Download AH Glasgow Order of Topics

N7
Partial Fractions
MAC 1.1
Unit standard and added value
Applying algebraic skills to partial fractions
 Expressing proper rational functions as a sum of partial fractions (denominator of degree at
most 3 and easily factorised)
 Express a proper rational function as a sum of partial fractions where the
denominator is of the type
a)
b)
c)

7𝑥+1
𝑥 2 +𝑥−6
5𝑥 2 −𝑥+6
𝑥 3 +3𝑥
3𝑥+10
𝑥 2 +6𝑥+9
(linear factors)
(irreducible quadratic factor)
(repeated factor)
Reduce an improper rational function to a polynomial and a proper rational function
by division or otherwise e.g.
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Nelson 1
Chapter 1.2
𝑥 3 +2𝑥 2 −2𝑥+2
(𝑥−1)(𝑥+3)
N7
Solving a quadratic with Complex roots
This skill will be needed before teaching Differential Equations. One period only.
N7
Differentiation
MAC 1.2
Unit standard and added value
Applying calculus skills through techniques of differentiation
 Differentiating functions using the chain rule
 Differentiating functions using the product rule
 Differentiating functions using the quotient rule
 Differentiate functions which require more than one application of the above
𝑑𝑦
1

Know that 𝑑𝑥 =

Use logarithmic differentiation for
 extended products and quotients
 indices involving the variable
Differentiating inverse trigonometric functions
Finding the derivative of functions defined implicitly
 Use implicit differentiation to find the first derivative
 Apply differentiation to related rates of change in problems where the functional
relationship is given implicitly
 Apply implicit differentiation to find the second derivative
Finding the derivative of functions defined parametrically
 Use parametric differentiation to find the first derivative
 Apply parametric differentiation to motion on a plane
 Use parametric differentiation to find the second derivative
 Apply differentiation to related rates in problems where the functional relationships is
given explicitly
 Solve related rates by first establishing a functional relationship between appropriate
variables



𝑑𝑥
𝑑𝑦
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Nelson 1
Nelson 2
Chapter 2.1
Chapter 2.1
Chapter 2.2 Ex2A,2B
N7
Integration
MAC 1.3
Unit standard and added value
Applying calculus skills through techniques of integration
 Integrating expressions using standard results

Integrate simple functions on sight e.g. ∫ 𝑥𝑒 𝑥 𝑑𝑥 , ∫ 𝑓(3𝑥 + 2)𝑑𝑥

Know the integrals of ∫ √1−𝑥 2 𝑑𝑥, ∫ 1+𝑥 2 𝑑𝑥

Recognise and integrate expressions of the form ∫


1
1
𝑓 ′ (𝑥)
𝑓(𝑥)
𝑑𝑥
Integrate when the substitution
 Integrate when the substitution is given


2

1
Use the substitution 𝑥 = 𝑎𝑡 to integrate functions of the form of ∫ √𝑎2
−𝑥 2
1
𝑑𝑥, ∫ 𝑎2 +𝑥 2 𝑑𝑥
Integrating proper rational functions
 Use partial fractions to integrate proper rational functions, where the denominator
has two separate or repeated linear factors.
 Use partial fractions to integrate proper and improper rational functions where the
denominator may have:
a) two separate or repeated linear factors
b) three linear factors with constant or non-constant numerator
c) a linear factor and an irreducible quadratic factor of the form x2+a
Integrating by parts
 Use one application of integration by parts
 Involve repeated applications of integration by parts
Apply integration to the evaluation of areas including integration with respect to y
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Nelson 1
Nelson 2
Chapter 3
Ex1A, Ex1B, Ex2A, Ex2B, Ex3, Ex4A, Ex4B, Ex5, Ex5B, Ex6A, Ex6B,
Ex10A, Ex10B (be selective looking for area questions only)
Chapter 3
Ex1A, Ex1B, Ex2, Ex3A, Ex3B, Ex4, Ex5A, Ex5B
N7
Differential Equation
MAC 1.4
Unit standard and added value
Applying calculus skills to solving differential equations
 Solving first order differential equations with variables separable


Solve equations that can be written in the form
𝑑𝑦
𝑑𝑥
𝑔(𝑥)
= ℎ(𝑦)
Solving first order linear differential equations using the integrating factor

𝑑𝑦
Solve equations by writing linear equations 𝑎(𝑥) 𝑑𝑥 + 𝑏(𝑥)𝑦 = 𝑔(𝑥) etc


Find general solutions and solve initial value problems eg:
a) mixing problems, such as salt water entering a tank of clear water which is
then draining at a given rate
b) growth and decay problems, an alternative method of solution to separation
of variables
c) simple electronic circuits
Solving second order differential equations
 Find the general solution of a second order homogeneous ordinary differential
𝑑2 𝑦
𝑑𝑦
equation 𝑎 𝑑𝑥 2 + 𝑏 𝑑𝑥 + 𝑐 = 0 with constant coefficients where the roots of the


auxiliary equation
a) are real and distinct
b) coincide (are equal)
c) are complex conjugates
Solve initial value problems for second order homogeneous ordinary differential
equations with constant coefficients
Solve second order non-homogeneous ordinary equations with constant coefficients
𝑑2 𝑦
𝑑𝑦
𝑎 𝑑𝑥 2 + 𝑏 𝑑𝑥 + 𝑐 = 𝑓(𝑥) using the auxiliary equation and particular integral method.
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Nelson 2
Nelson 3
Chapter 3
Chapter 4
Ex6, Ex7, Ex8,Ex9A, Ex9B
N7
Pascal’s Triangle
Expanding brackets using Pascal’s Triangle is a useful skill. One period only.
N7
Binomial Theorem and Complex Numbers
AAC 1.1
Unit standard and added value
Applying algebraic skills to the binomial theorem and to complex numbers.
 Expanding expressions
 Know and use the binomial theorem
𝑛
(𝑎 + 𝑏)𝑛 = ∑𝑛𝑟=0 ( ) 𝑎𝑛−𝑟 𝑏 𝑟 for 𝑟, 𝑛 ∈ ℕ
𝑟
4
 Expand (x+3)
 Find the term x5 in (x+3)7
 Expand (2u-3v)5



2 9
Find the term in x7 in (𝑥 + 𝑥)
Performing operations on complex numbers
 Perform algebraic operations on complex numbers:
a) equality (equating real and imaginary parts),
b) addition,
c) subtraction,
d) multiplication
e) division.
Evaluating the modulus and argument
 Evaluate the modulus, argument and conjugate of complex numbers.
 Find the roots of a quartic when one complex root is given.
 Solve 𝑧 + 𝑖 = 2𝑧 + 1
 Solve 𝑧 2 = 2𝑧
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Introduce Pascal’s Triangle
 Expand (1+x)n

1 4
More complicated expansions e.g. (3x2-1)6, (𝑥 + 𝑥)
Introduce the factorial function
 Define n! and 0!
𝑛
Define nCr=( ) (explain r combinations from n)
𝑟
 Show quick method for evaluating coefficients
𝑛
𝑛
𝑛
𝑛
𝑛+1
 Show ( ) = (
) and (
)+( )=(
) prove if time
𝑛−𝑟
𝑟
𝑟−1
𝑟
𝑟
Relate binomial coefficients to pascals triangle
𝑛 𝑛−𝑟 𝑟
𝑛
𝑛
𝑛
 Develop (a+b)n=( )an+( )an-1b1+...+( )b8 and (𝑎 + 𝑏)𝑛 = ∑𝑛
(
)𝑎 𝑏
𝑟=0
0
𝑛
1
𝑟
 Find individual terms using general term
Expand (1+2x-3x2)5 up to x3, use rainbows for (1+2x-3x2)5(1+2x)6 , find (0.97)6 correct to 3dp
Detail
Nelson 1
Nelson 2
Chapter 1.1
Chapter 4
Ex1, Ex2, Ex3, Ex8
N7
Sequences and Series
AAC 1.2
Unit standard and added value
Applying algebraic skills to sequences and series.
 Finding the general term and summing arithmetic and geometric sequences
 Apply the rules on sequences and series to find the n th term, sum to n terms (partial
sum), limit, sum to infinity (limit to infinity of the sequence of partial sums), common
difference, and common ratio of arithmetic and geometric sequences.
 Using the Maclaurin series expansion to find a stated number of terms of the power series
for a simple function.
 Expand power series:
a) 𝑒 2𝑥
b) 𝑒 𝑠𝑖𝑛𝑥
c) 𝑒 𝑥 𝑐𝑜𝑠3𝑥
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Arithmetic Series
 Find nth term
 Find sum to n terms (partial sum)
Geometric Series
 Find nth term
 Find sum to n terms
 Find sum to infinity
Maclaurin Series: Use the Maclaurin series expansion to find a stated number of terms of the
power series for a simple function
 Expand power series
𝑓(0)
𝑓 ′ (0)
𝑓 ′′ (0)
𝑓 ′′′ (0)

Show 𝑓(𝑥) =



Use to find ex, e2x, esinx, ln(1+x), sinx, cosx, tan-1x
Combine expansions
Show how to expand e.g. e-2xsin3x, excos3x, ln(cosx) etc
0!
+
1!
𝑥+
2!
𝑥2 +
Detail
Nelson 2
Nelson 3
Chapter 5
Chapter 3
Ex3,Ex4,Ex6
3!
𝑥3 + ⋯
N7
Summation and Mathematical Proof
Unit standard and added value
Applying algebraic skills to summation and mathematical proof.
 Applying summation formulae
 ∑𝑛𝑟=1(𝑎𝑟 + 𝑏) = 𝑎 ∑𝑛𝑟=1 𝑟 + ∑𝑛𝑟=1 𝑏

1

∑𝑛𝑟=1 𝑟(𝑟 + 1) = 𝑛(𝑛 + 1)(𝑛 + 2)
3

∑𝑛𝑟=1 𝑟(𝑟 + 1)(𝑟 + 2) = 𝑛(𝑛 + 1)(𝑛 + 2)(𝑛 + 3)
4
1
 ∑𝑛𝑟=1 𝑟(𝑟 2 + 2)
Using proof by induction
 Use proof by mathematical induction in simple examples
a) 1+2+22+...+2n=2n+1-1, for all nϵN
b) 8n is a factor of (4n)! for all nϵN
c) n<2n for all nϵN
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
Reintroduce ∑
notation to show ∑𝑛𝑘=1 𝑘 = 1 + 2 + 2 + ⋯ + 𝑛 =


𝑛(𝑛+1)
2
Prove sums of certain series and other straightforward results:
a) ∑𝑛𝑟=1(𝑎𝑟 + 𝑏) = 𝑎 ∑𝑛𝑟=1 𝑟 + ∑𝑛𝑟=1 𝑏
b) Show how to find ∑ 𝑘 + 2, ∑(4𝑘 − 1) etc
Use proof by induction
 1+2+22+...+2n=2n+1-1, for all nϵN
 8n is a factor of (4n)! for all nϵN
 n<2n for all nϵN

1
∑𝑛𝑟=1 𝑟 2 = 𝑛(𝑛 + 1)(2𝑛 + 1)
6
Detail
Nelson 2
Nelson 3
Chapter 5
Chapter 5
Ex8
Ex3A, Ex3B
AAC 1.3
N7
Properties of Functions
AAC 1.4
Unit standard and added value
Applying algebraic and calculus skills to properties of functions.
 Finding the asymptotes of rational functions
 Find the vertical asymptote of a rational function
 Find the non-vertical asymptote of a rational function
 Sketching the graph of a rational function including appropriate analysis of stationary points.
 Sketch graphs of real rational functions using available information, derived from
calculus and/or algebraic arguments, on zeros, asymptotes (vertical and nonvertical), critical points, symmetry.
 For rational functions, the degree of the numerator will be less than or equal to three
and the denominator will be less than or equal to two.
 Extrema of functions: the maximum and minimum values of a continuous function f
defined on a closed interval [a,b] can occur at stationary points, end points or points
where f ' is not defined:
a) Reflection in the line y=x
b) f(x)=ex, -∞<x<∞
c) f-1(x)=lnx, x>0
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








Go over how to find an inverse function and sketch its graph.
Graphs of inverse trig functions
Graph of the modulus function
Determine if a function is odd, even or neither ( both algebraically and graphically)
Find vertical and non - vertical asymptotes of a rational function and investigate asymptotic
behaviour.
Algebraic division of a rational function
Find and interpret the second derivative
Sketch graphs of rational functions
Sketch related graphs
Detail
Nelson 1
Chapter 2.2 Ex2
Chapter 4
N7
Motion and Optimisation
AAC 1.5
Unit standard and added value
Applying algebraic and calculus skills to motion and optimisation
 Applying differentiation to rectilinear motion
 Find the acceleration of a particle whose displacement s metres from a certain point
at time t seconds is given by s=8-75t+t3
 Applying differentiation to optimisation
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







Go over displacement, velocity and acceleration.
Apply differentiation to problems of rectilinear motion
Apply new differentiation to optimisation
Apply integration to problems of rectilinear motion
Revise area under curves and between curves
Area between curve and y-axis
Volume of revolution about x-axis
Volume of revolution about y-axis
Detail
Nelson 1
Chapter 2.2 Ex1, Ex4A, Ex4B
Chapter 3
Ex10A, Ex10B (be selective, looking for rectilinear motion ad volume
questions only)
N7
Matrices and Systems of Equations
GPS 1.1
Unit standard and added value
Applying algebraic skills to matrices and systems of equations
 Using Guassian Elimination to solve a 3x3 system of linear equations
 Find the solution to a system of equations Ax=B, where A is a 3x3 matrix and where
the solution is unique.
 Show that a system of equations is inconsistent i.e. it has no solutions.
 Show that a system of equations is redundant, i.e. it has an infinite number of
solutions.
 Compare the solutions of related systems of two equations in two unknowns and
recognise ill-conditioning.
 Performing matrix operations of addition, subtraction and multiplication
 Perform matrix operations (at most order 3): addition, subtraction, multiplication.
 Know and apply the properties of matrix addition and multiplication: A+B=B+A;
AB≠BA; A(B+C)=AB+AC; (AB)C=A(BC)
 Know and apply key properties of the transpose, determinant and inverse: (A’)’=A;
(A+B)’=A’+B’; (AB)’=B’A’; (AB)-1=B-1A-1; det(AB)=detAdetB
 Calculating the determinant of a matrix
 Find the determinant of a 2x2 matrix and a 3x3 matrix without aid of CAS.
 Finding the inverse of a matrix
 Find the inverse of a 2x2 matrix.
 Know the relationship of the determinant to invertibility.
 Using 2x2 matrices to carry out geometric transformations in the plane
 The transformations should include rotations, reflections and dilations.
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Detail
Nelson 1
Nelson 3
Chapter 5
Chapter 1
N7
Vectors
GPS 1.2
Unit standard and added value
Applying algebraic and geometric skills to vectors
 Calculating a vector product
 Find a vector product in three dimensions.
 Evaluate a.bxc
 Finding the equation of a line in three dimensions
 Find the equation of a line in parametric form, given a point on the line and a direction
vector.
 Find the angle between two lines in three dimensions.
 Determine whether or not two lines intersect and, where possible, find the point of
intersection.
 Finding the equation of a plane
 Find the equation of a plane in Cartesian form, given a normal to and a point on the
plane.
 Find the equation of a plane in parametric and vector form, given suitable defining
information.
 Find the point of intersection of a plane with a line which is not parallel to the plane.
 Determine the intersection of two or three planes.
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Nelson 3
Chapter 2
N7
Complex Numbers
GPS 1.3
Unit standard and added value
Applying geometric skills to complex numbers
 Converting complex numbers form Cartesian to polar form
 Convert a given complex number from Cartesian to polar form and vice-versa.
 Know and use de Moivre’s theorem with integer and fractional indices e.g. expand
(cosθ+isinθ)4.
 Apply de Moivre’s theorem to multiple angle trigonometric formulae e.g. express
sin5θ in terms of sinθ.
 Apply de Moirve’s theorem to find the nth root of unity e.g. solve z6=1
 Plotting a complex number on an Argand diagram
 Plot complex numbers as points in the complex plane.
 Interpret geometrically certain equations or inequalities in the complex plane
e.g. |z-i|=|z-2|, |z-a|>b
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Nelson 2
Chapter 4
Exercises 3-7
N7
Number Theory
GPS 1.4
Unit standard and added value
Applying number skills to number theory
 Using Euclid’s algorithm to find the greatest common divisor of two positive numbers
 Using Euclid’s algorithm to find the greatest common divisor of two positive integers i.e.
use the division algorithm repeatedly.
 Express the greatest common divisor (of two positive integers) as a linear combination
of the two.
 Use the division algorithm to express integers in bases other than ten.
 Know and use the Fundamental Theorem of Arithmetic.
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Detail
Nelson 3
Chapter 5
Exercises 4-5
N7
Methods of Proof
GPS 1.5
Unit standard and added value
Applying algebraic and geometric skills to methods of proof
 Disproving a conjecture by providing a counter example
 Disprove a conjecture by providing a counter example e.g. for all real values a and b, a
is greater than b implies a squared is greater than b squared
 Be able to use the symbols for there exists (backwards E) and for all (upside down A)
 Write down the negation of a statement
 Using indirect proof
 Prove a conditional statement by contradiction e.g. root 2 is irrational
 Prove an unconditional statement by contradiction e.g if a and b are real then a squared
plus b squared is greater than or equal to 2ab
 Use further proof by contradiction e.g. if x, y are real numbers such that x+y is irrational
then at least one of x, y is irrational.
 Use proof by contrapositive
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Detail
Nelson 2
Chapter 1
Nelson 3
Chapter 5
Exercises 12B
Exercises1A – 3B