Standard +0.8 This Newton-Raphson question requires differentiating a function with a fractional power (x^{-1/2}), applying the iterative formula, and careful arithmetic with multiple terms. While mechanically straightforward for Further Maths students, the derivative and calculation are more involved than standard C3 questions, placing it moderately above average difficulty.
3.
$$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$
A root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,5 ]\).
Taking 4 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 2 decimal places.
4.53 cso. Note errors in differentiating sometimes give 4.53 but final A1 is cso and should not be awarded in these cases
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = x^2 + \frac{3}{4}x^{-\frac{1}{2}} - 3x - 7$ | | |
| $f'(x) = 2x - \frac{3}{8}x^{-\frac{3}{2}} - 3$ | M1 | $x^n \to nx^{n-1}$ on at least one term |
| | A1 | Correct differentiation |
| $f(4) = -2.625 = -\frac{21}{8} = -2\frac{5}{8}$ | B1 | A correct evaluation of $f(4)$ or correct numerical expression for $f(4)$. In all other cases $f(4)$ must be seen explicitly evaluated |
| $f'(4) = 4.953125 = \frac{317}{64} = 4\frac{61}{64}$ | M1 | Attempt to insert $x = 4$ into their $f'(x)$. Not dependent on first M but must be what they think is $f'(x)$ |
| $\alpha_2 = 4 - \left(\frac{-2.625}{4.953125}\right)$ | M1 | Correct application of Newton-Raphson using their values |
| $= 4.529968454... \left(= \frac{1436}{317} = 4\frac{168}{317}\right)$ | | |
| $= 4.53$ (2 dp) | A1 cao | 4.53 cso. Note errors in differentiating sometimes give 4.53 but final A1 is cso and should not be awarded in these cases |
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3.
$$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$
A root $\alpha$ of the equation $\mathrm { f } ( x ) = 0$ lies in the interval $[ 3,5 ]$.\\
Taking 4 as a first approximation to $\alpha$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to obtain a second approximation to $\alpha$. Give your answer to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel FP1 2012 Q3 [6]}}