Numerical Methods

This lesson covers Numerical Methods as required by the A-Level Mathematics Pure 1 specification. Numerical methods are techniques for finding approximate solutions to equations that cannot be solved algebraically. These methods are particularly useful for equations where no exact analytical solution exists.

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Locating Roots

The Change of Sign Method

If f(x) is a continuous function and f(a) and f(b) have opposite signs (i.e., f(a) × f(b) < 0), then there is at least one root of f(x) = 0 in the interval (a, b).

This is a consequence of the Intermediate Value Theorem.

Example: Show that f(x) = x³ − 3x − 1 has a root between x = 1 and x = 2.

f(1) = 1 − 3 − 1 = −3      (negative)
f(2) = 8 − 6 − 1 = 1        (positive)

Since f(1) < 0 and f(2) > 0, and f(x) is continuous, there is a root in the interval (1, 2) by the change of sign rule. ∎