Numerical Solution of Equations

The diagram shows the graph of $y = 3 - e^{-\frac{1}{2}x}$.

On the diagram, sketch the graph of $y = |5x - 4|$, and show that the equation $3 - e^{-\frac{1}{2}x} = |5x - 4|$ has exactly two real roots.

It is given that the two roots of $3 - e^{-\frac{1}{2}x} = |5x - 4|$ are denoted by $\alpha$ and $\beta$, where $\alpha < \beta$.

Show by calculation that $\alpha$ lies between 0.36 and 0.37.

Use the iterative formula $x_{n+1} = \frac{1}{5}(7 - e^{-\frac{1}{2}x_n})$ to find $\beta$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

By sketching a suitable pair of graphs on the same diagram, show that the equation

has exactly one real root.

Use the iterative formula $x_{n+1} = \sqrt[5]{e^{-\frac{1}{2}x_n}}$ to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

By sketching a suitable pair of graphs on the same diagram, show that the equation

has exactly one real root.

Use the iterative formula $x_{n+1} = \sqrt[5]{e^{-\frac{1}{2}x_n}}$ to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

By sketching a suitable pair of graphs on the same diagram, show that the equation

has exactly one root.

Verify by calculation that the root lies between 1.5 and 1.6.

Show that if a sequence of values given by the iterative formula

converges, then it converges to the root of the equation in part (a).

Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

Use the iterative formula to find the value of $\alpha$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

State an equation satisfied by $\alpha$ and hence determine the exact value of $\alpha$.

Use the iterative formula to find the value of $\alpha$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

State an equation satisfied by $\alpha$ and hence determine the exact value of $\alpha$.

Use the iterative formula to find the value of $\alpha$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

State an equation satisfied by $\alpha$ and hence determine the exact value of $\alpha$.

Find the value of $\alpha$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.

Determine the exact value of $\alpha$.

By sketching a suitable pair of graphs, show that the equation

has exactly one real root.

Use the iterative formula

to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

By sketching a suitable pair of graphs, show that the equation

has exactly one real root.

Use the iterative formula

to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.