Numerical methods
Locating roots: sketches and sign changes
Most equations have no exact solution. Before you can approximate a root you need to know how many there are and roughly where: sketch two graphs to count them, then trap each one with a change of sign.
≈ 35 minFixed-point iteration
Feed a number into a formula, feed the answer back in, and repeat. When the values settle down they settle on a root. This lesson is the engine of every P2 and P3 numerical question.
≈ 40 minRearrangements, staircases and cobwebs
One equation can be rearranged into in many ways, and they do not all work. See why some iterations close in on a root, some zigzag towards it, and some run away, and learn to find the equation a limit satisfies.
≈ 40 minThe trapezium rule
When an integral can't be found exactly, estimate it: replace the curve by straight lines, add up the trapezia, and use the shape of the curve to say whether the answer is too big or too small.
≈ 35 minNewton–Raphson, bisection and interpolation
Slide down the tangent to the axis and repeat: the Newton–Raphson method roughly doubles the number of correct digits at every step. Also the two simpler interval methods in Edexcel FP1, and the ways all of them can go wrong. For AQA and Edexcel Further Pure 1; not examined in Cambridge 9709.
≈ 40 min