Mathematical Methods - a MaS Tutor.

Expansion
(or Multiplying Out Brackets) $a (b + c) = a b + a c$

Factorisation

Factor
Theorem

Rearranging Equations
$4 x - 2 x \equiv 2 x$

Graph Plots
and
Sketches

Numerical
Methods

Factor Theorem

For a polynomial $f (x)$ the factor theorem states that:

If $f (p) = 0$ , then $(x - p)$ is a factor of $f (x)$ ,

AND

If $(x - p)$ is a factor of $f (x)$ , then $f (p) = 0$ .

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Expansion (or Multiplying out Brackets)

Expanding Brackets Video:: Practice Questions;; Textbook Exercise.
Expanding Two Brackets Video:: Practice Questions;; Textbook Exercise.
Expanding Three Brackets Video:: Practice Questions;; Textbook Exercise.
Surds: expanding brackets Video:: Practice Questions;; Textbook Exercise.

Further Reading:

Binomial Expansion

Factorisation

Warm-up Material:

Prime Factor Decomposition

Factorisation: Practice Questions; Textbook Exercise
Factorisation: Quadratics: $a x^{2} + b x + c = (p x + q) (r x + s)$; Practice Questions; Textbook Exercise
Factorisation: Quadratics Harder: Practice Questions; Textbook Exercise
Factorisation: Splitting the Middle: Textbook Exercise
Factorisation: Difference of Two Squares: Practice Questions; Textbook Exercise; Worksheet; (Care of corbettmaths.com)

Factorising quadratics
(Care of revisionmaths.com)

Basics
(from www.savemyexams.co.uk)

General Factorising
(from www.savemyexams.co.uk)

More Factorising
(from thirdspacelearning.com)

Numerical Methods

Iterations

Locating Roots

Iteration

Iteration Video:: Practice Questions.
Iterative Processes Video.: (Care of corbettmaths.com)
Iteration:: Exam Questions;; Solving Equations using Iteration;; Solutions.; (Care of www.mathsgenie.co.uk)
Iterative methods revision and worksheets: (Care of mmerevise.co.uk)
Iteration using the ANS button on you calculator: (Care of @CanfordMaths)

Locating Roots

Newton-Raphson Method for Locating Roots $x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}$

Warm up:

Gradient of a Line

Differential Calculus

Iterations

Formula Proof

Graphical representation of Newton-Ramphson Method including a tangent to the function, f(x)

The gradient of the tangent, $A B$ to the curve $f (x)$ at point $P$ is equal to $f^{'} (x)$ . So,

f^{'} (x_{n}) = \frac{f (x_{n}) - 0}{x_{n} - x_{n + 1}}

Therefore,

f^{'} (x_{n}) (x_{n} - x_{n + 1}) = f (x_{n})

x_{n} - x_{n + 1} = \frac{f (x_{n})}{f^{'} (x_{n})}

x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}

As required.

Method to use the Newton-Raphson formula:

Rearrange the given formula into the form $f (x) = 0$ ;
Differentiate $f (x)$ to find $f^{'} (x)$ ;
Substitute $f (x)$ and $f^{'} (x)$ into the Newton-Raphson formula;
Subsitute the value of $x_{0}$ ;
Repeat the substitution until you have completed the expect number of iterations or an answer is gained to the appropriate degree of accuracy.

Example:

Solve,

π = 2 sin x + x, x_{0} = 3

Giving your answer to four significant figures.

π - 2 sin x - x = 0

f (x) = π - 2 sin x - x

f^{'} (x) = - 2 cos x - 1

x_{n + 1} = x_{n} - \frac{π - 2 sin x_{n} - x_{n}}{- 2 cos x_{n} - 1}

x_{1} = x_{0} + \frac{π - 2 sin x_{0} - x_{0}}{2 cos x_{0} + 1}

x_{1} = 3 + \frac{π - 2 sin 3 - 3}{2 cos 3 + 1} = 3.143519915...

x_{2} = 3.143519915... + \frac{π - 2 sin 3.143519915... - 3.143519915...}{2 cos 3.143519915... + 1}

= 3.141592649...

x_{3} = 3.141592654...

$x_{2}$ and $x_{3}$ are the same to four significant figures,

∴ x = 3.142 to 4 s.f.

Questions:

A Level Maths: The Newton-Raphson Method Exam Questions Sheet 1
(Care of alevelmathsrevision.com)

Exam Questions – Newton-Raphson
(Care of www.examsolutions.net)

The Newton-Raphson Method
(Care of mathsgenie.co.uk)

Trapezium Rule

part of the derivation of the trapezium rule for integration

\int_{a}^{b} y d x \approx \frac{h}{2} (y_{0} + y_{n} + 2 (y_{1} + y_{2} + ... + y_{n - 2} + y_{n - 1}))

= \frac{h}{2} (y_{0} + y_{n} + 2 Σ_{i = 1}^{n - 1} y_{i})

h = \frac{b - a}{n}

Questions:: Exam Questions Trapezium Rule
(www.examsolutions.net); Trapezium Rule 1
(www.physicsandmathstutor.com); Trapezium Rule 2
(www.physicsandmathstutor.com)

Graph Plotting and Sketching

Drawing Lines Video:: Practice Questions
Linear graphs: drawing (xy table) Video:: Practice Questions;; Textbook Exercise.
Linear graphs: drawing using $y = m x + c$ Video :: Practice Questions;; Textbook Exercise.
Linear graphs: (across one method) Video:: Practice Questions;; Textbook Exercise.
Quadratic graphs: drawing Video:: Practice Questions;; Textbook Exercise.
Quadratic graphs: sketching using key points Video:: Textbook Exercise.
Drawing Linear Graphs:: Questions [ MS ]
Drawing Quadratic Graphs:: Questions [ MS ]
Drawing Other Graphs: Cubic/Reciprocal:: Questions [ MS ]; (Care of www.mathsgenie.co.uk)

Rearranging Equations

Rearranging Equations
(Care of thirdspacelearning.com)

Including Revision Sheets
(Care of mmerevise.co.uk)

Algebraic Fractions

Changing the subject Video: Practice Questions; Textbook Exercise
Changing the subject advanced Video: Practice Questions; Textbook Exercise
Collecting like terms Video: Practice Questions; Textbook Exercise
Dividing terms Video: Practice Questions; Textbook Exercise
Algebra: multiplying terms Video:: Practice Questions;; Textbook Exercise.
Algebraic fractions: addition Video:: Practice Questions;; Textbook Exercise.
Algebraic fractions: division Video:: Practice Questions;; Textbook Exercise.
Algebraic fractions: multiplication Video:: Practice Questions;; Textbook Exercise.
Algebraic fractions: simplifying Video:: Practice Questions;; Textbook Exercise.; (Care of corbettmaths.com)

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