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Numerical Methods : Key Stage 5 (A2-Level) : Mathematics Content Knowledge


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Key Stage 5 (A2-Level)
Numerical Methods
Question 1 of 3

1. How confident are you that you can locate:

a. roots by considering change of sign of f(x)?


Example

If f(x) is continuous between x = a and x = b and the sign of f(x) changes between x = a and xb then a root of f(x) exists between a and b.

Root of f(x)

For example, in the diagram above f(a) > 0 and f(b) < 0 so a root lies between a and b.

Example:

Show that a root of f(x) = x^2 - \frac{1} {x + 3} - 1 lies between x = 1 and x = 2.

f(1) =1^2 - \frac{1} {1 + 3} - 1 = -\frac{1} {4}

f(2) =2^2 - \frac{1} {2 + 3} - 1 = \frac{14} {5}

The sign changes between x = 1 and x = 2 and therefore you will find a solution to f(x) = 0 between those two points.

Once you have found the bounds of a root you can use a decimal search or a binary search to find the actual value of the root to the required accuracy.

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