# Further Pure

Autograph file: Volumes of Revolutions (setup page)

Axes etc setup & ready for doing Volumes of Revolutions. My best example and another example.

Formula for Volume of Revolution. PDF here.

Notes.

Notes and proofs of standard integrals for AQA Core 3. Suggested use in classroom; pupils to read and understand proof of integral of 1/(a2+x2) and then construct a similar proof for integral of 1/[(a2+x2)1/2]. Pdf version.

Summary notes on intersection of three planes; all parallel, two parallel, tri-prism, sheaf and single point. Includes summary table plus extra notes and links to a couple of good youtube videos. Pdf here.

Questions to get you thinking about the crtieria for planes to intersect, be parallel etc. A few great tasks here with answers included. Works great together with dynamic graphing software such as Autograph, Geogebra etc. Pdf here.

Lots of useful formulae to do with vectors including dot and cross products, volume of parallelopiped, vector equations of lines and planes. Pdf here.

Derivation of the eipi formula from Maclaurin Expansions. Pdf here.

A set of questions to really check understanding of several vector concepts such as magnitudes of vectors, vector equations of lines, scalar product and closest distance between a point and a line. Answers included. PDF here.

An example each of undefined at one of the limits, undefined between the limits and infinity at one of the limits.

Bulletpoint style recipe on how to find the gradient of curves using the differentiation from first principles process. PDF here.

Notes on the general format of the expansion, some standard expansions and a couple of initial teaching ideas. PDF here.

Short set of key notes on sums of series.

Notes on finding single and more than one stationary point(s) on rational functions.

Powerpoint, with answers as you go, of questions to ease students into working with complex numbers. Covers adding, multiplying, dividing, Argand diagram, argument-modulus form, solving complex quadratics and complex equations. PDF here.

See page 3 for six (or so) harder trig questions to find the general solution of.

Concise notes and recipe for finding general solutions of trig equations.

Conics Autographs

Some pre-made Autograph files for teaching conics. Use the constant controller to change the constants and vary the functions.

 Parabola Hyperbola Ellipse Circle Rectangular Hyperbola

Includes eccentricity, focus, directrix and transformations of the standard equations.

List of standard matrix transformations; enlargement, rotation, reflection and shear. (Translations omitted as these are created by matrix addition rather than matrix multiplication). PDF here.

Matrix multiplication, finding the determinant and inverse matrices.

Set of three differentiated worksheets on matrix multiplication, answers included. Lesson starter questions on matrix addition and scalar multiplication.

Students matchup the images, words and matrices. Deep questions could include "Why did you match up that matrix with that transformation?" or "How do you know which matrices represent rotations and which represent reflections?"

What are the matrices representing each of these rotations and reflections? Pdf version.

Notes on Interval Bisection, Linear Interpolation, Newton Raphson and Numerical Solutions of Differential Equations.

Designed to highlight finding orientation of rational graphs and necessityfor checking either side of the asymptotes. Six graphs and eight equations, all with linear numerator and linear denominator. Match them, and draw the remaining two graphs that are given. Starter questions. Use this Autograph file.

Notes on how to draw graphs of rational functions, particularly for combinations of linear/quadratic numerator and denominators.

Rational Function Autographs

Some pre-made Autograph files for teaching rational functions. Use the constant controller to change the constants and vary the functions.