Locating roots by change of sign

Describe a sequence of two transformations which maps the graph with equation

onto the graph with equation

State the equation of the vertical asymptote of the graph with equation

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

The diagram shows a sector of a circle OAB.

The point C lies on OB such that AC is perpendicular to OB.

Angle AOB is radians.

Given the area of the triangle OAC is half the area of the sector OAB, show that

Use a suitable change of sign to show that a solution to the equation

lies in the interval given by

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

Using ₁ as a first approximation for apply the Newton-Raphson method twice

Give your answer to three decimal places.

Explain how a more accurate approximation for can be found using the Newton-Raphson method.

Explain why using ₁ as a first approximation in the Newton-Raphson method does not lead to a solution for .

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.

The function h is defined by

where h has its maximum possible domain.

Find the domain of h.

Give your answer using set notation.

Alice correctly calculates

She then argues that since there is a change of sign there must be a value of x in the interval 1 x 4 that gives h(x) 0

Explain the error in Alice’s argument.

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.