About this topic
Before you begin
Locate and approximate roots, approximate integrals by the trapezium and Simpson's rules, and step through differential equations by Euler methods.
Learning goals
- Locate a root by graphical or sign-change reasoning
- Approximate roots by linear interpolation and the Newton–Raphson method and explain failure
- Use fixed-point iteration and decide whether it converges
- Approximate integrals with the trapezium rule and Simpson’s rule
- Approximate solutions of first-order differential equations with Euler and improved Euler methods
Syllabus statements covered
- Include: location of roots of an equation by simple graphical or numerical methods; approximation of roots of equations using linear interpolation and Newton-Raphson method, with the aid of a computer, including cases where each method fails to converge to the required root; iterations involving recurrence relations of the form xₙ₊₁ = F(xₙ), including cases where the method fails to converge; approximation of integral of a function using the trapezium rule and Simpson’s rule, with the aid of a computer; approximation of solutions of first order differential equations using Euler method (including the use of the improved Euler formula), with the aid of a computer.
Choose a lesson
Work through them in order.
- Not startedLocating and approximating rootsNextLocate roots by sign change, approximate them by interpolation, Newton–Raphson and fixed-point iteration, and explain when each fails.
- Not startedNumerical integration and Euler methodsNextApproximate integrals with the trapezium rule and Simpson's rule, and differential equations with Euler and improved Euler steps.