Iterative methods Quiz

1. $x_{n+1}=\sqrt{3x_n+2}$, $x_0=1$. Find $x_2$ to 3 decimal places.

• A. $2.951$
• B. $2.236$
• C. $3.000$
• D. $8.708$

2. A sequence $x_{n+1}=g(x_n)$ converges to $L$. Which statement is true?

• A. $L=g(0)$
• B. $L=g(L)$
• C. $L=x_0$
• D. $g(L)=0$

3. Which is a correct rearrangement of $x^3-x-1=0$?

• A. $x=\sqrt[3]{x-1}$
• B. $x=x^3+1$
• C. $x=\sqrt[3]{x}+1$
• D. $x=\sqrt[3]{x+1}$

4. $f(x)=x^3-5x+1$ is continuous, with $f(2)=-1$ and $f(3)=13$. Which conclusion is justified?

• A. The root is nearer 2 because $|f(2)|<|f(3)|$
• B. There are no roots
• C. $x=2.5$ is a root
• D. A root lies between 2 and 3

5. $x_{n+1}=\frac12\left(x_n+\frac{6}{x_n}\right)$, $x_0=2$. Find $x_2$.

• A. $2.500$
• B. $2.450$
• C. $2.449$
• D. $2.400$