I: Numerical methods — AQA A-Level Mathematics
Test yourself on I: Numerical methods with AQA A-Level practice questions.
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I: Numerical methods explained
To locate a root of f(x) = 0, evaluate f at two points a and b.
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If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then by the intermediate value theorem there is at least one root between a and b. You then repeatedly bisect the interval, testing the midpoint, to narrow down the root. However, change of sign methods can fail: if f is not continuous, a sign change may not indicate a root; if there is an even number of roots in the interval, the signs at the endpoints may be the same; or if the interval is too large, roots may be missed. Also, a root may exist without a sign change if the graph touches the axis.
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 Understand how such methods can fail.
This topic covers approximate equation solving using iteration. You rearrange f(x)=0 into x=g(x) and use x_{n+1}=g(x_n). Cobweb and staircase diagrams show convergence or divergence: staircase when iterates approach a root monotonically, cobweb when they alternate. The Newton-Raphson method uses x_{n+1}=x_n - f(x_n)/f'(x_n) for faster convergence. You must understand failure: divergence when |g'(x)|>1, oscillation, or Newton-Raphson failing when f'(x_n)=0 or cycling. For example, solve x^3 - x - 1 = 0 using x_{n+1} = (x_n + 1)^{1/3} from x_0=1.5, and compare with Newton-Raphson. Diagrams help interpret convergence.
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.
This topic covers numerical integration using the trapezium rule to estimate the area under a curve y=f(x) between x=a and x=b. The rule divides the interval into n strips of equal width h=(b-a)/n and sums trapezium areas: integral ≈ (h/2)[y_0 + 2(y_1+...+y_{n-1}) + y_n]. You must understand that the estimate is approximate and can be an over- or under-estimate depending on whether the curve is concave or convex over the interval. You should also be able to determine upper and lower bounds for the estimate, for example by using rectangles or considering the nature of the curve. For example, estimate ∫_0^1 √(1+x^3) dx using 4 strips and state whether it is an over- or under-estimate.
Use numerical methods to solve problems in context.
Numerical methods approximate solutions when exact algebra is impractical. You must choose a suitable method, carry it out accurately, and interpret the result in the problem's context. For example, to solve x^3 - 2x - 5 = 0, use sign changes to locate a root, then apply interval bisection or the Newton-Raphson iteration x_{n+1} = x_n - f(x_n)/f'(x_n). Continue until the required accuracy is reached, then round appropriately and state the solution with units or meaning. In context, you may need to set up the equation from given information, decide on a sensible starting value, and explain why the approximation is valid. Always relate the final value back to the original problem, checking it lies within the domain and makes sense practically.
Your focus
- Use a change of sign to locate a root of an equation in a given interval.
- Apply the bisection method to find a root to a specified degree of accuracy.
- Explain circumstances under which change of sign methods can fail to locate a root.
Show all 12 objectives
- Rearrange an equation into the form x=g(x) and perform iterations to find an approximate root.
- Draw and interpret cobweb and staircase diagrams for iterative processes.
- Apply the Newton-Raphson method and explain circumstances under which iterative methods may fail.
- Apply the trapezium rule to estimate the area under a curve.
- Determine whether a trapezium rule estimate is an over- or under-estimate based on the curve's concavity.
- Establish upper and lower bounds for the true area using simple geometric shapes.
- Select and apply an appropriate numerical method to find approximate solutions to equations.
- Carry out iterative processes accurately and determine when a required accuracy has been achieved.
- Interpret numerical solutions within the context of a real-world problem, including units and validity.
I: Numerical methods exam tips
Marking Points
- Evaluates f(a) and f(b) correctly and states that a change of sign indicates a root in (a, b) provided f is continuous.
- Performs at least one iteration of the bisection method correctly, computing the midpoint and evaluating f there.
- States the new interval containing the root after an iteration, ensuring the sign change is maintained.
- Explains a situation where the change of sign method fails, such as a discontinuity, an even number of roots, or a repeated root.
- Uses appropriate accuracy and rounding when evaluating f, and continues until the interval is sufficiently small.
- Justifies the existence of a root using the intermediate value theorem, referring to continuity.
- Rearranges an equation f(x)=0 into the iterative form x=g(x) correctly, ensuring the rearrangement is suitable for the starting value.
- Carries out iterations accurately, showing sufficient decimal places and continuing until a required degree of accuracy is reached.
- Sketches or describes a cobweb diagram when successive iterates alternate around the root, and a staircase diagram when they approach monotonically.
- Applies the Newton-Raphson formula x_{n+1}=x_n - f(x_n)/f'(x_n) correctly, including differentiation and substitution.
- Explains how an iterative method can fail, for example divergence when the derivative of g has magnitude greater than 1 near the root, or Newton-Raphson failure when f'(x)=0 or when iterates cycle.
- Applies the trapezium rule formula correctly, identifying h, the number of strips, and the ordinates y_0, y_1, ..., y_n.
- Calculates the ordinates accurately, often using a table, and substitutes them into the formula without arithmetic errors.
- Determines whether the trapezium rule gives an over- or under-estimate by considering the concavity of the curve over the interval.
- Estimates limits between which the true area must lie, for example by using inscribed and circumscribed rectangles or by comparing with other approximations.
- Interprets the result in context, including units if applicable, and recognises that increasing the number of strips generally improves the approximation.
- Correctly rearrange or define a function f(x) so that the equation to solve is f(x) = 0, showing clear working.
- Use a sign change over an interval to establish a root lies between two values, stating the interval.
- Apply a valid iterative formula (such as interval bisection or Newton-Raphson) accurately for at least two iterations, showing substituted values.
- Continue the method until the required degree of accuracy is achieved, then round and state the final approximate solution.
- Interpret the numerical answer in the context of the problem, including correct units or a concluding statement.
Examiner Tips
- 💡Show the values of f(a) and f(b) clearly, and state that there is a change of sign, so a root lies between a and b.
- 💡When using bisection, present your working in a table with columns for a, b, midpoint, f(midpoint), and the new interval.
- 💡If asked to explain why the method might fail, give a specific example or describe a scenario such as a discontinuity or an even number of roots.
- 💡Use the language of the intermediate value theorem correctly: mention continuity on the interval and the sign change.
- 💡Show the iterative formula you are using and at least three iterations, even if the answer converges quickly, to demonstrate the method.
- 💡When asked to justify a rearrangement, refer to the condition |g'(x)|<1 near the root or show that the iteration converges.
- 💡For Newton-Raphson, write down f(x) and f'(x) clearly before substituting into the formula.
- 💡If an iteration fails, explain why in terms of the gradient or the behaviour of the iterates, not just that it does not work.
- 💡Draw a sketch of the curve and the trapezia to help decide whether the estimate is an over- or under-estimate.
- 💡Show the table of ordinates and the substitution into the formula clearly, even if you use a calculator.
- 💡If asked for limits, consider using the minimum and maximum values of f(x) on each subinterval to bound the area.
- 💡State the number of strips and the value of h explicitly before calculating.
- 💡Show every step of your iteration, including the formula used and the substituted values, so that method marks can be awarded even if an arithmetic slip occurs.
- 💡When using Newton-Raphson, clearly state your starting value and derivative, and be prepared to justify why that starting value is suitable.
- 💡If the question asks for a solution to a given accuracy, state the final answer explicitly and indicate the interval or error bound to demonstrate the accuracy.
Common Mistakes
- Assuming a sign change always means a root without checking continuity; for example, f(x) = 1/x has a sign change at x = 0 but no root.
- Stopping after one iteration without narrowing the interval sufficiently; continue until the required accuracy is reached.
- Failing to maintain the sign change when choosing the new interval; always check which half contains the sign change.
- Ignoring the possibility of multiple roots in the interval; if there are two roots, the signs at the endpoints may be the same, so no sign change is detected.
- Incorrect rearrangement leading to a divergent iteration: for example, rearranging x^3 - x - 1 = 0 as x = x^3 - 1 instead of x = (x+1)^{1/3}. Correction: choose a rearrangement where |g'(x)|<1 near the root.
- Using degrees instead of radians when differentiating trigonometric functions in Newton-Raphson. Correction: always use radians in calculus.
- Stopping iterations too early or rounding intermediate values excessively. Correction: keep full calculator precision and only round the final answer.
- Confusing cobweb and staircase diagrams. Correction: staircase occurs when iterates move steadily towards the root; cobweb occurs when they alternate sides.
- Forgetting to multiply the interior ordinates by 2 in the trapezium rule. Correction: remember the pattern y_0 + 2(y_1+...+y_{n-1}) + y_n.
- Using the wrong strip width, for example taking h=b-a instead of h=(b-a)/n. Correction: always divide the interval width by the number of strips.
- Assuming the trapezium rule always underestimates. Correction: check the concavity; for a convex curve (concave up) the trapezia lie above the curve, giving an over-estimate, and vice versa.
- Rounding ordinates too early, leading to loss of accuracy. Correction: keep full precision until the final answer.
- Error: using degrees instead of radians when applying trigonometric functions in numerical methods. Correction: always check the mode and use radians for calculus-based iterations unless the context explicitly requires degrees.
- Error: stopping after one iteration and assuming the answer is accurate enough. Correction: continue until two successive approximations agree to the required accuracy, or until the interval width is sufficiently small.
- Error: ignoring the context and giving a purely mathematical answer without units or explanation. Correction: always relate the final value back to the original problem, stating what it represents and including units where appropriate.