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We are concerned with two main types of series or sequences of numbers: Arithmetic series were the difference between any two consecutive terms is a constant, and geometric series were the ratio of any two consecutive terms is always the same.
A well-known example of a geometric series is the ‘Wheat and Chessboard Problem’: Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains are placed on an 8×8 chessboard? Since this is a geometric series, the answer for n squares is so for n = 8 x 8= 64 squares, the total number of grains of wheat = 264 - 1.
Where the Greek letter sigma ∑ is the mathematical symbol for a sum, or the summation operator.
The Fibonacci series is the series where each subsequent number is the sum of the previous two or Fn = Fn-1 + Fn-2 . The Fibonacci series is used in computer search algorithms, and in biology they describe the pattern of pine cones and flowering of artichokes.
An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares whose side lengths are successive Fibonacci numbers.
Public domain image from the Wikimedia Commons
Sequences and Series are used in many fields of engineering.
Sequences and Series is part of the national engineering unit MEM23004A Apply technical mathematics. More details on unit content from training.gov.au
Required skill: using the techniques of sequences and series to solve simple mathematical problems.
There are no prerequisites required for this module; however, to complete the lessons and quiz below you should have an understanding of arithmetic, fractions, positive and negative numbers.
Electronic and Electrical Engineers use the Fourier series to represent the properties of electrical signals.
Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, S(f) (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies and their amplitudes. Public domain image from the Wikimedia Commons
Mechanical Engineers use Series to predict the life of machine components subject to a random sequence of repeated loading, known as fatigue loading. Watch this video for some examples.
Assumed prerequisite knowledge: Arithmetic, fractions, positive and negative numbers. See reference sites for Sequences and Series if revision is required for prerequisite topics.
To get a better understanding of Sequences and Series review these lessons and quiz:
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry, registration and login required to access the exercises below.
From the integralmaths.org ‘My home’ home page:
Select:
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. AlgebraPower Demand Planning
- Student PDF (Power demand planning): complete this exercise
- Power Demand Planning Interactive: provides a graphical calculator of the power demand over time.
Temperature sensing
- Student PDF (Temperature sensing): complete this exercise
- View the Temperature Sensing video
On completion of the exercises above you may wish to review the solutions contained in the Teacher PDF files below, after logging into the integralmaths.org site.
From the integralmaths.org ‘My home’ home page:
Select:
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Algebra
- Teacher PDF (Power demand planning)
- Teacher PDF (Temperature Sensing)
Interactive Maths exercises with solutions
Sequences exercises from Math Is Fun - Maths Resources
Worked examples on Algebra: Sequences of numbers, series and how to sum them and free tutoring at Algebra.com
When you have completed the lessons, verify your knowledge with this multiple choice Sequences & Series Quiz
PLEASE NOTE:
Calculus is one of the greatest achievements of the human mind separately invented by Newton and Leibniz. Calculus allows us to understand things that change, such as the motion of an object, or the change in position of an object over time.
Calculus is made up of 2 main branches: The first is differentiation (or derivatives), which help us to find the rate of change of one quantity compared to another.
Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.
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The second branch is integration, which is the reverse of differentiation. We may be given a rate of change, such as acceleration and we need to work backwards to find the original relationship (or equation) between the two quantities. The volume of wine barrels was one of the early problems solved using integration.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (a and b above).
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Calculus is used in all fields of engineering: see this TEDx talk ‘What is Calculus Used For’ by Chemical engineer Jeff Heys.
Calculus is part of the national engineering unit MEM23007A Apply calculus to engineering tasks. More details on unit content from training.gov.au
Required skills: using differentiation to find rates of change and applying special calculus techniques such as: method of substitution and using trigonometric identities to solve integrals; identifying key points to find constants of integration; finding integrals of algebraic, trigonometric and exponential functions.
The prerequisite for this module is MEM23004A Apply technical mathematics. More details on unit content from training.gov.au
Electrical & Electronic Engineers use calculus to model electromagnetic fields based on Maxwell's Equations, which describe the interaction of electrical and magnetic fields.
Mechanical Engineers use calculus to calculate the motion of machines and their components, in terms of their velocity and acceleration, see this Learn Engineering video.
Structural Engineers use calculus to find the bending stress and deflection of beams in building structures, see this Learn Engineering video.
Civil Engineers use calculus to find the compression of columns in tall buildings, see this example from +plus magazine...living mathematicson the 46 storey Heron Tower in London.
Chemical Engineers use calculus to calculate the rate of mass transfer in a chemical plant, see this example from the LearnChemE YouTube channel.
Assumed prerequisite knowledge: Algebra, functions, trigonometry, sequences and series. See reference sites above if revision is required for prerequisite topics.
To get a better understanding of Calculus, review these lessons and quiz:
Gradient or slope of a function, limits:
and
Derivatives:
and/or
Integration:
and/or
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry complete at least 2 of the exercises below, registration and login is required to access the exercises.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. CalculusOptimal gutter design
- Student PDF (Optimal gutter design): complete this exercise
Gas compression and expansion
- Student PDF (Gas compression and expansion): complete this exercise to find energy to compress a gas and view,
- Gas compression and expansion Interactive animation
Natural gas storage
- Student PDF (Natural gas storage vessel): complete this exercise
Regenerative braking
- Student PDF (Regenerative braking): complete this exercise to find energy stored in the system and view,
- Regenerative braking animation and interactive calculator.
On completion of the exercises above you may wish to review the solutions contained in the Teacher PDF files below, after logging into the integralmaths.org site.
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Calculus
- Teacher PDF (Optimal gutter design)
- Teacher PDF (Gas compression and expansion)
- Teacher PDF (Natural gas storage)
- Teacher PDF (Regenerative braking)
Open Course Library Calculus 1
Paul's Online Math Notes
Interactive Maths Calculus with solutions
SOS Mathematics exercises with solutions
Coursera Calculus One course from Ohio State University
When you have completed the lessons, verify your knowledge with this multiple choice Introduction to Calculus Quiz
PLEASE NOTE:
Functions and graphs are used in all fields of engineering.
This series of lectures “The Most IMPORTANT Video You'll Ever See” explains some example uses of the exponential function, a function that describes the size of anything with steady growth or decline, and the ‘rule of 70’ used to estimate growth from compound interest.
A graph of the A-, B-, C- and D-weightings across the frequency range used in the measurement of sound pressure level or noise.
Public domain image from the Wikimedia Commons
This module covers a number of more advanced algebra topics that are part of the Australian Curriculum Assessment and Reporting Authority ACARA senior secondary curriculum.
The ‘solve simultaneous equations’ topic is also part of the national engineering unit MEM30012A Apply mathematical techniques in a manufacturing engineering or related environment. More details on unit content from training.gov.au
Civil Engineers use functions to model the flow of water in storm water and irrigation channels, see this example video.
Watch the following video from the LearnChemE youtube channel for an example calculation of the heat generated by an insulated wall.
Electronic and electrical engineers use functions to model electrical signals, see this introductory MIT lecture for more details.
Structural engineers use functions to predict the deflection of beams and columns in building structures, see this example video.
Mechanical engineers use functions to predict the life of ball bearings, see an example calculation here.
Assumed prerequisite knowledge: Algebraic expression manipulation, substitution, transposition, polynomials, quadratic and cubic equations. See the Algebra module or reference sites on this page if revision is required for prerequisite topics.
Simultaneous equations
To get a better understanding of Simultaneous equations review these lessons and quiz:
and/or
Graphing functions
To get a better understanding of graphing functions review these lessons and quiz:
Khan Academy Graphing solutions to equations series of there are 9 short video lectures with 3 practice sections
and
and/or
Introduction to hyperbolic graphs
Matrices
To get a better understanding of Matrices review these lessons and quiz:
and/or
Exponential and Logarithmic functions
To get a better understanding of exponents and logarithms review these lessons and quiz:
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry complete at least 2 of the exercises below, registration and login is required to access the exercises.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Algebra
Capstans, also known as a capstan winch.
Heat loss from building
Profiling power lines
The AC Transformer
On completion of the exercises above you may wish to review the solutions contained in the Teacher PDF files below, after logging into the integralmaths.org site.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Algebra
Open Course Library Intermediate Algebra
Paul's Online Math Notes
Interactive Maths exercises on Simultaneous Linear Equations with solutions
SOS Mathematics Algebra exercises with solutions
Worked examples and free tutoring at Algebra.com
Functions and Graphs Quiz
When you have completed the lessons, verify your knowledge with this multiple choice Functions and Graphs Quiz
PLEASE NOTE:
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2 May, 2018
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