Root finding and numerical integration
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| iteration/ˌɪtəˈreɪʃn/ |
How reliable is an approximation?
- A nonlinear equation has no convenient exact solution. An approximation still needs evidence that it is reliable.
- This lesson studies iteration 迭代: A repeated update in which each new approximation is calculated from the previous one.
Choose the mathematical structure
- A continuous function with opposite signs at two endpoints has a root between them. Newton's method uses x next=x-f(x)/f prime(x), with a nonzero derivative. The trapezium rule approximates a definite integral using endpoint heights.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines iteration?
A repeated update in which each new approximation is calculated from the previous one.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For f(x)=x²-2 and x₀=1.5, Newton gives x₁=1.5-(2.25-2)/3=1.4166667. With y=x² on [0,2] and two equal strips, h=1 and trapezium area=(1/2)(0+2×1+4)=3; the exact area is 8/3.
Root finding and numerical integration
A continuous function with opposite signs at two endpoints has a root between them
Compare the model with the worked case and explain one change.
After one Newton step for x²-2 from x₀=1.5, find x₁.
x₁=1.5-(1.5²-2)/(2×1.5)=17/12≈1.416667.
Test a tempting shortcut
- A sign change across a discontinuity does not prove a root. Iteration can diverge or cycle. The trapezium rule's overestimate or underestimate depends on curvature, not just whether the function increases.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every sign change proves a root, including one across a discontinuity. This claim is false. Explain which definition or assumption it violates.
Use two strips for the trapezium estimate of integral x² from 0 to 2.
h=1 with heights 0,1,4 gives (1/2)(0+2×1+4)=3.
Every sign change proves a root, including one across a discontinuity.
A sign change across a discontinuity does not prove a root. Iteration can diverge or cycle. The trapezium rule's overestimate or underestimate depends on curvature, not just whether the function increases.
Interpret a new situation
- Give a stopping criterion and report sensible accuracy. Confirm the approximated root by a sign bracket around the stated rounded answer; explain any failure of the chosen iteration.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the exact integral of x² from 0 to 2.
The exact integral is [x³/3] from 0 to 2=8/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · I. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A repeated update in which each new approximation is calculated from the previous one. Choose the relationship, show the method, check its assumptions and interpret the result.