The Newton-Raphson processEdexcel International A Level Further Maths: Subtopic test

1. 1
   The equation $x^3-2x-5=0$ has a root $\alpha$ close to $2$. Let $f(x)=x^3-2x-5$ and use the Newton-Raphson method with $x_0=2$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$x^2-2$
   • B$3x^2$
   • C$\frac{x^4}{4}-x^2-5x$
   • D$3x^2-2$

   (b)

   Find $x_1$.

   [1 mark]

   • A$1.9$
   • B$2.1$
   • C$12$
   • D$-0.1$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   The Newton-Raphson method is applied with $x_0=1$. Which statement is correct?

   [1 mark]

   • AThe iteration converges quickly to $\alpha$
   • BThe iteration converges to a different root
   • C$x_1$ cannot be found, because $f'(1)=0$
   • D$x_1=0$, because $f(1)=-1$ and $f'(1)=-1$

   (b)

   The method is applied with $x_0=0$. Find $x_1$.

   [1 mark]

   • A$\frac13$
   • B$-\frac13$
   • C$1$
   • D$3$

   (c)

   Taking $x_0=\frac13$, find $x_1$ to 4 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\cos x-x$, where $x$ is in radians. The equation $f(x)=0$ has a root $\alpha$ near $0.7$.

   (a)

   Show that the Newton-Raphson iteration for this equation can be written as $x_{n+1}=x_n+\frac{\cos x_n-x_n}{1+\sin x_n}$.

   [3 marks]

   (b)

   Taking $x_0=1$, use the iteration to find $x_1$ and $x_2$, giving each answer to 4 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The equation $x^3-4x-2=0$ has three real roots. Let $f(x)=x^3-4x-2$ and let $\alpha$ be the largest root.

   (a)

   (i) Show that $\alpha$ lies in the interval $[2,3]$.
   (ii) Taking $x_0=2.5$, use the Newton-Raphson method to find $x_1$ and $x_2$, giving each answer to 3 decimal places.

   [6 marks]

   (b)

   A student takes $x_0=0$.
   (i) Find $x_1$ and $x_2$ to 3 decimal places.
   (ii) Describe what happens to the iteration, and explain why this starting value does not find $\alpha$.

   [6 marks]

   Total for question 4: 12 marks

End of questions