C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

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

(a) Show that there is a root of f (x) = 0 in the interval [1.2, 1.3]. (2)

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

41

sin1

+=x

x

(2)

(c) Use the iterative formula

,25.1,41

sin1

01 =+=+ xxx

nn

to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places. (3)

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

(2) (Total 9 marks)

2. f(x) = x3 + 2x2 3x 11

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

++

=2113

xxx , .2x

The equation f(x) = 0 has one positive root . (2)

Edexcel Internal Review 1

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The iterative formula

++

=+ 2113

1n

nn x

xx is used to find an approximation to .

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

(c) Show that = 2.057 correct to 3 decimal places. (3)

(Total 8 marks)

3.

The diagram above shows part of the curve with equation y = x3 + 2x2 + 2, which intersects the x-axis at the point A where x = .

To find an approximation to , the iterative formula

2

)(2

21 +=+n

n xx

is used.

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

(3)

Edexcel Internal Review 2

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(b) Show that = 2.359 correct to 3 decimal places. (3)

(Total 6 marks)

4. f(x) = 3xex 1

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

(a) Find the exact coordinates of P. (5)

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

(b) Use the iterative formula

nxnx

+ = e31

1

with x0 = 0.25 to find, to 4 decimal places, the values of x1, x2 and x3. (3)

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

(3) (Total 11 marks)

5. f(x) = 3x3 2x 6

(a) Show that f(x) = 0 has a root, , between x = 1.4 and x = 1.45 (2)

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

0,322

+= x

xx .

(3)

Edexcel Internal Review 3

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(c) Starting with x0 = 1.43, use the iteration

+=+ 3

221

nn x

x

to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places. (3)

(d) By choosing a suitable interval, show that = 1.435 is correct to 3 decimal places. (3)

(Total 11 marks)

6. f(x) = ln(x + 2) x + 1, x > 2, x .

(a) Show that there is a root of f(x) = 0 in the interval 2 < x < 3. (2)

(b) Use the iterative formula

xn+1 = 1n(xn + 2) + 1, x0 = 2.5

to calculate the values of x1, x2 and x3 giving your answers to 5 decimal places. (3)

(c) Show that x = 2.505 is a root of f(x) = 0 correct to 3 decimal places. (2)

(Total 7 marks)

Edexcel Internal Review 4

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7.

O

y

x

P

4

The figure above shows part of the curve with equation

y = (2x 1) tan 2x, 0 x < 4

The curve has a minimum at the point P. The x-coordinate of P is k.

(a) Show that k satisfies the equation

4k + sin 4k 2 = 0. (6)

The iterative formula

,3.0),4sin2(41

01 ==+ xxx nn

is used to find an approximate value for k.

(b) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places. (3)

(c) Show that k = 0.277, correct to 3 significant figures. (2)

(Total 11 marks)

Edexcel Internal Review 5

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8.

f(x) = 2x3 x 4.

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

+=

212

xx

(3)

The equation 2x3 x 4 = 0 has a root between 1.35 and 1.4.

(b) Use the iteration formula

+=+ 2

121 x

xn ,

with x0 = 1.35, to find, to 2 decimal places, the values of x1, x2 and x3. (3)

The only real root of f(x) = 0 is .

(c) By choosing a suitable interval, prove that = 1.392, to 3 decimal places. (3)

(Total 9 marks)

9. f(x) = 3ex 21 ln x 2, x > 0.

(a) Differentiate to find f (x). (3)

The curve with equation y = f(x) has a turning point at P. The x-coordinate of P is .

(b) Show that = 61 e.

(2)

Edexcel Internal Review 6

C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

The iterative formula

xn + 1 = nxe61 , x0 = 1,

is used to find an approximate value for .

(c) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places. (2)

(d) By considering the change of sign of f (x) in a suitable interval, prove that = 0.1443 correct to 4 decimal places.

(2) (Total 9 marks)

10.

y

xO A

C

B

f(x) = x2

1 1 + ln 2x , x > 0.

The diagram above shows part of the curve with equation y = f(x). The curve crosses the x-axis at the points A and B, and has a minimum at the point C.

(a) Show that the x-coordinate of C is 21 .

(5)

(b) Find the y-coordinate of C in the form k ln 2, where k is a constant. (2)

(c) Verify that the x-coordinate of B lies between 4.905 and 4.915. (2)

Edexcel Internal Review 7

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(d) Show that the equation x2

1 1 + ln

2x

= 0 can be rearranged into the form x =

( )x- 211e2 . (2)

The x-coordinate of B is to be found using the iterative formula

xn + 1 = ( )

nx- 2

11e2 , with x0 = 5.

(e) Calculate, to 4 decimal places, the values of x1, x2 and x3. (2)

(Total 13 marks)

11.

f(x) = x3 2 x1

, x 0.

(a) Show that the equation f(x) = 0 has a root between 1 and 2. (2)

An approximation for this root is found using the iteration formula

xn + 1 = 31

12

+

nx, with 0x = 1.5.

(b) By calculating the values of x1, x2, x3 and x4, find an approximation to this root, giving your answer to 3 decimal places.

(4)

(c) By considering the change of sign of f(x) in a suitable interval, verify that your answer to part (b) is correct to 3 decimal places.

(2) (Total 8 marks)

Edexcel Internal Review 8

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12.

f(x) = x3 + x2 4x 1.

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

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

x =

++114

xx , x 1.

(2)

The iterative formula xn + 1 =

++114

n

n

xx

is used to find an approximation to .

(b) Taking x1 = 1, find, to 2 decimal places, the values of x2, x3 and x4. (3)

(c) By choosing values of x in a suitable interval, prove that = 1.70, correct to 2 decimal places.

(3)

(d) Write down a value of x1 for which the iteration formula xn + 1 =

++114

n

n

xx

does not

produce a valid value for x2.

Justify your answer. (2)

(Total 10 marks)

13. (a) Sketch, on the same set of axes, the graphs of

y = 2 ex and y = x.

[It is not necessary to find the coordinates of any points of intersection with the axes.] (3)

Edexcel Internal Review 9

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Given that f(x) = ex + x 2, x 0,

(b) explain how your graphs show that the equation f(x) = 0 has only one solution, (1)

(c) show that the solution of f(x) = 0 lies between x = 3 and x = 4. (2)

The iterative formula xn + 1 = (2 nxe )2 is used to solve the equation f(x) = 0.

(d) Taking x0 = 4, write down the values of x1, x2, x3 and x4, and hence find an approximation to the solution of f(x) = 0, giving your answer to 3 decimal places.

(4) (Total 10 marks)

14. The curve with equation y = ln 3x crosses the x-axis at the point P (p, 0).

(a) Sketch the graph of y = ln 3x, showing the exact value of p. (2)

The normal to the curve at the point Q, with x-coordinate q, passes through the origin.

(b) Show that x = q is a solution of the equation x2 + ln 3x = 0. (4)

(c) Show that the equation in part (b) can be rearranged in the form x = 2

e31x .

(2)

(d) Use the iteration formula xn + 1 = 2

e31 nx , with x0 = 3

1 , to find x1, x2, x3 and x4. Hence write down, to 3 decimal places, an approximation for q.

(3) (Total 11 marks)

Edexcel Internal Review 10

C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

15.

x

y

O A

B

The diagram above shows a sketch of the curve with equation y = f(x) where the function f is given by

f: x ex 2 1, x .

The curve meets the x-axis at the point A and the y-axis at the point B.

(a) Write down the coordinates of A and B. (2)

(b) Find, in the form f 1(x): x . . ., the inverse func