Maths - Higher Index
Online Lessons for Higher Maths Students in Scotland
- Straight line
- Recurrence Relations
- Graphs of related functions
- Graphs of Trigonometric
- The Circle
- Further Calculus
- Logs and Exponentials
- The Wave Function
Straight Line: The distance between points. Points on horizontal or vertical lines. It is relatively straightforward to work out the distance between two points which lie on a line parallel to the x- or y-axis. The distance formula gives us a method for working out the length of the straight line between any two points. It is based on Pythagoras’s Theorem.
Calculating the gradient when given an angle: calculating a slope using the width and height to find the percentage, angle or length of a slope. The slope corresponds to the inclination of a surface or a line in relation to the horizontal. It can be measured as an angle in degrees, radians or gradian.
Collinearity: in statistics, multicollinearity (also collinearity) is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy.
Finding the equation of the median: the learning intention in this lesson to be able find the equation of the median from a vertex in a triangle. Find the mid-point of two points, calculate the gradient between two points and more.
Finding the equations of the altitude: find the gradient, equations and intersections of medians, altitudes and perpendicular bisectors using our knowledge of the mid-point as well as parallel and perpendicular lines.
Finding the equation of the perpendicular bisector: to find the perpendicular bisector of two points, all you need to do is find their midpoint and negative reciprocal and apply these answers into the equation. This lesson will demonstrate how it’s done.
Finding the point of intersection between two lines: how do I find the point of intersection of two lines? Get the two equations for the lines into slope-intercept form. Set the two equations for y equal to each other. Solve for x. Use this x-coordinate and substitute it into either of the original equations for the lines and solve for y.
Functions: a function relates inputs to outputs. A function takes elements from a set (the domain) and relates them to elements in a set (the codomain). In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set.
Composite function: generally a function that is written inside another function. Composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x).
Calculating inverse functions: finding the inverse of a function. First, replace f(x) with y. Replace every x with a y and replace every y with an x . Solve the equation from Step 2 for y. Replace y with f−1(x) f − 1 ( x ) Verify your work by checking that (f∘f−1)(x)=x ( f ∘ f − 1 ) ( x ) = x and (f−1∘f)(x)=x ( f − 1 ∘ f ) ( x ) = x are both true.
Finding the domain and range: the way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis.
Graphs of related functions: given the graph of a common function, (such as a simple polynomial, quadratic or trig function) you should be able to draw the graph of its related function. The graph of the related function can be sketched without knowing the formula of the original function. The following changes to a function will produce a similar effect on the graph regardless of the type of function involved. You should be familiar with the general effect of each change. You can also consider the effect on a few key points on each graph to help determine the related graph.
Graphs of functions: given the graph of a common function, (such as a simple polynomial, quadratic or trig function) you should be able to draw the graph of its related function. The graph of the related function can be sketched without knowing the formula of the original function. The following changes to a function will produce a similar effect on the graph regardless of the type of function involved. You should be familiar with the general effect of each change. You can also consider the effect on a few key points on each graph to help determine the related graph.
Graphs of logarithmic functions: the graph of a logarithmic function has a vertical asymptote at x = 0. The graph of a logarithmic function will decrease from left to right if 0 < b < 1. And if the base of the function is greater than 1, b> 1, then the graph will increase from left to right.
Graphs of trignonmetric functions: Trigonometric graphs. The sine and cosine graphs. The sine and cosine graphs are very similar as they both have the same curve only shifted along the x-axis, have an amplitude (half the distance between the maximum and minimum values) of 1 and have a period (size of one wave) of 360˚
Changing into radians and exact value: Degrees and radians are two units for measuring angles. A circle contains 360 degrees, which is the equivalent of 2π radians, so 360° and 2π radians represent the numerical values for going “once around” a circle. This leads us to the rule to convert degree measure to radian measure.
Sketching trigonometric graphs: start with the basic shape. Use the number of cycles to calculate the period. Use the horizontal shift to slide the graph into position. Use the amplitude to scale vertically and calculate the starting minimum and maximum values.
Differentiation: differentiation is used in maths for calculating rates of change. For example in mechanics, the rate of change of displacement (with respect to time) is the velocity. The rate of change of velocity (with respect to time) is the acceleration. The rate of change of a function can be found by finding the derived function . For an equation beginning, the rate of change can be found by differentiating. This is also known as ‘Leibniz Notation’.
Basic rules of differentiation: general rule for differentiation: the derivative of a constant is equal to zero. The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. The derivative of a sum is equal to the sum of the derivatives.
Finding the rate of change: to find the average rate of change, we divide the change in y (output) by the change in x (input).
Finding the equation of a tangent to a curve: a Tangent Line is a line which locally touches a curve at one and only one point. The slope-intercept formula for a line is y = mx + b.
Equation of a tangent: we can calculate the gradient of a tangent to a curve by differentiating. In order to find the equation of a tangent, we: differentiate the equation of the curve. Substitute the value into the differentiated equation to find the gradient. Substitute the value into the original equation of the curve to find the y-coordinate. Substitute your point on the line and the gradient.
Increasing and decreasing functions: if we draw in the tangents to the curve, you will notice that if the gradient of the tangent is positive, then the function is increasing and if the gradient is negative then the function is decreasing. To calculate the gradient of the tangents, we differentiate in order to substitute the relevant ‘x’ value in.
Stationary points: a stationary point of a function f(x) is a point where the derivative of f(x) is equal to 0. These points are called “stationary” because at these points the function is neither increasing nor decreasing. Graphically, this corresponds to points on the graph of f(x) where the tangent to the curve is a horizontal line.
The graph of the derived function.
Optimisation: Optimisation is used to find the greatest/least value(s) a function can take. This can involve creating the expression first. Also find the rate of change by differentiating then substituting. To find the maximum or minimum values of a function, we would usually draw the graph in order to see the shape of the curve. Now, using our knowledge from differentiation, we can find these greatest and least values of a function without plotting the graph in a given interval.
Quadratics: quadratic equations contain terms which have a highest power of two. This type of equation can be used to solve different problems including modelling the flight of objects through the air.
Completing the square: some quadratics cannot be factorised. An alternative method to solve a quadratic equation is to complete the square. This lesson explains what completing the square means.
Quadratics inequalities: a quadratic inequality is an equation of second degree that uses an inequality sign instead of an equal sign. Examples of quadratic inequalities are: x2 – 6x – 16 ≤ 0, 2×2 – 11x + 12 > 0, x2 + 4 > 0, x2 – 3x + 2 ≤ 0 etc. Solving a quadratic inequality in Algebra is similar to solving a quadratic equation.
Using the discriminant: in a quadratic equation, the discriminant helps tell you the number of real solutions to a quadratic equation. The expression used to find the discriminant is the expression located under the radical in the quadratic formula.
Intersection of lines and curves: say you are given the equations of both a line and a curve, for example y=x4+3x-1 and y=4x-9, and asked to find where these two intersect. This just means where the two lines would cross or touch if drawn on the same graph. To find these points you simply have to equate the equations of the two lines, where they equal each other must be the points of intersection.
Polynomials: in mathematics, a polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Factorising polynomials when given a factor: In this lesson, you will learn how to factor out common factors from polynomials.
Factorising polynomials when not given a factor: a polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial.
Finding missing values of polynomials: Enter the polynomial and the value of x for which you want to find the value of the polynomial. The value of polynomial P(x) at x=a is P(a) .
Finding the equation of cubic graphs: cubic graphs can be drawn by finding the x and y intercepts. Because cubic graphs do not have axes of symmetry.
Trigonometry: a branch of mathematics that studies relationships between side lengths and angles of triangles. The three basic functions in trigonometry are sine, cosine and tangent.
Double angle formula: a double angle formula is a trigonometric identity which expresses a trigonometric function of 2θ in terms of trigonometric functions of θ
Addition formulae: In these lessons, we learn the cosine addition formulae; how to derive the cosine of a sum and difference of two angles; the sine addition formula – how to derive the sine of a sum and difference of two angles; how to use the sine and cosine addition and subtraction formulas to prove identities; how to use the sine and cosine addition and subtraction formulas to determine function values.
Solving trig equations: learn how to solve trigonometric equations in Higher Maths involving multiple or compound angles and the wave function in degrees or radians.
Trig identities; the Trigonometric Identities are equations that are true for Right Angled Triangles. The eight fundamental trigonometric identities are: cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ; sin2θ + cos2θ = 1; tanθ = sinθ/cos θ; 1+ tan2θ = sec2θ
Integration: in mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.
Indefinite integrals: the process of finding the indefinite integral is called integration or integrating f(x) f ( x ). An indefinite integral is a function that takes the antiderivative of another function. It is visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to symbolize taking the antiderivative.
Definite integrals: definite integrals are integrals which have limits (upper and lower) and can be evaluated to give a definite answer. A question of this type may look like: ∫ a b a x n d x = [ a x n + 1 n + 1 ] a b.
Area under a curve: the area under a curve between two points can be found by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. Areas under the x-axis will come out negative and areas above the x-axis will be positive.
The area between two curves: the process for calculating the area between two curves is the same as finding the area between a curve and a straight line.
The circle: a circle is all points in the same plane that lie at an equal distance from a center point. The circle is only composed of the points on the border. … A circle is the same as 360°. You can divide a circle into smaller portions. A part of a circle is called an arc and an arc is named according to its angle.
The equation of a circle: any point P with coordinates (x,y) on the circumference of a circle can be joined to the centre (0, 0) by a straight line that forms the hypotenuse of a right angle triangle with sides of length x and y. This means that, using Pythagoras’ theorem, the equation of a circle with radius r and centre (0, 0) is given by the formula x2 + y2 = r2 .
Equation of tangents to circles: the tangent is perpendicular to the radius which joins the centre of the circle to the point P. As the tangent is a straight line, the equation of the tangent will be of the form y = m x + c.
Intersection of line and circles: There are three ways a line and a circle can be associated, ie the line cuts the circle at two distinct points, the line is a tangent to the circle or the line misses the circle.
Further Calculus: Further Calculus at Higher Maths involves differentiating and integrating sinax & cosax/
Further differentiation including trig functions: this lesson integrates functions involving brackets and raised to a power. It also demonstrates how to integrate trigonometric functions.
Further integration including trig functions: this lesson integrates functions involving brackets and raised to a power. It also demonstrates how to integrate trigonometric functions.
Logs and exponentials: this section talks about the relationship between logs and exponentials. The graph of y = logax is symmetrical to the graph of y = ax with respect to the line y = x. This relationship is true for any function and its inverse.
Evaluating logarithmic functions: you learn how to evaluate logarithmic expression over the following lessons:
Solving logarithmic equations: this lesson demonstrates how to solve logarithmic equations.
Solving exponential equations to: this lesson demonstrates how to solve exponential equations.
Solving problems involving half-life: students learn to solve problems related to half-life.
The wave function: a wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it.
Writing the wave function form: What is the wave function equation? To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form y(x,t)=Asin(kx−ωt+ϕ).
Maximum and minimum values: this lesson demonstrates how to calculate the maximum and minimum values of a single trig function and find the values of x at which this occurs.
Sketching the graph: this lesson demonstrates how to sketch the graph of a single trig function.