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Numerical Methods Practice Problems with Solutions - A Level Year 13

A Level Year 13 | Math | Numerical Methods
Name: _________________________ Date: _________________________ Score: _______ / 40

The iteration formula xₙ₊₁ = f(xₙ) is used to find a root of an equation, and the Newton–Raphson method uses the derivative of the function, so this Year 13 sheet opens with iterative root-location tools. A sign change answer records the existence of a root between a and b.

Key Takeaways

Section A: Multiple Choice Questions (1 Mark Each)

Choose the correct option for each question.
1. The iteration formula xₙ₊₁ = f(xₙ) is used to
(a) find a root of an equation(b) find the maximum of a function(c) differentiate a function(d) integrate a function
2. The Newton–Raphson method uses
(a) the derivative of the function(b) the integral of the function(c) the second derivative only(d) random numbers
3. If f(a) and f(b) have opposite signs, then
(a) there is at least one root between a and b(b) there is no root(c) there are exactly two roots(d) the function is undefined
4. The Newton–Raphson iteration formula is
(a) xₙ₊₁ = xₙ − f(xₙ) ÷ f'(xₙ)(b) xₙ₊₁ = xₙ + f(xₙ) ÷ f'(xₙ)(c) xₙ₊₁ = f(xₙ) ÷ f'(xₙ)(d) xₙ₊₁ = f(xₙ)
5. A sign change between f(a) and f(b) shows that
(a) a root lies in the interval [a, b](b) the function is increasing(c) no root exists(d) the function is constant
6. Numerical methods are used when
(a) an equation cannot be solved algebraically(b) the answer is negative(c) only one root exists(d) the function is linear

Section B: Short Answer Type Questions (2 Marks Each)

Show all steps clearly.
7. What is an iteration formula?
8. What does a sign change of a function indicate?
9. State the Newton–Raphson formula.
10. What does the Newton–Raphson method require?
11. When are numerical methods used?
12. What is a root of an equation?
13. Why might the Newton–Raphson method fail?
14. Give one example of an equation that needs to be solved numerically.

Section C: Numericals & Word Problems (3 Marks Each)

Apply the concepts to solve the problems. Show all working.
15. For a function, f(1) = −1 and f(2) = 3. What can you conclude about a root of f(x) = 0?
16. Using the iteration xₙ₊₁ = √(xₙ + 1) starting with x₀ = 1, find x₁.
17. Using the iteration xₙ₊₁ = xₙ + 1 starting with x₀ = 2, find x₁.
18. Use one Newton–Raphson step with x₀ = 1 for f(x) = x² − 2. First find f(1) and f'(x) = 2x, then use x₁ = x₀ − f(x₀) ÷ f'(x₀) to find x₁.
19. For a function, f(3) = 2 and f(4) = −1. What can you conclude about a root?
20. Using the iteration xₙ₊₁ = 1 + 1 ÷ xₙ starting with x₀ = 1, find x₁.

Answer Key

1. a) find a root of an equation
2. a) the derivative of the function
3. a) there is at least one root between a and b
4. a) xₙ₊₁ = xₙ − f(xₙ) ÷ f'(xₙ)
5. a) a root lies in the interval [a, b]
6. a) an equation cannot be solved algebraically
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet

People Also Ask

What is the iteration formula xₙ₊₁ = f(xₙ) used to find?

A root of an equation, option (a) in this worksheet (Q1).

What does the Newton–Raphson method use?

The derivative of the function, option (a) in this worksheet (Q2).

What does a change of sign guarantee?

At least one root between a and b, option (a) in this worksheet (Q3).

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