A-Level Mathematics Revision — Numerical Methods
Revise Numerical Methods for A-Level Mathematics with a topic explanation, worked example and common mistakes. Check the board notes for specification differences.
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What is Numerical Methods?
Numerical methods at A-Level are techniques used to find approximate solutions to mathematical problems that are difficult or impossible to solve analytically. This includes methods for finding roots of equations (e.g., the Newton-Raphson method), and for approximating definite integrals (e.g., the trapezium rule).
Board notes: The specific numerical methods covered can vary between exam boards. For example, some boards may include the Newton-Raphson method while others focus on interval bisection. The trapezium rule is a standard component for all boards (AQA, Edexcel, OCR).
Step-by-step explanationWorked examples
Worked example
Use the trapezium rule with 4 strips to find an approximate value for the integral of 1/x from x=1 to x=3. The width of each strip h = (3-1)/4 = 0.5. The ordinates are y0=1/1=1, y1=1/1.5=2/3, y2=1/2=0.5, y3=1/2.5=2/5, y4=1/3. The integral is approximately 0.5 * [ (1+1/3)/2 + (2/3+0.5)/2 + (0.5+2/5)/2 + (2/5+1/3)/2 ] = 1.1. To be more precise, using the formula: 0.5 * [ (1 + 1/3) + 2*(2/3 + 0.5 + 2/5) ] = 1.1166...
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Common mistakes
- 1Choosing a poor initial approximation for iterative methods, which can lead to slow convergence or failure to find a root.
- 2Incorrectly applying the formula for the trapezium rule, especially with the first and last ordinates.
- 3Not giving the answer to the required degree of accuracy. Numerical methods provide approximations, so it's important to state the level of accuracy.
Numerical Methods exam questions
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Frequently asked questions
When should I use numerical methods?
Numerical methods are used when it is difficult or impossible to find an exact solution to a problem. For example, you might use a numerical method to find the roots of a complicated equation or to approximate the area under a curve for which you cannot find an antiderivative.
What is an iterative method?
An iterative method is a process that generates a sequence of improving approximate solutions to a problem. The process is repeated until a desired level of accuracy is reached.