Iterative methods

A curve has equation

The point Q has coordinates (3, 2.5)

The point P on the curve which is closest to the point Q is shown on the diagram below.

Show that the x-coordinate of P satisfies the equation

The Newton–Raphson method is to be used to find an approximate solution to the equation

Show that the Newton–Raphson method generates the iterative formula

Starting with x₀ = 3, use the iterative formula given in part (b) to find the value of x₃

Give your answer to three decimal places.

Hence find the distance PQ

Give your answer to two decimal places.

The equation x 3 = e 6 – 2 x

has a single solution, x = α

By considering a suitable change of sign, show that α lies between 0 and 4

Show that the equation x3 = e6–2x can be rearranged to give

Explain why the iterative formula

fails to converge to α when the starting value is x₁ = 0

2 3 7 The equation x x x 3 has a single solution, x

By considering a suitable change of sign, show that lies between 1.5 and 1.6

Show that the equation x² x³ x 3 can be rearranged into the form

Hence, deduce an interval of width 0.001 in which lies.

equation

has exactly one solution for x 0

By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6

Show that the equation can be rearranged to give

Use the iterative formula

with x 04, to find x, x and x, giving your answers to four decimal places. 1 2 3 4

On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of x, x and x.

The daily world production of oil can be modelled using

where V is volume of oil in millions of barrels, and t is time in years since 1 January 1980.

The model is used to predict the time, T, when oil production will fall to zero.

Show that T satisfies the equation

values of T, T, and T, giving your answers to three decimal places. 1 2 3

Explain the relevance of using T₀ 38

From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as

Use the models to show that the country’s use of oil and the world production of oil will be equal during the year 2029.