Iterative methods

Figure 1 shows a sketch of the curves with equations y = f(x) and y = g(x) where

Find

f′(x)

g′(x)

Given that f′(x) = g′(x) at x = α

show that α satisfies the equation

The iterative formula

is used withx₁ = 0.6 to find an approximate value for α

Calculate, giving each answer to 4 decimal places,

the value of x₂

the value of α

A curve has equation y = f(x), where

Show that

where A and B are constants to be found.

Hence show that the x coordinates of the turning points of the curve are solutions of the equation

The equation x has two positive roots α and β where β α

A student uses the iteration formula

in an attempt to find approximations for α and β Diagram 1 shows a plot of part of the curve with equation y and part of the line with equation y = x

Using Diagram 1 on page 42

draw a staircase diagram to show that the iteration formula starting withx₁ = 1 can be used to find an approximation for β

Use the iteration formula with x₁ = 1, to find, to 3 decimal places,

the value of x₂

the value of β

Using a suitable interval and a suitable function that should be stated

show thatα = 0.432 to 3 decimal places.

Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

Diagram 1
copy of Diagram 1

The curve with equation y = f (x) where

has a single turning point at x = α

Show that α is a solution of the equation

The iterative formula

is used to find an approximate value for α.

Starting with x₁ = 0.3

calculate, giving each answer to 4 decimal places,

the value of x₂

the value of x₄

Using a suitable interval and a suitable function that should be stated,

show that α is 0.341 to 3 decimal places.

The curve with equation y = f (x) where

has a single turning point at x = α

Show that α is a solution of the equation

The iterative formula

is used to find an approximate value for α. Starting with x₁ = 0.3

calculate, giving each answer to 4 decimal places,

the value of x₂

the value of x₄ Using a suitable interval and a suitable function that should be stated,

show that α is 0.341 to 3 decimal places.

Figure 1 shows a sketch of the curve C with equation

Show that

The point P, shown in Figure 1, is the minimum turning point on C.

Show that the x coordinate of P is a solution of

Use the iteration formula

to find (i) the value of x₂ to 5 decimal places,

the x coordinate of P to 5 decimal places.

Figure 8 shows a sketch of the curve C with equation y = x^x, x 0

Find, by firstly taking logarithms, the x coordinate of the turning point of C. ((6ROXWLRQV EDVHG HQWLUHO\ RQ JUDSKLFDO RU QXPHULFDO PHWKRGV DUH QRW DFFHSWDEOH ))

The point P(, 2) lies on C.

Show that 1.5 < < 1.6

A possible iteration formula that could be used in an attempt to find α is

Using this formula with x₁ = 1.5

find x₄ to 3 decimal places,

describe the long-term behaviour of x_n

Figure 1 is a sketch showing part of the curve with equation y = 2^x+1 – 3 and part of the line with equation y = 17 – x. The curve and the line intersect at the point A.

Show that the x coordinate of satisfies the equation

Use the iterative formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 3 decimal places.

Use your answer to part (b) to deduce the coordinates of the point A, giving your answers to one decimal place.

A curve C has equation y = e^4x + x⁴ + 8x + 5

Show that the x coordinate of any turning point of C satisfies the equation

On the axes given on page 5, sketch, on a single diagram, the curves with equations

y = –2 – e^4x On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes.

Explain how your diagram illustrates that the equation x³ = –2 – e^4x has only one root.

The iteration formula

can be used to find an approximate value for this root.

Calculate the values of x₁ and x₂, giving your answers to 5 decimal places.

Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

Figure 2 shows a sketch of part of the curve with equation

The curve crosses the x-axis at the point Q and has a minimum turning point at R.

Show that the x coordinate of Q lies between 2.1 and 2.2

Show that the x coordinate of R is a solution of the equation

Using the iterative formula

find the values of x₁ and x₂ to 3 decimal places.

f(x) = 25x²e^2x – 16, x

Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x).

Show that the equation f(x) = 0 can be written as x = ± e^–x The equation f(x) = 0 has a root , where = 0.5 to 1 decimal place.

Starting with x₀ = 0.5, use the iteration formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 3 decimal places.

Give an accurate estimate for to 2 decimal places, and justify your answer.

The curve cuts the x-axis at points A and B as shown in Figure 2.

Calculate the x coordinate of A and the x coordinate of B, giving your answers to 3 decimal places. (2)

Find f/(x). (3) The curve has a minimum turning point at the point P as shown in Figure 2.

Show that the x coordinate of P is the solution of

Use the iteration formula

to calculate the values of x), x. and x3, giving your answers to 3 decimal places. (3)

The x coordinate of P is a.

g(x) = e^x1 + x − 6

Show that the equation g(x) = 0 can be written as

The root of g(x) 0 is . The iterative formula

is used to find an approximate value for .

Calculate the values of x₁, x₂ and x₃ to 4 decimal places.

By choosing a suitable interval, show that = 2.307 correct to 3 decimal places.

f (x) = x³ + 3x² + 4x −12

Show that the equation f (x) = 0 can be written as

The equation x³ + 3x² + 4x −12 = 0 has a single root which is between 1 and 2

Use the iteration formula

with x₀ = 1 to find, to 2 decimal places, the value of x₁, x₂ and x₃.

The root of f (x) = 0 is .

By choosing a suitable interval, prove that α 1 272. to 3 decimal places.

f (x) = 2sinn(x²) + x − 2, 0 x 2

Show that f (x) = 0 has a root between x = 0.75 and x = 0.85

The equation f (x) = 0 can be written as x = arcsin 1− 0 5. x¹².

Use the iterative formula₁

to find the values of x₁, x₂ and x₃, giving your answers to 5 decimal places.

Show that = 0.80157 is correct to 5 decimal places.

f (x) = 4cosec x 4x +1, where x is in radians.

Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3].

Show that the equation f (x) = 0 can be written in the form

Use the iterative formula

to calculate the values of x₁, x₂ and x₃, giving your answers to 4 decimal places.

By considering the change of sign of f (x) in a suitable interval, verify that α = 1.291 correct to 3 decimal places.

Show that f(x) = 0 can be rearranged as

The equation f(x) = 0 has one positive root α.

Taking x₁ = 0, find, to 3 decimal places, the values of x₂, x₃ and x₄.

Show that α = 2.057 correct to 3 decimal places.

Figure 1 shows part of the curve with equation y x³ + 2x² 2, which intersects the x-axis at the point A where x = α. To find an approximation to α, the iterative formula

is used.

Taking x₀ = 2.5, find the values of x₁, x₂, x₃ and x₄. Give your answers to 3 decimal places where appropriate.

Show that α = 2.359 correct to 3 decimal places.

f (x) 3xe^x 1 The curve with equation y = f (x) has a turning point P.

Find the exact coordinates of P.

The equation f (x) = 0 has a root between x = 0.25 and x = 0.3

Use the iterative formula

with x₀ = 0.25 to find, to 4 decimal places, the values of x₁, x₂ and x₃.

By choosing a suitable interval, show that a root of f (x) = 0 is x = 0.2576 correct to 4 decimal places.