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  • 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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    (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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    (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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    (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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    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

  • C3 Numerical Methods - Iterative equations PhysicsAndMathsTutor.com

    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