Mathematics and Maximum Number

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Topic assessment

1. Use the midpoint rule, the trapezium rule and Simpson’s rule to approximate the integral of the function f between and , using the values of f given in the following table. In each case use the maximum number of strips that you can with the given information.

x
1
1.4
1.8
2.2
2.6
3.0
3.4
3.8
4.2
4.6
5.0
f(x)
23
26
28
27
28
32
56
78
90
94
99

(15 marks)

2. Let f (x) be a function and consider . Let be the approximations given by the midpoint rule, the trapezium law and Simpson’s rule respectively all with n strips.

(i) Write down a formula which gives in terms of and .

(ii) Write down a formula which gives in terms of and .

(iii) Calculate and for the integral .

(iv) Calculate directly for the same integral.

(v) Verify that your answers to (iii) and (iv) agree with the formula you gave in (ii).

(20 marks)

3. Consider the integral .

(i) What is the exact value of this integral?

Let .

(ii) Calculate .

(iii) Compare your answers to (ii) to your answer to (i). Calculate the absolute error in each case. (iv) Complete the following table.

No of strips, to calculate Mn and Tn
4
8
10
Strip Width, h
1.25
0.625
0.5
Absolute Error, in

(v) From your table, to which power of h would you say the absolute error is proportional?

(25 marks)
Topic assessment Solutions

1. The midpoint rule is:

The maximum number of strips we can use with the available information is 5, each of width .

The trapezium rule is:

The maximum number of trapezia we can use with the available information is 10, each of width .

Simpson’s rule is:

The maximum number of strips is 10, each of width .

2. (i)

(ii)

(iii)

(iv)

(v)

3. (i)

(ii)

(iii)
Absolute error in

(iv)

(v)The row looks...
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