- Locate roots of f(x)=0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well-behaved
- Understand how change of sign methods can fail
- Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams
- Solve equations using the Newton-Raphson method and other recurrence relations of the form xn+1 = g(xn) and understand how such methods can fail
- Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between
- Use numerical methods to solve problems in context
Links and Resources
This excel resource looks at various methods of evaluating pi, including Vièta’s formula and using the trapezium rule.
The final series of sheets demonstrate how the trapezium rule can give an approximate value of pi equal to the area under a curve for the functions 4/(1+x^2 ) , 6/√(1-x^2 ) and√(1-x^2 ).
Question 2 of this Rich Starting Point task requires students to compare the area beneath a curve calculated by integration with the area found by using the trapezium rule.
The activity sheet "Coastal erosion A" shows how the area beneath a curve can be estimated using the trapezium rule. There is clear explanation of how the trapezium rule is derived for use in this topical subject.
The final interactive sheet of this Excel file requires students to use trapezia to find an estimate of the area beneath a (time,velocity) curve. New curves can be generated and results can be hidden or shown.
Iterative Solutions of Quadratic Equations
This interactive spreadsheet shows calculations and graphs for the iterative method of solving quadratic equations. The graphs can show the stages of the cobweb diagram for iterations for both of the solutions. There is also a sheet of questions which may be suitable for use in the classroom.
Chapter 17 (pdf pages 153-165) deals with iterative methods leading to staircase or cobweb diagrams.
Chapter 18 (pdf pages 166-170) deals with Newton's method for solving equations.
This Rich Starting Point activity Polynomial Equations with Unit Coefficients sets students the task of finding the roots of polynomials with an increasing number of terms. The task leads to a numerical method, iteration, to find roots of the polynominals.
Students may use a graph plotter to compare the graphs of several polynomials looking for common points and differences.
The teacher's notes provide clear explantation of the task which can be adapted to make it more approachable. The summation of a geometric progression could be used to simplify the main part of the type of equations in the activity.