J1 Mid-ordinate and Simpson's rules Quiz: AQA Further Maths, Unit 2

The 20 questions

1. In the mid-ordinate rule, at which points is the function evaluated for each strip?

   • At the two endpoints of each strip
   • At the left endpoint of each strip only
   • At the mean of the function values at the ends
   • At the midpoint of each strip

2. Using n strips of width h over [a, b], what is the mid-ordinate approximation to the integral?

   • h times the sum of f at the endpoints
   • (h/3) times the sum of f at the midpoints
   • (h/2) times f(a) + 2 f(x1) + ... + f(b)
   • h times the sum of f at the midpoints of the strips

3. What condition on the number of strips n is required to apply Simpson's rule?

   • n must be a multiple of 3
   • n must be a prime number
   • n must be even
   • n must be odd

4. In Simpson's rule with n strips of width h, which weight multiplies the interior ordinates with odd index?

   • 4
   • 2
   • 3
   • 1

5. In the numerical rules for integration, what does h represent?

   • The width of each strip, (b - a)/n
   • The length b - a
   • The difference f(b) - f(a)
   • The number of strips

6. Simpson's rule is exact for which class of polynomials?

   • Quadratic polynomials only
   • Polynomials of degree 3 or less
   • Linear polynomials only
   • Polynomials of degree 4 or less

7. Which functions are integrated exactly by the mid-ordinate rule?

   • No functions exactly
   • Linear functions
   • Cubic functions
   • Constant functions only

8. Use Simpson's rule with n = 2 to estimate the integral of x^2 from 0 to 2.

   • 2
   • 2.5
   • 8/3
   • 3

9. Use Simpson's rule with n = 2 to estimate the integral of x^3 from 0 to 2.

   • 4.5
   • 4
   • 3
   • 5

10. Use the mid-ordinate rule with n = 2 to estimate the integral of 1/(1 + x) from 0 to 1, to 3 decimal places.

    • 0.693
    • 0.750
    • 0.686
    • 0.861

11. Use Simpson's rule with n = 2 to estimate the integral of 1/(1 + x) from 0 to 1, to 3 decimal places.

    • 0.693
    • 0.686
    • 0.861
    • 0.750

12. Use the mid-ordinate rule with n = 2 to estimate the integral of x^2 from 0 to 2.

    • 2
    • 8/3
    • 2.5
    • 3

13. Use Simpson's rule with n = 2 to estimate the integral of e^x from 0 to 2, to 2 decimal places.

    • 7.39
    • 6.39
    • 6.00
    • 6.42

14. Use the mid-ordinate rule with n = 2 to estimate the integral of x^2 from 0 to 1.

    • 5/16
    • 1/4
    • 1/3
    • 3/8

15. Use Simpson's rule with n = 2 to estimate the integral of x^3 from 0 to 1.

    • 1/3
    • 0.5
    • 1/4
    • 3/16

16. Use Simpson's rule with n = 4 to estimate the integral of x^2 from 0 to 1.

    • 1/3
    • 0.375
    • 0.25
    • 0.3125

17. A table gives f(0) = 1, f(1) = 3 and f(2) = 5. Estimate the integral from 0 to 2 using Simpson's rule.

    • 5
    • 9
    • 4.5
    • 6

18. Strips of width 0.5 cover [0, 1.5], with mid-ordinate values 2, 3 and 4. Estimate the integral using the mid-ordinate rule.

    • 4.5
    • 9
    • 5.5
    • 3

19. How many ordinates are needed when Simpson's rule uses 4 strips?

    • 6
    • 5
    • 4
    • 3

20. Use Simpson's rule with n = 2 to estimate the integral of 1/x from 1 to 3, as an exact fraction.

    • 4/3
    • 10/9
    • 1.5
    • ln 3

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