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Higher Mathematics Home Page

Scholar Check List Formulae


Alphabetical Listing of Topics

Algebra Differentiation Exponentials and Logarithms Functions and Graphs Integration The Circle The Straight Line Trigonometry Vectors

Course Overview

The course is based on : Algebra, Calculus, Geometry and Trigonometry
An external SQA exam is sat at the end of the course, with calculator and non-calculator papers.


Higher Maths Past Papers SQA Higher Maths

Prerequisite Knowledge

You must have a good grasp of the following :

  • Completing the Square.
  • Changing the subject of the formula.
  • Manipulating Algebraic fractions.
  • Factorising.
  • Manipulating Surds and Indices.
  • Manipulating quadratics.
  • Basic Trigonometry.

Algebra

Manipulating algebraic expressions
Identifying and sketching related functions
Determining composite and inverse functions
Solving algebraic equations

Calculus

Differentiating functions
Using differentiation to investigate the nature and properties of functions
Integrating functions
Using integration to calculate definite integrals
Applying differential calculus
Applying integral calculus

Geometry

Applying algebraic skills to rectilinear shapes
Applying algebraic skills to circles and graphs
Modelling situations using sequences
Determining vector connections
Working with vectors

Trigonometry

Solving trigonometric equations
Application of the addition or double angle formulae
Application of trigonometric identities
Manipulating trigonometric expressions




Algebra

Algebraic Manipulation

Polynomials

Coefficients etc. Nested form. Dividing polynomials
The factor theorem The remainder theorem
Synthetic division Dividing by ax+b
Solving polynomial equations Sketching polynomial equations

Quadratics

Completing the square - solving quadratic equations The Discriminant Solving quadratics Finding roots of quadratics - derivation Tangency

Sequences

nth term of sequence Recurrence relations Divergence / convergence Linear recurrence relations Formula for linear recurrence relations Solving recurrence relations Special sequences


Key subskills
Factorising a polynomial expression .
Solving polynomial equations.
Using the discriminant.
Solving quadratics
Find the co-ordinates of intersections of curves and straight lines.
Create and use recurrence relations.
Find and use limits of sequences.


  • A2 Completing the square in a quadratic expression
  • A6 Determining and using a recurrence relation
  • A7 Finding and interpreting the limit of a sequence, where it exists
  • A8 Solve quadratic inequalities
  • A9 Use the discriminant
  • A10 Factorising a cubic or quartic polynomial expression
  • A11 Solving a cubic or quartic polynomial equation
  • A12 Finding the coordinates of the point(s) of the intersection of a straight line and a curve or of two curves

Older Learning Outcomes

R&C 1.1 Applying algebraic skills to solve equations.
Apps 1.3 Applying algebraic skills to sequences.

Old Higher Course
Unit 2 LO1 Use the factor/remainder theorem and apply quadratic theory.


Exponentials and Logarithms

Exponentials

Exponentials Growth functions Decay functions

Logarithms

Logarithms Logarithms - converting bases logarithm laws Log and LnLogarithmic equations

Graphs

Graphing Exponential and Logarithmic functions Logarithmic - linear scales on graphs
Log - log graphs Excel -Exponential / log functions - graphs


Key subskills
Simplifying a numerical expression, using the laws of logarithms and exponents.
Solving logarithmic and exponential equations.
Using the laws of logarithms and exponents.


  • A13 Simplifying a numerical expression using the laws of logarithms and exponents
  • A14 Using the laws of logarithms and exponents
  • A15 Solving logarithmic and exponential equations
  • A16 Solve equations involving y = axb and y = abx
  • A17 Use a straight line graph to confirm relationships of the form y = axb and y = abx
  • A18 Model mathematically situations involving the logarithmic or exponential function
  • A19 Sketching the inverse of a logarithmic or an exponential function

E&F 1.1 Applying algebraic skills to logarithms and exponentials.
E&F 1.3 Applying algebraic and trigonometric skills to functions.

Old Higher Course
Unit 3 LO3 Logarithmic and exponential functions.


Functions and Graphs

Functions

Functions Set notation Domain, codomain & range One to one correspondence Inverse functionInverse function - algebraic method Inverse function - graphical method composite functions Simple graphs of functions Reflections, translations, etc

Graphs

Simple graphs of functions Reflections, translations, etc Sketching polynomial equations Graphing Exponential and Logarithmic functions Logarithmic - linear scales on graphs Log - log graphs Excel -Exponential / log functions - graphs


Key subskills
Identifying and sketching related functions.
Determining composite and inverse functions.


  • A1 Determining a composite function
  • A3 Identifying or sketching a function after a transformation
  • A4 Determining f-1 (x) of functions
  • A5 Sketch f'(x) given the graph of y = f(x)
  • A17 Use a straight line graph to confirm relationships of the form y = axb and y = abx
  • A18 Model mathematically situations involving the logarithmic or exponential function
  • A19 Sketching the inverse of a logarithmic or an exponential function

E&F 1.3 Applying algebraic and trigonometric skills to functions.

Old Higher Course
Unit 1 LO2 Associate functions and graphs and solve related equations.


Calculus

Differentiation

Main Differentiation

Differentiation Gradients Notation

xn axn Derivatives of sums
Differentiating Sin and Cos functions Differentiating (x + a)n Differentiating (ax + b)n The chain rule

Stationary Points and Tangents

Increasing / decreasing functions Stationary points finding stationary points
maxima and minima closed intervals
Tangents- gradients Tangents- equations

Optimisation and Curve Sketching

Applications curve sketching graphing derived functions Optimisation problems Newton's equations of motion Rectilinear motion


Key subskills
Differentiating functions.
Determining the equation of a tangent to a curve at a given point by differentiation.
Determining where a function is strictly increasing/decreasing.
Using nature tables.
Using differentiation to determine the optimal solution for a given problem.
Solve problems using rate of change.

  • C1 Differentiating an algebraic function
  • C2 Use differentiation to find the equation of a tangent to a curve at a given point
  • C3 Determining where a function is strictly increasing/decreasing
  • C4 Use stationary points to sketch graphs
  • C5 Differentiate ksin(x) and kcos(x)
  • C6 Differentiating a composite function using the chain rule
  • C7 Determining the optimal solution for a given problem
  • C8 Solving problems using rate of change

R&C 1.3 Applying calculus skills of differentiation.
Apps 1.4 Applying calculus skills to optimisation and area.

Old Higher Course
Unit 1 LO3 Use basic differentiation .
Unit 3 LO2 Use further differentiation and integration .


Integration

Basic Integration

Integral calculus indefinite integrals basic rules of integrationIntegration and the area function Definite integrals Fundamental theorem of calculus Integrating trig functions Indefinite integral of (ax+b)n Differential equations

Area under curves

Areas enclosed by graph and x-axis Area between two graphs Formula for area between two graphs Areas of graphs Areas under curves


Key subskills
Integrating functions.
Calculating definite integrals of polynomial functions.
Finding the area between a curve and the x axis.
Finding area under and between curves.

  • C9 Integrating an algebraic function
  • C10 Integrating functions of form f(x)=(x + q)n n ≠-1
  • C11 Integrating ksin(x) and kcos(x)
  • C12 Integrating functions of form f(x)=(px + q)n n ≠-1
  • C13 Integrate psin(qx+r) and pcos(qx+r)
  • C14 Solve differential eqns of form dy/dx=f(x)
  • C15 Calculating definite integrals of functions
  • C16 Finding the area between a curve and the x-axis
  • C17 Finding the area between a straight line and a curve or two curves
  • C18 Determine and use a function from a given rate of change and initial conditions

R&C 1.4 Applying calculus skills of integration.
Apps 1.4 Applying calculus skills to optimisation and area.
Unit 2 LO2 Use basic integration .

Old Higher Course
Unit 3 LO2 Use further differentiation and integration .


Geometry





The Circle

Circles

The circle - higher course examples equation of circle : x2 + y2 = r2
Equation of circle : given centre, radius Equation of circle : line joining diameter General equation of circle Intersection of line & circle Tangents to a circle Equation of a circle from chords Touching circles


Key subskills
Determining and using the equation of a circle.
Using properties of tangency.
Determining intersections.

  • G5 Determining and using the equation of a circle
  • G6 Using properties of tangency
  • G7 Determining the intersection of circles or a line and a circle

Apps 1.2    Applying algebraic skills to circles.

Old Higher Course
Unit 2 LO4 Use the equation of the circle.


The Straight Line

General

y - b = m(x-a) general equation of a straight line gradient = tan θ distance formula midpoint formula perpendicular lines collinearity

Straight lines and circles

Altitudes Centroid Circumcentre Circumcircle Concurrency Median Orthocentre Perpendicular bisector


Key subskills
Find the equation of parallel and perpendicular lines.
Using m = tan θ to calculate a gradient or angle.
Using properties of medians, altitudes and perpendicular bisectors.

  • G1 Finding equations of parallel and perpendicular ines
  • G2 Use m = tanθ
  • G3 Use properties of medians, altitudes and perpendicular bisectors
  • G4 Determine whether or not two lines are perpendicular

Apps 1.1 Applying algebraic skills to rectilinear shapes.

Old Higher Course
Unit 1 - LO1-Use the properties of the Straight Line


Vectors

Basic Vectors

Scalar and vector quantities Components Magnitude Position vectors Unit vector Zero vector Equal vectors Adding vectors Subtracting vectors Perpendicular vectors

Other Vectors

Collinearity Angle between vectors Scalar product Section formula


Key subskills
Working with 2D vectors.
Working with 3D coordinates.
Using vector components .

  • G8 Determining the resultant of vector pathways in three dimensions
  • G9 Working with collinearity
  • G10 Determining the coordinates of an internal division point of a line
  • G11 Evaluate a scalar product and determine the angle between two vectors
  • G12 Use the scalar product
  • G13 Use unit vectors i, j, k

E&F 1.4 Applying geometric skills to vectors

Old Higher Course
Unit 3 LO1 Use vectors in three dimensions.



Trigonometry

Formulae

Trig refresher Trigonometric identities sec, cosec & cot
Compound formulae Double Angle Half Angle Product formulae Multiple Angle Formulae

Deriving formulae

Deriving addition angle formulae Deriving double angle formulae Deriving multiple angle formulae Deriving half angle formulae
Algebra - Solving Trig equations
Amplitude and Period Sketching trig graphs
Compound angle formulae - Examples

Radians and Exact Values

Radians Converting degrees to Radians Exact values Radians & Degrees Using exact values : Radians & Degrees Conversion table

Wave Theory

Adding two waves Subtracting two waves Express in the form cos(x+a) Express in the form sin(x-a) Express in the form sin(x+a)
Multiple Angles Maximum / Miminum Value Sketching wave functions Wave Theory- Solving equations


Key subskills
Application of the addition or double angle formulae.
Application of trigonometric identities.
Converting acos x± bsin x to k cos(x ±α) or k sin(x ±α, k > 0

  • T1 Solving trigonometric equations in degrees or radians
  • T2 Application of the addition or double angle formulae
  • T3 Application of trigonometric identities
  • T4 Converting acos x± bsin x to k cos(x ±α) or k sin(x ±α, k > 0

E&F 1.2 Applying trigonometric skills to manipulating expressions.
E&F 1.3 Applying algebraic and trigonometric skills to functions.
R&C 1.2 Applying trigonometric skills to solve equations.
Unit 1 LO2 Associate functions and graphs and solve related equations.

Old Higher Course
Unit 2 LO3 Apply trigonometric formulae.
Unit 3 LO4 Apply further trigonometric relationships .


Check List



Books

A selection of resources available at Amazon

Leckie: Higher Maths Student Book TeeJay Higher Maths
Heinemann Higher Mathematics Student Book

Maths Mutt at Amazon

Higher Mathematics Revision Questions
With Worked Solutions Basic Differentiation Basic Integration

Amazon:Higher Maths MM Amazon Store UK school subjects Maths Mutt Publications

As an Amazon Associate I earn from qualifying purchases.

© Alexander Forrest