10.1.1 Change of Sign

Numerical Methods for Solving Equations

In A Level Maths, the change of sign method is used to find the roots of an equation $f(x) = 0$ by identifying where the function changes from positive to negative (or vice versa).

Step-by-step process:

Locating Roots via Change of Sign

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Proving a Root Correct to 3 Decimal Places

Given Function: $f(x) = 3 + x^2 - x^3$

Objective: Prove that $x = 1.864$ is a root correct to 3 decimal places.

1. Identify the Interval:

• To prove that $x = 1.864$ is correct to 3 decimal places, we need to show that any value that rounds to $1.864$ lies within an interval containing the root.
• Since we're rounding to 3dp, $x$ must lie within the interval $[1.8635, 1.8645]$.

2. Check the Function Values at the Interval Boundaries:

• Calculate $f(1.8635)\ and\ f(1.8645)$:
• $f(1.8635) = 1.98 \times 10^{-3} > 0$
• $f(1.8645) = -5.13 \times 10^{-3} < 0$
• Notice the change of sign between these two values.

3. Conclusion:

• The change of sign and the continuity of the function within the interval $[1.8635, 1.8645]$ indicate that the root lies within this interval.
• Therefore, $x = 1.864$ is indeed a root correct to 3dp.

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