Newton-Raphson methodEdexcel A-Level Maths: Subtopic test

1. 1
   The equation $f(x)=0$, where $f(x)=x^3-2x-5$, has a root $\alpha$ near $x=2$. The Newton-Raphson method is used with $x_0=2$.

   (a)

   Which formula gives $x_{n+1}$ in terms of $x_n$?

   [1 mark]

   • A$x_{n+1}=x_n+\frac{x_n^3-2x_n-5}{3x_n^2-2}$
   • B$x_{n+1}=x_n-\frac{3x_n^2-2}{x_n^3-2x_n-5}$
   • C$x_{n+1}=x_n-\frac{x_n^3-2x_n-5}{3x_n^2}$
   • D$x_{n+1}=x_n-\frac{x_n^3-2x_n-5}{3x_n^2-2}$

   (b)

   Find $x_1$.

   [1 mark]

   • A$1.9$
   • B$2.1$
   • C$2.25$
   • D$3$

   (c)

   Find $x_2$, giving your answer to 4 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=\mathrm{e}^x-3x$. The equation $f(x)=0$ has a root $\alpha$ close to $0.6$, and the Newton-Raphson method is used with $x_0=0.5$.

   (a)

   Find $x_1$, correct to 3 decimal places.

   [1 mark]

   • A$0.390$
   • B$0.351$
   • C$0.610$
   • D$0.410$

   (b)

   A student starts instead with $x_0=\ln3$. What happens?

   [1 mark]

   • A$x_1$ cannot be found, because $f'(\ln3)=0$.
   • BThe method converges to $\alpha$ in one step.
   • C$x_1=x_0$, so the method has converged.
   • DThe iteration diverges to infinity.

   (c)

   Given that $x_1=0.610060\ldots$, find $x_2$ to 4 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The Newton-Raphson method is used to find $\sqrt7$, as the positive root of $x^2-7=0$, starting from $x_0=3$.

   (a)

   Show that the Newton-Raphson formula for this equation simplifies to $x_{n+1}=\frac12\left(x_n+\frac{7}{x_n}\right)$, and find $x_1$ as an exact fraction.

   [3 marks]

   (b)

   Find the equation of the tangent to $y=x^2-7$ at the point where $x=3$. Show that this tangent meets the $x$-axis at $x_1$, and hence explain geometrically how the method finds $x_{n+1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=x^3-2x+2$. The equation $f(x)=0$ has a single real root $\alpha$.

   (a)

   (i) Show that $\alpha$ lies between $-2$ and $-1$.
   (ii) Use the Newton-Raphson method with $x_0=0$ to find $x_1$ and $x_2$.
   (iii) Describe what happens to the sequence and explain why, in terms of tangents to the curve.

   [6 marks]

   (b)

   A student starts with $x_0=0.8$. Find $f'(0.8)$ and $x_1$, and explain why this is a poor choice of starting value. Show that $x_0=-1.5$ is a better choice by finding $x_1$ to 4 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions