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The addition and subtraction of matrices is important across many applications, from computer graphics and image processing to statistics. Use this resource to learn the rules for adding and subtracting matrices. Addition of matrices There is one key rule for adding matrices. Matrices of the same shape (same number of...
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Angles are commonly measured in degrees but sometimes, it is useful to define them in terms of the length around the unit circle. This resource introduces radians as a measure of angle. Being able to use radians is crucial for solving problems in engineering, such as analysing the motion of...
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Arithmetic forms the foundation of mathematics, focusing on the basic operations and properties of numbers. You need arithmetic to perform calculations and solve everyday problems. Use this resource to enhance your numerical skills. Numbers Use this resource to learn the basics for working with integers. Fractions Use this resource to...
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Matrices can be used to solve systems of equations using elementary row operations and the augmented matrix. This method of solving systems of equations is handy when you have many variables and equations. You may see it being used in fields like economics, statistics and machine learning. Systems of equations...
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Trigonometric functions such as sin, cos and tan are usually defined as the ratios of sides in a right-angled triangle. These ratios can be extended to angles greater than \(90^{\circ}\), using angles in a unit circle. Circular functions have applications in many STEM disciplines, such modelling oscillating motion in engineering,...
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Complex numbers are a group of numbers that help us to get mathematical solutions where real numbers (which includes positive and negative, counting numbers, fractions and decimals) just can’t work. They can be useful in engineering and physical sciences. They have two parts: one part is an imaginary number. (For...
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The limit of a function means finding the value of a curve at a particular point. But what is that point? A value to many decimal places can be very long, to define a value that is very precise. The precise point can be infinitely small. We can only ask...
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If there is a relationship between two or more variables (like, area and radius of a circle (A = πr2 ), or pressure, volume and temperature of a gas), then there will also be a relationship between how these variables change. You may need to find how fast one variable...
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Sometimes a small change in one variable can render a big change in a larger value. For example, a small increase (or error) in the radius of a sphere means a lot more volume is added! If you estimate the small error in one variable, you can calculate the significant...
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Implicit differentiation enables you to find the derivative of y with respect to x without having to solve the original equation for y. If we are able to write an equation relating \(x\) and \(y\) explicitly, that is in the form \(y=f(x)\), then we can find the derivative function \(y=f'\left(x\right)\)...
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Partial derivatives reveal how a function with many variables changes when you adjust just one of the variables in the input. Let us suppose that we have the equation for a paraboloid with an elliptical cross-section such as \(z=x^{2}+4y^{2}.\) In this case we have a function of two independent variables,...
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Higher derivatives are used in many mathematical, scientific and engineering subjects. Consider a function \(y=f(x)=4x^{3}-6x^{2}+7\). If we differentiate this function we obtain \(f^{\prime}\left(x\right)=12x^{2}-12x\). This is the first derivative of the function \(y=f\left(x\right)\). It is possible to find second, third and subsequent derivatives by continuing to differentiate and these are called...
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Sometimes it is easier to differentiate the logarithm of a function than the original function. This is called logarithmic differentiation and this module provides an overview of the method and provides some examples. Suppose you have to differentiate \[\begin{align*} y & =\frac{x^{2}-1}{x^{8}\sqrt{x^{4}+1}}. \end{align*}\] At first sight, you need to use...
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A tangent is a line that touches a curve at only one point. Where that point sits along the function curve, determines the slope (i.e. the gradient) of the tangent to that point. A derivative of a function gives you the gradient of a tangent at a certain point on...
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Learn how to take a derivative of a function using first principles. Using this method is the best way to understand the concepts around differentiation. Start here to really appreciate what you are doing when you differentiate, before you start differentiating using other methods in later modules. Definition The derivative...
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Learn about the rules for differentiation and the different notations that are used. This section includes algebraic, exponential, logarithmic and trigonometric examples. It is not always convenient to use differentiation from first principles to find a derivative function. The “rules” shown below have been established from first principles and can...
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How do you differentiate a larger function that has components that are smaller functions? This module will show you how to package these functions and work them out separately, before plugging them into the Chain Rule formula. The derivatives of functions such as \(y=\sin\left(x^{3}\right)\)and \(f(x)=\left(x^{2}-1\right)^{4}\) can be found using the...
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What is the product rule? It is useful when you want to differentiate a function that comprises one function multiplied by another function. The derivatives of functions such as \(y=f(x)=2x\sin\left(x\right)\) and \(y=f(x)=xe^{x}\) can be found using the product rule. Definition If \[\begin{align*} y & =f\left(x\right)\\ & =u\left(x\right)\cdot v\left(x\right) \end{align*}\] then...
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What is the quotient rule? The quotient rule is like the product rule but this time it is for one function that is divided by another (rather than multiplied). Review this section to learn how to differentiate using the quotient rule. Functions such as \(y=f(x)=\frac{1}{x^{2}+x}\), \(y=f(x)=\frac{\sin x}{x}\) and \(y=f(x)=\frac{x^{2}+1}{x+1}\) may...
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How do you find the maximum (highest) or minimum (lowest) value of a curve? The maximum or minimum values of a function occur where the derivative is zero. That is where the graph of the function has a horizontal tangent. If you go looking for the horizontal tangents (i.e. where...
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If you find some key points of a function such as: maxima, minima, or turning points; x and y axis intercepts; and regions where the gradient is positive or negative, you can put together a sketch of a curve. Read this section to find examples of this being done. To...
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Decimals are used to express fractions in a more precise and easily understandable form. They are essential for handling money, measuring quantities, and applying mathematical concepts in everyday life and various professional fields. Use this resource to learn how to add, subtract, multiply, divide and convert decimals. A decimal is...
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A determinant is a number that can be calculated for any square matrix. The determinant is used in calculating vector cross products, eigenvalues, eigenvectors and solving simultaneous equations. Use this resource to learn how to find the determinant of \(2\times2\) and \(3\times3\) matrices. Determinant of a \(2\times2\) matrix The determinant...
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Differential equations provide calculations where things happen with a changing rate. The rate that things change may depend on another value which is changing too. For example, compound interest or rabbit populations both get bigger and bigger. Moreover, they get bigger faster! as time goes along. Their rate of change...
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The value of a function changes when one of the variables (x, y, a, or b etc) changes. It may change like the variable, (both doubling) or it may change more, or less. Differentiation means finding the derivative of an expression. This means you are finding a rate of change...
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Eigenvalues and eigenvectors are important parts of an engineer’s mathematical toolbox. They give us an understanding of how buildings, structures, automobiles and materials behave in real life. But that's not all they are used for; you can see their use in many other areas of STEM, including colour theory, electric...
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Solving algebraic equations involves finding the values of variables that satisfy mathematical statements. Build on your foundational understanding of algebra by tackling how to solve various types of equations, from linear to quadratic and beyond. Getting started Solving linear equations Use this resource to learn how to solve linear equations....
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Understanding functions and relations is crucial for exploring more advanced topics in mathematics and their applications in various scientific and engineering domains. In this module, you'll learn about what functions and relations are and you can practice with the provided exercises. Relations A relation is a set of ordered pairs....
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Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians. The functions \(y=\sin x\) and \(y=\cos x\) have a domain of \(\mathbb{R}\)...
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Often the domain of a function will be restricted to a subset of R. This subset is called an interval, and the end points are a and b. Intervals Often the domain of a function will be restricted to a subset of the set of real numbers, \(\mathbb{R}.\) This subset...
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If f-1(x) is the inverse function of a one-to-one function f(x) then f-1(x) is the set of ordered pairs obtained by interchanging the first and second elements in each ordered pair. Definition of an inverse function If \(f^{-1}(x)\) is the inverse function of a one-to-one function \(f(x)\) then \(f^{-1}(x)\) is...
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The absolute value of a number x gives a measure of its size or magnitude regardless of whether it is positive or negative. If a number is plotted on a number line then its absolute value can be considered to be the distance from zero. Introduction The absolute value of...
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Functions which have different rules for each subset of the domain are called hybrid functions. Sometimes they are referred to as piecewise defined functions. An example of a hybrid function is: \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1. \end{cases} \end{align*}\] Note that this hybrid...
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Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians. The functions \(y=\sin x\) and \(y=\cos x\) have a domain of \(\mathbb{R}\)...
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A linear graph is the simplest way of representing data or a functional relationship. Consequently an understanding of linear graphs is an essential requirement for many courses in science, engineering and mathematics. The Cartesian plane The Cartesian plane is defined by a pair of mutually perpendicular coordinate axes. The horizontal...
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A quadratic graph is the graph of a quadratic function. These graphs have applications in a wide range of fields. Keep reading to learn about parabolas and how to sketch them, then test your knowledge with some exercises. A quadratic function has the form \(y=ax^{2}+bx+c\) where \(a\neq0\) . Parabolas The...
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The known graphs of some simple functions and relations can be used to sketch related, but more complicated functions. If you know the graph of a function, you can often transform it to a graph of a more complex but related function. A simple example is the graph of the...
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An index can be an integer – a counting number - either positive or negative. It can also be a fraction such as \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), or \(2.5\). Use this resource to learn how the laws of indices apply to fractional indices. Index laws Remember the basic index laws: \(a^{m}\times a^{n}=a^{m+n}\)...
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Mastering fractions is crucial for tackling a wide range of mathematical problems. The skill is applicable to a huge range of real-world contexts, from cooking and construction to science and finance. Use this resource to learn how to simplify, add, subtract, multiply and divide fractions. A fraction is the easiest...
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Functions and graphs provide a way to represent, analyse, and understand relationships between different quantities. If you are looking to improve your knowledge of this area, the linked resources are a great way to start. Plotting points on a graph Graphs usually have points or markers on them. The location...
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Maths uses many symbols, or notations, which represent different functions and operations. You will come across lots of these as you explore the Learning Lab. Use this glossary to learn what any unfamiliar symbols mean. Foundational notations Foundational notations appear across different areas of maths and science. Symbol Explanation \(+\)...
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The hyperbolic functions are analogous to the circular (trigonometric) functions and are widely used in engineering, science and mathematics. This module introduces hyperbolic functions, their graphs and similarities to the circular functions. Whereas circular functions are defined on a unit circle, the hyperbolic functions are defined on a hyperbola. Hyperbolic...
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The differentiation and integration of hyperbolic functions allow us to understand how these functions change and accumulate values which teaches us the essential tools for analysing complex mathematical and real-world systems. Keep reading to learn about these functions and put your new knowledge to the test with some exercises. See...
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Providing a function is one to one, it is possible to find an inverse function. This module discusses inverse hyperbolic functions, which are used in advanced calculus for integration and in the solution of differential equations. See Hyperbolic functions for a list of definitions. Inverse hyperbolic functions may also be...
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Information about functions and graphs to improve your maths skills in these areas. HF1 Hyperbolic functions This module introduces hyperbolic functions, their graphs and similarities to the circular functions. HF2 Derivatives and integrals of hyperbolic functions The hyperbolic functions are widely used in engineering, science and mathematics. HF3 Inverse hyperbolic...
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We know that 3 to the power of 2 is 9 and 3 to the power of 3 is 27. But what is the power of 3 that is equal to something in between, such as 20? It would be 3 to the power of something greater than 2 but...
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How do you antidifferentiate a function? Antidifferentiation (also called integration) is the opposite operation to differentiation. If you have the differentiation of a function, you can then obtain the original function via integration (antidifferentiation). Given a derivative \(f^{\prime}\left(x\right)\) of a function we want to find the original function \(f\left(x\right)\). The...
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How do you integrate a polynomial where x is raised to a power? We saw this in the previous section on antidifferentiation. But how do you integrate a linear expression in brackets where the whole bracket is raised to a power? This module shows how to integrate functions like :...
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How do you integrate a logarithm? How do you integrate an exponential function? How do you integrate a trigonometric function? Read this worksheet to see how this is done. This module looks at integrals such as \(\int\frac{1}{x}dx\) and\(\int\frac{1}{3x-1}dx.\) The power rule for integration One of the most important rules for...
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The integration of exponential functions is widely used in various fields such as physics, engineering, and economics to model and solve problems involving growth, decay, and other processes that change exponentially over time. Keep reading to find out more. This module deals with differentiation of exponential functions such as: \[\begin{align*}...
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The integration of trigonometric functions in solving problems is related to oscillatory motion, waves and other periodic phenomena in physics and engineering. This module deals with integration of trigonometric functions. These include: \[\begin{align*} & \int\sin\left(2x+3\right)dx\\ & \int\cos\left(5x\right)dx\\ & \int_{1}^{2}\sec^{2}\left(x-2\right)dx. \end{align*}\] Indefinite integral (antiderivative) of a trigonometric function Recall that: \[\begin{align*}...
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An area under a curve might be above the axis (and therefore positive). But sections might also be below the axis (and therefore negative). Read this worksheet to see how to deal with finding the integral for sections of graph which go above and below the x-axis (horizontal axis). This...
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An expression that is composed of two functions (say an algebraic expression nested within a trigonometric expression) can be complicated to integrate. You can simplify this by substituting a single pronumeral (say u) to represent one of the functions. The substitution rule for integration is like the chain rule for...
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How do you integrate an expression when there is an algebraic expression in the numerator and denominator of a fraction? Integrating using partial fractions helps you to solve this problem. Read this worksheet for several worked examples. Some times a complex function may be integrated by breaking it up into...
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If you can consider your expression to be a product (i.e. multiplication \(\times\)) of two functions, you can integrate this using Integration by parts. This reflects the product rule in differentiation and is applicable to logarithmic, exponential, trigonometric and algebraic functions. Integration by parts is a technique for integrating the...
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Integrating will find the area between the curve and the x-axis (horizontal axis). We've learned how to limit this to a section of the x axis. But what if you have an area that is bounded by limits on both the x and the y axes? You will need to...
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Index notation is a powerful mathematical tool for expressing repeated multiplication concisely. Learn how to work with indices, understand the laws governing them, and convert expressions involving powers and roots. This will help you simplify complex expressions and solve problems across a wide range of mathematical and scientific disciplines. Video...
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Indices, surds and logarithms extend our understanding of numbers and their relationships. Use these resources to enhance your comprehension of these powerful mathematical tools. Indices Use this resource to learn how to work with indices, understand the laws governing them, and convert expressions involving powers and roots. Surds Use this...
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Integration is vital in engineering. It is the key mathematical tool for finding the centre of mass or the surface area of a body. Integration is also called antidifferentiation. It is the reverse process of differentiation. If you differentiate an expression, you can integrate it to get back to the...
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Use this resource to learn about matrices, their order, and the different types of matrices. A sound understanding of matrices is required in a huge range of real-world applications, like machine learning, natural language processing, robotics, image processing, urban planning and infrastructure development. Matrices Matrices are arrays of numbers or...
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Vectors are used to represent forces in physics, handle 2D and 3D manipulation with computer graphics, and to calculate the forces acting on materials in textiles. You also deal with vector quantities in your everyday life, from driving your car down the road to planning the shortest route to get...
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In matrix algebra, we can add, subtract and multiply matrices as long as the matrix order is correct. Unlike traditional arithmetic, we cannot divide matrices. Instead, we multiply by the inverse matrix. Inverse matrices have many applications, including computer animation, encryption and digital image transformations. Inverse matrices An inverse matrix...
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Being able to find the inverse of a \(3\times3\) matrix will help to simplify complex problems and enhances your ability to perform matrix operations efficiently. This is crucial in fields like engineering, physics and computer science. Before you read further, make sure that you are familiar with augmented matrices and...
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The Inverse Square Law is a mathematical concept that gives the relationship between intensity and the distance from an energy source. \[{intensity} \ \propto \ \frac{1}{{distance}^2} \] This is relevant to the energy of wave phenomena whether it be sound, light waves or other forms of radiation. The video will...
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Transforms are another means of solving some differential equations that may prove too difficult to solve using other methods. Download the laplace transforms worksheets to improve your skills in these areas. LT1 Basic definition of laplace transforms (PDF) LT2 Table of transforms (PDF) LT3 Solving differential equations (PDF) LT4 Convolutional...
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The modeling of growth and decay in areas such as finance, epidemiology and science makes use of equations with logarithms and exponentials. The laws for working with logarithms enable us to solve equations that cannot be solved with other algebraic techniques. Use this resource to learn about them. Video tutorial...
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For any system of equations, there may be: Infinitely many solutions No solution A unique solution Coefficient matrix Consider the following system of equations: \[\begin{align*} 2x+4y-z & =9\\ x-y+2z & =-4\\ -x+y-z & =3 \end{align*}\] These may be written in the matrix form: \[\begin{align*} \left[\begin{array}{ccc} 2 & 4 & -1\\...
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Algebra Arithmetic Complex numbers Differential equations Differentiation Equation solving Functions and graphs Hyperbolic functions Indices, logs, surds Integration Inverse square law Laplace transforms Matrices Numerical methods Statistics Trigonometry Vectors Glossary of maths symbols This glossary of maths symbols explains common notations you might encounter in the Learning Lab....
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Matrices are used in a wide range of fields and applications to represent and manipulate data in a structured way. Use these resources to increase your understanding of matrices. Getting started Introduction to matrices Use this resource to learn about matrices, their order, and the different types of matrices. Addition...
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A strong understanding of how to multiply matrices is critical in many fields, like computer science, robotics, statistics and economics. Just as there are rules for adding and subtracting matrices, we can multiply matrices with each other when certain requirements are met. Use this resource to learn how to multiply...
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Measurement forms the basis of many practical applications in STEM fields. Use this resource to develop a solid understanding of fundamental measurement concepts. Working through these pages will equip you with the skills you need for further exploration in science, technology, engineering, and mathematics. Dimensions (PDF) All measurements need a...
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You are used to solving equations using basic algebraic operations and perhaps the quadratic formula. However, some equations cannot be solved using these methods. For example the equations: \[\begin{align*} e^{-x} & =x & \left(1\right) \end{align*}\] or \[\begin{align*} -\ln\left(x\right) & =x^{3} & \left(2\right) \end{align*}\] cannot be solved using conventional methods. However...
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Some integrals cannot be evaluated in terms of the rules of integration or elementary functions. Simpson’s rule is a numerical method that calculates a numerical value for a definite integral. You will have evaluated definite integrals such as \[\begin{align*} \int_{1}^{3} & \left(x^{2}\right)dx \end{align*}\] before. In doing this, you are evaluating...
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The trapezoidal rule is a numerical method that calculates a numerical value for a definite integral. Some integrals cannot be evaluated in terms of the rules of integration or elementary functions. You will have evaluated definite integrals such as \[\begin{align*} \int_{1}^{3} & \left(x^{2}\right)dx \end{align*}\] before. In doing this, you are...
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Learn the basics for working with integers, including negative numbers. Numerical reasoning skills you will learn here are very fundamental to maths; you will apply them to everything from everyday situations to more complex mathematical challenges. Integers Counting or whole numbers, also known as integers, can be used in many...
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Numerical methods are a collection of techniques used to solve mathematical problems that are difficult or impossible to solve with exact solutions. These worksheets are filled with information that will improve your skills in numerical methods. NM1 Newton’s method Some equations cannot be solved using algebra or other mathematical techniques....
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Further resources Also, the Learning Lab has sections that will help nursing students with important academic concepts:...
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Percentages are a powerful tool for expressing proportions, comparisons and changes. Learn how to convert fractions and decimals into percentages using this resource. This will enhance your ability to interpret data and make informed decisions when faced with quantities. A percentage is another way to express a part of a...
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An object in motion tends to stay in motion and an object at rest tends to stay at rest unless acted upon by an unbalanced force. Force equals mass times acceleration. For every action there is an equal and opposite reaction. Do you need to learn about Newton's first three...
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We will look at the gravitational force acting on an object on a slope. These can be divided into two components, the normal (resisting) force pushing into the slope which produces friction and the shear or driving force pushing the block down the slope. So we must consider forces parallel...
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Momentum is a vector quantity, so it is important to remember that direction must be taken into account when doing problems on momentum. Let's explore how we can do this. The moment p of an object is the product of its mass \(m\) and velocity \(v\), or: \[\begin{align*} p &...
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Graphs usually have points or markers on them. The location of this point is given by an ordered pair. Watch this video to find out more....
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Linear motion refers to the motion of an object in a straight line. Describing these motions require some technical terms such as displacement, distance, velocity, speed and acceleration. The terms and their relationships to one another are described in this module. Scalar and vector quantities Quantities that have only a...
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Many concepts in physics may be represented by vectors. A vector has both a size (called its magnitude) and a direction. This module explains how vectors may be added together. Scalars and Vectors Many physical quantities can be classified into one of two groups: scalars or vectors. Scalar quantities are...
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Pythagoras’ theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. Understanding Pythagoras' theorem helps you solve problems in fields such as architecture, engineering, and computer graphs, where precise calculations...
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The quadratic formula provides a straightforward way to find the solutions of any quadratic equation, especially when factoring is difficult or impossible. By becoming familiar with this formula, you'll gain a reliable tool for solving quadratic equations and deepen your understanding of algebraic relationships. Solving quadratic equations The solutions to...
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Breaking vectors down into their components—or resolving them—makes it easier to add or subtract them, especially when dealing with vectors that don't act along the same line. You will encounter this in many areas of STEM, like when analysing forces involved in robotics, studying the projectile motion of objects launched...
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Explore the relationships between the angles and side lengths of a right-angled triangle. You will learn how to use trigonometric ratios to find unknown angles and lengths, enhancing your ability to solve geometric problems. Mastering this concept is vital for fields such as engineering, physics, and architecture, where precise calculations...
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Summation notation or sigma notation is a shorthand method of writing the sum or addition of a string of similar terms. This module explains the use of this notation. The basic idea We use the Greek symbol sigma \(\Sigma\) to denote summation. \(\Sigma\) is called the summation sign. A typical...
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A normal distribution with a mean of zero and a standard deviation of one is called the standard normal distribution. Areas under the standard normal distribution curve represent probabilities which can be found via a calculator or a “z-table”. The standard normal distribution (sometimes called a z-distribution) has a mean...
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In any normal distribution the mean and standard deviation can be used to convert it to a standard normal distribution and when can then compute probabilities. Even when data follows a normal distribution, different data sets will have their own mean and standard deviation and a different bell shaped curve...
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Learn how we can sample distributions. The distribution of the means of all the possible samples of a certain size tend to follow a normal distribution. A sampling distribution is the probability distribution for the means of all samples of size \(n\) from a given population. The sampling distribution will...
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We can use the mean of a sample to estimate the mean of the entire population. It is more appropriate to give an interval estimate rather than a point estimate. We use the statistics we obtain from samples to make inferences or estimates about the population from which the sample...
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This module explains how to set up and test hypotheses to see if a difference between a sample mean and a population mean is significant. Consider statements such as Teenagers aged \(13\)-\(15\) spend no more than \(10\) hours a week on Facebook. The average weight of Australian men is the...
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Hypothesis testing usually uses the population standard deviation to calculate a “z” value. If the population standard deviation is unknown, we use the sample standard deviation to calculate a “t” value. T-distribution In conducting a hypothesis test the population standard deviation \(\sigma\) may be unknown. In that case we approximate...
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Hypotheses can be tested by comparing the test statistic to the critical value or by comparing the p-value to the significance level, α. In hypothesis testing two approaches are possible when making the decision as to whether to reject the null hypothesis. So far we have compared the test statistic...
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How do we apply a test of proportions? Rather than comparing a sample mean to a population mean, we can compare a sample proportion to a population proportion. In the hypothesis tests we have looked at so far, we have been concerned to find evidence that there has been a...
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Hypothesis tests can be either two-tailed (non-directional) suggesting that the sample mean is different to the population mean, or one- tailed (directional) suggesting that the sample mean is greater than (or alternatively, less than) the population mean. A test of proportion is used to determine whether or not a sample...
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The Poisson distribution deals with the number of random occurrences over a period of time (or distance or area or volume), such as the number of people who enter a shop every hour, or the number of flaws in a sheet of glass. The Poisson distribution is a discrete distribution...
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Data is everywhere and increasingly drives many aspects of our day-to-day lives. Here we explain the different types of data that can be collected and some ways of illustrating this data. Definitions \(\mathbf{Population}\): the total group of individuals or items under consideration. \(\mathbf{Sample}\): a group of individuals or items chosen...
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The mean, mode and median are measures of the centre or middle of a set of data. They are sometimes called measures of central tendency and they provide a single value that is typical of the data. Definitions The mode is the value that occurs most often. The median is...
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The range, the interquartile range and the standard deviation are three different measures of the spread of a set of data. This module shows three different ways to calculate a number to represent the spread of a set of data. Measuring spread or dispersion in data Consider the two sets...
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This module covers the rules of basic probability, including the multiplication and addition principles and complementary events. The probability of an event \(A\) is expressed as a number between zero and one: \(0\leq Pr(A)\leq1\) . \(Pr(A)=0\) means that event \(A\) is impossible. \(Pr(A)=1\) means that event \(A\) is certain. When...
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A sample space is a list of all the possible outcomes. There are a number of techniques that can be used to list the sample space. A list or diagram showing all possible outcomes in a probability experiment is called a sample space. The probability that an event \(E\) occurs...
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If two events are not independent then the outcome of one event can change the probability of the second event occurring. Dependent events Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed. Example...
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The binomial distribution is a discrete distribution consisting of repeated trials, where each trial has two possible outcomes. Introduction A random variable may be described as having a binomial distribution when there are a number of repeated trials and there are only two possible outcomes on each trial. The following...
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This module introduces the normal distribution. Data that is normally distributed is characterized by a bell shaped curve when displayed graphically. Graphical representations of data may look quite different as shown below: But many things that can be measured, such as heights of people blood pressure errors in measurement scores...
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There are two ways to multiply two vectors. Here, we will learn about the scalar product. It has many applications in STEM. For example, scalar products are used to calculate the work done by a system, in computer graphics to calculate the amount of light hitting surfaces, and in engineering...
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Learn how to solve cubic equations, where the highest power of the variable is three. These types of equations can model various real-world scenarios, from physics to engineering. The factor theorem The long division of polynomials is an important tool for solving cubic equations. However, the factor theorem is equally...
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Exponential equations are common in fields such as science, finance, and population dynamics, where exponential growth or decay is observed. By learning how to isolate variables and find solutions to exponential equations, you will be able to handle a variety of real-world problems involving exponential relationships. Exponential equations Indicial (or...
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Earlier, we looked at how to transpose formulas. Now, let's combine those skills with arithmetic to solve linear equations. By mastering the techniques for solving these equations, you'll be equipped to tackle a wide range of problems and build a strong basis for more advanced mathematical concepts. Video tutorial –...
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Just as linear equations can be solved to find the value of an unknown variable, we can do the same with quadratic equations. Here, we will learn how to solve quadratic equations using the null factor law. By applying this principle, you can factorise quadratic equations and determine their solutions...
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Delve into solving simultaneous equations, where a solution works for multiple equations. This skill is essential for analysing systems with multiple variables and is widely used in fields such as engineering, economics, and science. Simultaneous equations Simultaneous equations are equations that share variables and must be solved at the same...
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A calculator may only give you one answer to an inverse trigonometry question between \(0\) and \(90^{\circ}\). When you use the unit circle, you can visualise the many different solutions for the angle. Understanding this concept will help you in many STEM fields, such as satellite navigation in physics and...
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There are a number of important special matrices to become familiar with if you are studying a STEM discipline. The transpose of a matrice is important for solving problems in structural mechanics, symmetric matrices can be applied to motion, dynamics and force calculations in physics, and orthogonal matrices have many...
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Do you need to brush up on statistics or learn some new skills? The links on this page lead to Learning Lab pages that will put you on the right track. S1 Summation notation Summation notation, also known as sigma notation, is a shorthand method of writing the sum or...
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How do you find a square root of any number that isn’t already a square? The square root of \(16\) is \(4\) and the square root of \(25\) is \(5\), but can you find the square root of a number between these, for example \(\sqrt{20}\)? Learning how to identify and...
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Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians. The functions \(y=\sin x\) and \(y=\cos x\) have a domain of \(\mathbb{R}\)...
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Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are essential tools for simplifying expressions and solving equations in mathematics. On this page, you'll learn about fundamental identities, double angle formulas, sums and difference, and other trigonometric functions. An algebraic expression...
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Like the sine rule, the cosine rule comes in handy for triangles that are not right-angled. Use this resource to learn about the cosine rule, how it is applied to find unknown sides and angles, and its importance in solving complex geometric problems involving any type of triangle. The cosine...
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The trigonometric ratios are handy for real-life situations where right-angled triangles are involves, but what about triangles that are not right-angled? We use the sine rule. Use this resource to learn what the sine rule is and how it is applied. The sine rule shows that the ratio of the...
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Expand your transposition skills by exploring how to rearrange formulas involving brackets and fractions. By doing this, you will develop the skills needed to isolate variables in more complex expressions and prepare yourself well for advanced problem-solving in STEM. You will have learned the necessary theory when you were first...
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Trigonometry is the study of triangles. Ancient builders and mariners used it for finding lengths that are not physically measurable (because they were so large) but they could be defined by angles. Use these resources to develop a solid understanding of trigonometry and its practical applications. Getting started Pythagoras’ theorem...
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The shortest distance from any point in the air to the ground is defined by a line sitting at right angles to the ground going straight down. Learn how to find the perpendicular (right angle) distance from a point to a plane. What do we mean when we talk about...
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If you are on the side of a hill, the gradient depends on the direction you look. So the directional derivative is the gradient in a particular direction. (See also Linear graphs) Learn how to find the directional derivative of a function of two variables f(x,y) or three variables g(x,y,z)...
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What is a vector product? What is a cross product? The vector product is a vector that is the result of multiplying the magnitudes (size) of two vectors. The magnitude is found using matrices and determinants). The result of the cross product is another vector, and the direction is perpendicular...
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In Resolution of vectors we learned how to resolve vectors in two dimensions along horizontal and vertical axes. It is also possible to resolve one vector along the line of another vector (instead of along the x-y axes). (See also Linear graphs) Learn how to find the projection (resolution) of...
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If you want to uniquely define a line, you need to pin it between two points in 3-dimensional space. You can also define a point with a 3-dimensional vector through it. This process uses three types of equations. Learn how to find the vector equation, the parametric equation, and the...
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Learn how to determine if two lines in three dimensions intersect (cross each other) and, if so, what their point of intersection is. (See also Linear graphs) In order to find the point of intersection of two lines in three dimensional space, it is best to have both equations in...
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Learn how to find the equation of a plane (a 2-dimensional space) a) through three points or b) given a normal (line at right angles) and a point on the plane or c) given a parallel plane and a point on the plane. (See also Linear graphs) A plane is...
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Two 2-dimensional planes will slice through each other (unless they are parallel). Where they slice will be defined by a straight line. There will also be an angle between the two planes. Learn how to determine the angle between two intersecting planes and the equation of the line of intersection....
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2 + 2 does not always equal 4 when the problem is translated into a 2 or 3 dimensional space. Vectors are quantities that have both magnitude (size) and direction. This branch of maths is fundamental to physics and engineering to represent physical quantities that have a direction. The University...
