9 Numerical methods — 9MA0 spec coverage

9.1

Locate roots of f(x) = 0 by considering changes of sign of f(x) in an interval on which f(x) is sufficiently well behaved; understand how change of sign methods can fail.

If a continuous function has values of opposite sign at the endpoints of an interval, the intermediate value theorem guarantees at least one root inside that interval.
Evaluate $f(a)$ and $f(b)$ accurately, record their signs, and conclude that a root lies in $(a,b)$ only after checking that $f$ is continuous there.
For example, $f(1)<0$ and $f(2)>0$ for a continuous $f$ , so the graph must cross the $x$ -axis at least once between $1$ and $2$ .
A sign change across a discontinuity need not contain a root, while a repeated root can touch the axis without changing sign; a common error is to treat the sign test as an equivalence.

Tier 1 · Easy

2 marks

ORIGINAL

Let $f(x)=x^2-x-1$ . Use endpoint values to establish that $(1,2)$ brackets a zero of $f$ .

Tier 2 · Standard

4 marks

ORIGINAL

For $f(x)=x^3+x-5$ , show that there is exactly one root in $(1,2)$ .

Tier 3 · Hard

4 marks

ORIGINAL

Two sign tests are proposed. For $h(x)=\frac{1}{x-2}$ , the values $h(1)$ and $h(3)$ have opposite signs. For $k(x)=(x-2)^2$ , the values $k(1.9)$ and $k(2.1)$ have the same sign. Explain why neither test gives the correct conclusion about roots in the stated intervals.

9.2

Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams.

A fixed-point iteration rewrites an equation as $x=g(x)$ and generates approximations using $x_{n+1}=g(x_n)$ from a chosen starting value.
Calculate successive values without premature rounding; on a graph of $y=g(x)$ and $y=x$ , move vertically to $y=g(x)$ and horizontally to $y=x$ to display each iteration.
Near a fixed point $\alpha$ , iterations usually converge when $|g'(\alpha)|<1$ : a positive gradient gives a staircase pattern and a negative gradient gives a cobweb pattern.
Different rearrangements of the same equation can converge at different rates or diverge; a common error is to assume that obtaining $x=g(x)$ automatically produces a useful recurrence.

Tier 1 · Easy

2 marks

ORIGINAL

The recurrence $x_{n+1}=\frac{1}{2}\left(x_n+\frac{6}{x_n}\right)$ is used with $x_0=2$ . Find $x_1$ and $x_2$ .

Tier 2 · Standard

4 marks

ORIGINAL

Use $x_{n+1}=\sqrt{7-x_n}$ with $x_0=2$ to calculate $x_1$ , $x_2$ and $x_3$ to $4$ decimal places. State the equation satisfied by any limiting value.

Tier 3 · Hard

5 marks

ORIGINAL

For $x_{n+1}=3-0.4x_n$ with $x_0=1$ , find $x_1$ to $x_4$ . Determine the limiting value and state whether the graphical construction is a cobweb or a staircase, giving a reason.

9.3

Solve equations using the Newton-Raphson method and other recurrence relations of the form xₙ₊₁ = g(xₙ); understand how such methods can fail.

Newton-Raphson replaces the curve locally by its tangent, giving $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$ for a root of $f(x)=0$ .
Differentiate first, substitute the current approximation into both $f$ and $f'$ , and retain extra calculator digits until the requested final accuracy.
For instance, applying Newton-Raphson to $f(x)=x^2-a$ gives $x_{n+1}=\frac12\left(x_n+\frac{a}{x_n}\right)$ .
The method can fail when $f'(x_n)=0$ , when a tangent sends the next value away from the desired root, or when the values enter a cycle; a common error is to continue without checking the iterates.

Tier 1 · Easy

2 marks

ORIGINAL

Use one Newton-Raphson step on $f(x)=x^2-7$ starting from $x_0=2.5$ . Find $x_1$ .

Tier 2 · Standard

5 marks

ORIGINAL

Derive a Newton-Raphson recurrence for $x^3+x-4=0$ . Starting with $x_0=1.3$ , find $x_1$ and $x_2$ to $5$ decimal places.

Tier 3 · Hard

4 marks

ORIGINAL

Newton-Raphson is applied to $f(x)=x^3-2x+2$ with $x_0=0$ . Calculate $x_1$ , $x_2$ and $x_3$ , then explain why the method fails from this starting value.

9.4

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.

The trapezium rule approximates an integral by replacing the curve over each strip with a straight chord and adding the resulting trapezium areas.
For ordinates $y_0,\ldots,y_n$ at equal spacing $h$ , use $\frac{h}{2}[y_0+y_n+2(y_1+\cdots+y_{n-1})]$ and include each interior ordinate twice.
A convex curve with $f''(x)>0$ lies below its chords, so the trapezium rule overestimates its integral; more, narrower strips usually improve the estimate.
Numerical area is not automatically the signed integral: a common error is to leave contributions below the $x$ -axis negative when the question asks for total area.

Tier 1 · Easy

2 marks

ORIGINAL

The values of a function are $y=1,3,9$ at $x=0,1,2$ respectively. Use two strips and the trapezium rule to estimate $\int_0^2 y\,dx$ .

Tier 2 · Standard

4 marks

ORIGINAL

Use four equal strips and the trapezium rule to estimate $\int_0^2(x^2+1)\,dx$ . State whether the estimate is an overestimate or an underestimate, with a reason.

Tier 3 · Hard

6 marks

ORIGINAL

For $f(x)=\frac1x$ on $1\leq x\leq2$ , use four equal strips to find a trapezium-rule estimate. By using left- and right-endpoint rectangles, also give lower and upper bounds for $\int_1^2\frac1x\,dx$ . Give decimals to $4$ decimal places.

9.5

Use numerical methods to solve problems in context.

A contextual numerical model translates a physical condition, such as a target value or accumulated quantity, into a root, recurrence or numerical integral.
Define the function and units before applying the method, choose an interval or starting value that is meaningful in the context, and state the requested accuracy.
For a threshold $M(t)=C$ , solve $M(t)-C=0$ numerically; for a varying rate $q(t)$ , approximate the accumulated change with $\int q(t)\,dt$ .
A calculator value is not a complete contextual answer: a common error is to omit units, ignore the model's domain or round in a way that makes a safety decision invalid.

Tier 1 · Easy

2 marks

ORIGINAL

A greenhouse temperature is modelled by $T(t)=18+4t-t^2$ degrees Celsius, where $t$ is hours after noon. Show that the temperature returns to $20^\circ\mathrm{C}$ at a time between $3$ and $4$ hours after noon.

Tier 2 · Standard

5 marks

ORIGINAL

A culture is modelled by $P(t)=200e^{0.3t}$ , where $t$ is in hours. Use Newton-Raphson on $f(t)=200e^{0.3t}-1000$ with $t_0=5$ to find $t_1$ and $t_2$ . Hence estimate when the culture reaches $1000$ cells.

Tier 3 · Hard

5 marks

ORIGINAL

Water enters a tank at rates $10,14,18,21,19,15,11$ litres per minute at times $0,5,10,15,20,25,30$ minutes. The tank initially contains $350$ litres and has capacity $800$ litres. Use the trapezium rule to estimate whether the tank overflows during the $30$ minutes, and estimate any excess at the end.