Iteration Practice Questions

Show that the equation $x^3 - 5x + 1 = 0$ has a solution between $x = 0$ and $x = 1$.

Show that the equation $x^3 - 5x + 1 = 0$ can be rearranged to give
$$x = \frac{x^3 + 1}{5}$$

Find the values of $x_1$, $x_2$ and $x_3$.

Give your answers correct to 4 decimal places.

Hence write down a solution of $x^3 + 2x = 9$ correct to 2 decimal places.

Show that one run through the process gives $x_1 = 4.5$

Run the process twice more to find the value of $x_3$.

Give your answer correct to 4 decimal places.

Write down, in the form $\sqrt{a}$, the number that Priya's process is finding an estimate for.

Show that the equation $2x^3 + x - 8 = 0$ can be rearranged to give
$$x = \sqrt[3]{\frac{8 - x}{2}}$$

Starting with $x_0 = 1.5$, use the iteration formula $x_{n+1} = \sqrt[3]{\dfrac{8 - x_n}{2}}$ three times to find an estimate for the solution of $2x^3 + x - 8 = 0$.

Give your answer correct to 4 decimal places.

By substituting your answer to part (b) into $2x^3 + x - 8$, comment on the accuracy of your estimate for the solution of $2x^3 + x - 8 = 0$.

Find the values of $x_1$, $x_2$ and $x_3$.

Give your answers correct to 3 decimal places.

Explain the relationship between the values of $x_1$, $x_2$ and $x_3$ and the equation
$$x^3 + 3x^2 - 10 = 0$$

Show that the equation $x^3 + 2x = 40$ has a solution between $x = 3$ and $x = 4$.

Ravi starts with the interval from $x = 3$ to $x = 4$.

Carry out Ravi's process twice and write down the interval he has after the second time.

Show that $f(x) = 0$ has a solution between $x = 1.4$ and $x = 1.5$.

Show that the equation $x^3 + 4x^2 - 11 = 0$ can be rearranged to give
$$x = \sqrt{\frac{11}{x + 4}}$$

Starting with $x_0 = 1.5$, use the iteration formula $x_{n+1} = \sqrt{\dfrac{11}{x_n + 4}}$ to find a solution of $x^3 + 4x^2 - 11 = 0$ correct to 3 decimal places.

Hence find a solution of the equation $8y^3 + 16y^2 - 11 = 0$, correct to 3 decimal places.