine the relationship between the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) and the graphs of 𝑦 = 𝑓(𝑥) + 𝑔(𝑥) and 𝑦 = 𝑓(𝑥)𝑔(𝑥) and hence sketch the graphs

• apply knowledge of graphical relationships to solve problems in practical and abstract contexts AAM solve quadratic inequalities using both algebraic and graphical techniques solve inequalities involving rational expressions, including those with the unknown in the denominator solve absolute value inequalities of the form |𝑎𝑥 + 𝑏| ≥ 𝑘, |𝑎𝑥 + 𝑏| ≤ 𝑘 , |𝑎𝑥 + 𝑏| < 𝑘 and |𝑎𝑥 + 𝑏| > 𝑘 • define the inverse relation of a function 𝑦 = 𝑓(𝑥) to be the relation obtained by reversing all the ordered pairs of the function • examine and use the reflection property of the graph of a function and the graph of its inverse (ACMSM096) o understand why the graph of the inverse relation is obtained by reflecting the graph of the function in the line 𝑦 = 𝑥 o using the fact that this reflection exchanges horizontal and vertical lines, recognise that the horizontal line test can be used to determine whether the inverse relation of a function is again a function • write the rule or rules for the inverse relation by exchanging 𝑥 and 𝑦 in the function rules, including any restrictions, and solve for 𝑦, if possible • when the inverse relation is a function, use the notation 𝑓 −1 (𝑥) and identify the relationships between the domains and ranges of 𝑓(𝑥) and 𝑓 −1 (𝑥) when the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques AAM • understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form o understand that linear and quadratic functions, and circles can be expressed in either parametric form or Cartesian form o convert linear and quadratic functions, and circles from parametric form to Cartesian form and vice versa • understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form sketch linear and quadratic functions, and circles expressed in parametric form

Content Students learn to:

• define a general polynomial in one variable, 𝑥 , of degree 𝑛 with real coefficients to be the expression: 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 , where 𝑎𝑛 ≠ 0 understand and use terminology relating to polynomials including degree, leading term, leading coefficient and constant term • A polynomial is a function defined for all real 𝑥 involving positive powers of 𝑥 in the form 𝑃(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 where 𝑛 is a positive integer or zero

• 𝑃(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 has degree 𝑛 where 𝑎𝑛 ≠ 0 – degree means the highest power • 𝑎𝑛 , 𝑎𝑛−1 , … 𝑎2 , 𝑎1 , 𝑎0 are the coefficients

• 𝑎𝑛 𝑥 𝑛 is known as the leading term, 𝑎𝑛 is the leading coefficient and 𝑎0 is the constant term

• if 𝑎𝑛 = 1 then 𝑃(𝑥) is called a monic polynomial • if the coefficients all equal to 0. For example

= 𝑎𝑛−1 = ⋯ = 𝑎2 = 𝑎1 = 𝑎0 = 0 then 𝑃(𝑥) is the zero polynomial

• use division of polynomials to express 𝑃(𝑥) in the form 𝑃(𝑥) = 𝐴(𝑥). 𝑄(𝑥) + 𝑅(𝑥) where deg 𝑅(𝑥) < deg 𝐴(𝑥) and 𝐴(𝑥) is a linear or quadratic divisor, 𝑄(𝑥) the quotient and 𝑅(𝑥) the remainder o

review the process of division with remainders for integers

describe the process of division using the terms: dividend, divisor, quotient, remainder prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations (ACMSM089, ACMSM091) Remainder theorem • If a polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑎 then the remainder is 𝑃(𝑎). This can be demonstrated using the long division method as well as the functions method of substituting 𝑥 = 𝑎 into the polynomial Example Di...