Newton-Raphson with verification

A question is this type if and only if it asks to apply Newton-Raphson and then verify the answer is correct to a specified number of decimal places using a change of sign or other method.

6 questions · Standard +0.3

1.07i Differentiate x^n: for rational n and sums 1.09d Newton-Raphson method

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Edexcel F1 2017 January Q6

7 marks Standard +0.3

6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root $\alpha$ of the equation $\mathrm { f } ( x ) = 0$ lies in the interval [0.6, 0.7].

Taking 0.6 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 3 decimal places.
Show that your answer to part (a) is correct to 3 decimal places.

Edexcel F1 2016 June Q3

7 marks Standard +0.3

3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, $\alpha$, of the equation $\mathrm { f } ( x ) = 0$ lies in the interval $[ - 2 , - 1 ]$.

Taking - 1.5 as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$, giving your answer to 2 decimal places.
Show that your answer to part (a) gives $\alpha$ correct to 2 decimal places.

tion 3continued -

Edexcel F1 2018 Specimen Q3

7 marks Standard +0.3

3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, $\alpha$, of the equation $\mathrm { f } ( x ) = 0$ lies in the interval $[ - 2 , - 1 ]$.

Taking - 1.5 as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$, giving your answer to 2 decimal places.
Show that your answer to part (a) gives $\alpha$ correct to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-06_2250_51_317_1980}

Edexcel FP1 Q3

7 marks Standard +0.3

$$f(x) = x^3 + x - 3.$$ The equation $f(x) = 0$ has a root, $\alpha$, between 1 and 2.

By considering $f'(x)$, show that $\alpha$ is the only real root of the equation $f(x) = 0$. [3]
Taking 1.2 as your first approximation to $\alpha$, apply the Newton-Raphson procedure once to $f(x)$ to obtain a second approximation to $\alpha$. Give your answer to 3 significant figures. [2]
Prove that your answer to part (b) gives the value of $\alpha$ correct to 3 significant figures. [2]

Edexcel FP1 Q37

11 marks Standard +0.3

$$f (x) = x^3 + 8x - 19.$$

Show that the equation $f(x) = 0$ has only one real root. [3]
Show that the real root of $f(x) = 0$ lies between 1 and 2. [2]
Obtain an approximation to the real root of $f(x) = 0$ by performing two applications of the Newton-Raphson procedure to $f(x)$ , using $x = 2$ as the first approximation. Give your answer to 3 decimal places. [4]
By considering the change of sign of $f(x)$ over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places. [2]

OCR H240/03 2020 November Q4

11 marks Standard +0.3

A curve has equation $y = 2\ln(k - 3x) + x^2 - 3x$, where $k$ is a positive constant.

Given that the curve has a point of inflection where $x = 1$, show that $k = 6$. [5] It is also given that the curve intersects the $x$-axis at exactly one point.
Show by calculation that the $x$-coordinate of this point lies between 0.5 and 1.5. [2]
Use the Newton-Raphson method, with initial value $x_0 = 1$, to find the $x$-coordinate of the point where the curve intersects the $x$-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places. [3]
By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places. [1]