Interval Bisection with Other Methods

Questions that require interval bisection followed by or combined with other numerical methods such as Newton-Raphson or linear interpolation.

3 questions · Standard +0.3

1.09a Sign change methods: locate roots 1.09b Sign change methods: understand failure cases

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Edexcel F1 2015 June Q5

7 marks Standard +0.3

In the interval \(2 < x < 3\), the equation

$$6 - x ^ { 2 } \cos \left( \frac { x } { 5 } \right) = 0 , \text { where } x \text { is measured in radians }$$ has exactly one root \(\alpha\).
[0pt]

Starting with the interval [2,3], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
[0pt]
Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 2 decimal places.

Edexcel FP1 2014 June Q2

9 marks Standard +0.3

2. $$\mathrm { f } ( x ) = 3 \cos 2 x + x - 2 , \quad - \pi \leqslant x < \pi$$

Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2,3].
[0pt]
Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval \([ - 1,0 ]\). Starting with this interval, use interval bisection to find an interval of width 0.25 which contains \(\beta\).

Edexcel FP1 2015 June Q2

7 marks Standard +0.3

2. In the interval \(13 < x < 14\), the equation $$3 + x \sin \left( \frac { x } { 4 } \right) = 0 , \text { where } x \text { is measured in radians, }$$ has exactly one root, \(\alpha\).
[0pt]

Starting with the interval [13,14], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
[0pt]
Use linear interpolation once on the interval [13,14] to find an approximate value for \(\alpha\). Give your answer to 3 decimal places.