Edexcel IAL Mathematics FP1.3.1 numerical methods for f(x)=0

For f(x)=0, the marks sit in the numerical step: test the signs, substitute into the chosen formula and round only after the approximation is found.

Exam points

Evaluate f at both endpoints to justify the sign change across the interval.
Carry out one bisection or linear-interpolation step using the given bounds.
Use x_{n+1}=x_n−f(x_n)/f′(x_n) and state x1 to the requested accuracy.

[Maximum number: 4]

\mathrm{f}(x)=10-2 x-\frac{1}{2 \sqrt{x}}-\frac{1}{x^{3}} \quad x>0

(a)

Using $x_{0}=0.5$ as a first approximation to $\alpha$ , apply the Newton-Raphson procedure once to f(x) to find a second approximation to $\alpha$ , giving your answer to 3 decimal places.

The equation f(x)=0 has another root $\beta$ in the interval [4.8, 4.9]

[ 2 ]

$x_{1}=0.5-\frac{\mathrm{f}(0.5)}{\mathrm{f}^{\prime}(0.5)}=0.5-\frac{0.29289321 \ldots}{46.70710678 \ldots}$

Correct application of Newton-Raphson

M1

=0.494

Correct value 3dp. A correct derivative
must have been used

A1

(2)

(b)

Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to $\beta$ , giving your answer to 3 decimal places.

[ 2 ]

$\frac{4.9-\beta}{|f(4.9)|}=\frac{\beta-4.8}{f(4.8)}$ $\beta=4.883$

M1 A1

ALT:
$\beta=\frac{4.8|f(4.9)|+4.9|f(4.8)|}{|f(4.8)|+|f(4.9)|}$ $\beta=\frac{4.8\times0.0344+4.9\times0.1627}{0.0344+0.1627}=4.883$

M1 A1

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FP1.3.1 - Equations of the form f(x) = 0 f(x) will involve only question 1

Question (a)

Question (b)