Lesson 1.09d

1.09d Newton-Raphson method Quiz: OCR Maths, Unit 9

20 questions

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Lesson 1.09d, Newton-Raphson method: 20 multiple choice questions for the OCR Maths (H240), Unit 9: Numerical Methods, written with Revision Ninja.

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The 20 questions

  1. Which formula gives the Newton-Raphson iteration for solving f(x) = 0?

    • x(n+1) = x(n) - f'(x(n)) / f(x(n))
    • x(n+1) = f(x(n)) / f'(x(n)) - x(n)
    • x(n+1) = x(n) + f(x(n)) / f'(x(n))
    • x(n+1) = x(n) - f(x(n)) / f'(x(n))

  2. The Newton-Raphson method fails if the starting value is at what type of point?

    • Y-intercept
    • Local root
    • Stationary point
    • Point of inflection

  3. Geometrically, a Newton-Raphson step finds where which line intersects the x-axis?

    • Chord line
    • Secant line
    • Normal line
    • Tangent line

  4. What general type of numerical method is the Newton-Raphson method?

    • Analytic method
    • Iterative method
    • Integration method
    • Direct method

  5. What is f'(x) for f(x) = x^3 - 2?

    • 3x^2
    • 3x
    • 3x^3
    • x^2

  6. What condition must f'(x_n) satisfy for a Newton-Raphson step to be defined?

    • Non-zero
    • Equal to x_n
    • Positive
    • Zero

  7. What rate of convergence does the Newton-Raphson method usually exhibit near a simple root?

    • Logarithmic convergence
    • Quadratic convergence
    • Cubic convergence
    • Linear convergence

  8. Starting at x0 = 1 for f(x) = x^2 - 2, what is x1?

    • 1.414
    • 2
    • 1.5
    • 1.25

  9. Continuing from x1 = 3/2 for f(x) = x^2 - 2, what is x2 as an exact fraction?

    • 10/7
    • 7/5
    • 17/12
    • 3/2

  10. For f(x) = x^3 - x - 1 with x0 = 1.5, what is x1 to 4 decimal places?

    • 1.3478
    • 1.2500
    • 1.6522
    • 1.3750

  11. For f(x) = cos x - x with x0 = 1, what is x1 to 3 decimal places?

    • 0.841
    • 0.540
    • 0.750
    • 1.250

  12. For f(x) = x^3 - 3x + 2, why does Newton-Raphson fail at x0 = 1?

    • f(1) = 0
    • f'(1) = 3
    • f''(1) = 0
    • f'(1) = 0

  13. For f(x) = x^2 - 10 with x0 = 3, what is x1 as an exact fraction?

    • 3
    • 19/6
    • 7/2
    • 10/3

  14. For f(x) = 1/x - 2, what is x(n+1) in terms of x(n)?

    • x(n)/2 + x(n)^2
    • 1/(2x(n))
    • 2x(n) + 2x(n)^2
    • 2x(n) - 2x(n)^2

  15. For f(x) = 2x^3 - 5x - 3 with x0 = 2, what is x1 as an exact fraction?

    • 37/19
    • 19/35
    • 35/19
    • 2/19

  16. Applying Newton-Raphson to x^2 - a = 0 gives which recurrence?

    • x(n+1) = a/x(n)
    • x(n+1) = (x(n) - a/x(n)) / 2
    • x(n+1) = x(n) + a/x(n)
    • x(n+1) = (x(n) + a/x(n)) / 2

  17. For f(x) = x^3 - 2x + 2 with x0 = 0, what is x1?

    • 0.5
    • -1
    • 2
    • 1

  18. For the same f(x) = x^3 - 2x + 2 with x0 = 0, what is x2?

    • 1
    • -1
    • 0
    • 2

  19. If successive Newton-Raphson iterates alternate endlessly between two values, what is happening?

    • It is converging
    • It is cycling
    • It is integrating
    • It reaches infinity

  20. How does the error behave at each step during quadratic convergence in Newton-Raphson?

    • It doubles
    • It roughly squares
    • It is halved
    • It stays constant

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