3.6 Numerical solution of equations

3 of 3 objectives have lessons · 0 studied · 0 secure

Lessons

Spotting where a root hidesUse a sign change (and the graph) to show that an equation has a root between two values, without solving it. 8 min, not started
When the sequence settles downUnderstand how a sequence of approximations converges to a root, and use the notation x₁, x₂, x₃… to track the values. 7 min, not started
Running an iteration to find a rootWhat an iteration formula is, why its values home in on a root, and how to run one to the accuracy a question asks for. 10 min, not started
Rearranging equations into iteration formLearn to rearrange f(x) = 0 into x = F(x) to create your own iterative formula, and understand why some rearrangements converge while others diverge. 9 min, not started

Syllabus objectives

3 objectives

What Cambridge says you should be able to do, in our words. Codes are the topic number plus the objective’s position.

3.6.1

Locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change.

For example, finding a pair of consecutive integers between which a root lies.

Taught in: Spotting where a root hides · 6 questions

3.6.2

Understand the idea of, and use the notation for, a sequence of approximations that converges to a root of an equation.

Taught in: When the sequence settles down · 4 questions

3.6.3

Understand how a given simple iterative formula of the form xₙ₊₁ = F(xₙ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

Knowledge of the condition for convergence is not included, but candidates are expected to understand that an iteration may fail to converge.

Taught in: Rearranging equations into iteration form, Running an iteration to find a root · 7 questions