Interval Bisection (Edexcel International A Level Further Maths)

Revision Note

Author

Mark Curtis

Expertise

Maths

Interval Bisection

What is interval bisection?

Interval Bisection means splitting an interval containing a root into two halves and using a sign change test to find out which half contains the root
For example, if $f (1) < 0$ and $f (2) > 0$ and $f (x)$ is continuous
- Then a root must lie in $[1, 2]$
- Find the midpoint of the interval
  - $x = 1.5$
- Check the sign at the midpoint
  - For example, $f (1.5) < 0$
- Write down the half-interval that has the sign change
  - $f (1) < 0$ , $f (1.5) < 0$ and $f (2) > 0$
  - So $[1.5, 2]$ has the sign change
The process can be repeated
- For example, if $f (1.75) > 0$ then $[1.5, 1.75]$ contains the sign change

How do I use interval bisection to estimate a root?

If a root lies in the interval $[1, 2]$ , then the first approximation of the root, $x_{1}$ , is simply the midpoint
- $x_{1} = 1.5$
The second approximation, $x_{2}$ , requires using Interval Bisection once
- For example, giving $[1.5, 2]$
- Then find the midpoint
  - $x_{2} = 1.75$
- And so on

Exam Tip

Read the question carefully to see if an approximation to the root is required, or the interval containing the root is required.

Worked Example

The equation $x^{3} + x - 16 = 0$ has a root in the interval $[2.1, 2.9]$ .

Use interval bisection to find an interval of width 0.2 that contains the root.

Find the width of the interval given in the question

$2.9 - 2.1 = 0.8$

Interval bisection halves the width
To get a width of 0.2, interval bisection is needed twice
Find the signs of the interval end-points

$\begin{array}{rcl} f (2.1) & = & - 4.639 < 0 \\ f (2.9) & = & 11.289 > 0 \end{array}$

Find the midpoint of the interval

$x = \frac{2.1 + 2.9}{2} = 2.5$

Find the sign of $f (x)$ at $x = 2.5$

$f (2.5) = 2.125 > 0$

Find the new halved interval containing the sign change
Check the signs: $f (2.1) < 0$ , $f (2.5) > 0$ and $f (2.9) > 0$

$[2.1, 2.5]$

Repeat the process
Find the midpoint of the interval

$x = \frac{2.1 + 2.5}{2} = 2.3$

Find the sign of $f (x)$ at $x = 2.3$

$f (2.3) = - 1.533 < 0$

Find the new halved interval containing the sign change
Check the signs: $f (2.1) < 0$ , $f (2.3) < 0$ and $f (2.5) > 0$

$[2.3, 2.5]$

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