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 in a step does Euler's method calculate the gradient?

   • End of step
   • Start of step
   • Previous step
   • Midpoint of 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. Which initial value must be calculated using standard Euler's method before using the two-step improved Euler method?

   • y3
   • y2
   • y0
   • y1

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 effect does decreasing the step size h have on Euler's method approximations?

    • Causes divergence
    • Decreases accuracy
    • Increases accuracy
    • No effect

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 is used to find numerical solutions to which type of equation?

    • Quadratic algebraic equations
    • First-order differential equations
    • Linear simultaneous equations
    • Second-order differential equations

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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