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10.4PureStretch

Applications to Modelling

Numerical methods are applied to real models where an exact solution is impossible. Typical exam questions combine locating a root, an iterative formula, and Newton-Raphson within a modelling context.

24 min Numerical Methods
Edexcel A level Maths: 10.4 Numerical Methods Application to ModellingCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Apply numerical methods to models
  • Locate roots in a modelling context
  • Use iteration and Newton-Raphson together
  • Interpret the result in context
1

A combined approach

A modelling question often asks you to: (1) show a root exists by a sign change, (2) use an iterative formula to estimate it, and (3) refine with Newton-Raphson. Each tool plays its part.

f(a)f(b)<0 → xn+1=g(xn) → Newton-Raphsonf(a)f(b)<0 \ \to \ x_{n+1}=g(x_n) \ \to \ \text{Newton-Raphson}
Typical workflow.
Example 1A model gives f(t)=t3−5t+1=0f(t) = t^3 - 5t + 1 = 0. Where is the root near t=0t = 0?
1f(0)=1>0f(0) = 1 > 0, f(1)=−3<0f(1) = -3 < 0.
2Sign change ⇒ a root in (0,1)(0, 1); refine numerically.
Answer
Root in (0, 1).

Tip — Read the context: state units and round sensibly for the real situation.

Formula recap

f(a)f(b)<0f(a)f(b) < 0
Locate the root.
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
Refine the estimate.

Common mistakes to avoid

Giving an unrounded or context-free answer.
Round appropriately and state units/meaning.
Skipping the sign-change justification.
Show the root exists before estimating it.

Key takeaways

  • Combine sign change, iteration and Newton-Raphson.
  • Justify the root exists, then estimate and refine.
  • Interpret the answer in the model’s context.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A typical numerical modelling question first asks you to:

    • show a root exists
    • integrate
    • differentiate twice
    • plot every point
    Show answer and explanation

    Answer: show a root exists

    Sign change.

  2. For f(t)=t3−5t+1f(t) = t^3 - 5t + 1, f(0)f(0) and f(1)f(1) are:

    • 11 and −3-3
    • 00 and −3-3
    • 11 and 33
    • −1-1 and 33
    Show answer and explanation

    Answer: 11 and −3-3

    Evaluate.

  3. The fastest refinement tool is:

    • Newton-Raphson
    • sign change
    • plotting
    • guessing
    Show answer and explanation

    Answer: Newton-Raphson

    Quadratic convergence.

  4. Final answers in a model should be:

    • rounded with units/context
    • left as fractions only
    • unrounded
    • in radians
    Show answer and explanation

    Answer: rounded with units/context

    Interpret the result.

  5. A sign change of f over [a,b] (f continuous) shows:

    • a root in (a,b)
    • a maximum
    • no root
    • an asymptote
    Show answer and explanation

    Answer: a root in (a,b)

    Locating rule.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.