J2-J3 Euler's method and the improved Euler method Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which formula gives the next value in Euler's step by step method for dy/dx = f(x, y)?

   • y(r+1) = y(r) + f(x(r), y(r))
   • y(r+1) = y(r) + h f(x(r), y(r))
   • y(r+1) = y(r) - h f(x(r), y(r))
   • y(r+1) = h y(r) + f(x(r), y(r))

2. In Euler's method, how is x(r+1) found from x(r)?

   • x(r+1) = x(r)/h
   • x(r+1) = x(r) + 1
   • x(r+1) = h x(r)
   • x(r+1) = x(r) + h

3. At which point does Euler's method take the gradient for each step?

   • At the average of all steps
   • At the start of the step
   • Always at x = 0
   • At the end of the step

4. The improved Euler method, as given in the specification, uses which two-step formula?

   • y(r+1) = y(r-1) + h f(x(r), y(r))
   • y(r+1) = y(r-1) + 2h f(x(r), y(r))
   • y(r+1) = y(r) + h f(x(r), y(r))
   • y(r+1) = y(r) + 2h f(x(r+1), y(r+1))

5. To start the improved Euler method, which extra value is needed beyond y(0)?

   • The exact solution at x = 1
   • dy/dx at x = 0 only
   • y(-1), found by reversing the equation
   • y(1), usually found by Euler's method

6. For dy/dx = x + y with y(0) = 1 and h = 0.1, what does Euler's method give for y(0.2)?

   • 1.2428
   • 1.22
   • 1.24
   • 1.1

7. For dy/dx = 2x with y(0) = 0 and h = 0.5, what does Euler's method give for y(1)?

   • 0
   • 0.5
   • 1
   • 0.25

8. For dy/dx = x + y with y(0) = 1 and h = 0.1, what does the improved Euler method give for y(0.2)?

   • 1.2428
   • 1.22
   • 1.24
   • 1.1

9. For dy/dx = y with y(0) = 1 and h = 0.1, what does Euler's method give for y(0.2)?

   • 1.1
   • 1.22
   • 1.21
   • 1.2

10. For dy/dx = 2x with y(0) = 0 and h = 1, what does the improved Euler method give for y(2)?

    • 3
    • 4
    • 0
    • 2

11. For dy/dx = -2y with y(0) = 1 and h = 0.1, what does Euler's method give for y(0.2)?

    • 0.64
    • 0.8
    • 0.4
    • 0.6703

12. For dy/dx = x + y with y(0) = 1 and h = 0.5, what does Euler's method give for y(1)?

    • 3.0
    • 3.44
    • 2.5
    • 2.0

13. For the same equation dy/dx = x + y with y(0) = 1 and h = 0.5, what does the improved Euler method give for y(1)?

    • 2.0
    • 3.0
    • 3.44
    • 2.5

14. What is the effect of using a smaller step size h in Euler's method?

    • It changes the sign of y
    • It makes the results worse
    • It generally improves accuracy, bringing the values closer to the exact solution
    • It has no effect on accuracy

15. For dy/dx = 2y with y(0) = 1 and h = 0.2, what does Euler's method give for y(0.4)?

    • 1.4
    • 2.0
    • 1.96
    • 1.8

16. For dy/dx = 2y with y(0) = 1 and h = 0.2, what does the improved Euler method give for y(0.4)?

    • 2.2255
    • 2.12
    • 1.96
    • 2.0

17. For dy/dx = x^2 with y(0) = 0 and h = 1, what does Euler's method give for y(2)?

    • 4
    • 8/3
    • 1
    • 2

18. For dy/dx = x^2 with y(0) = 0 and h = 1, what does the improved Euler method give for y(2)?

    • 4
    • 2
    • 8/3
    • 1

19. Euler's method gives approximate solutions to which kind of problem?

    • An integral with no differential equation
    • A first order differential equation with a given initial condition
    • A system of linear equations
    • A second order differential equation with no initial condition

20. For dy/dx = 3x^2 with y(0) = 0 and h = 1, what does Euler's method give for y(2)?

    • 3
    • 8
    • 0
    • 6

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