Specialist Maths — Question 13

QCAA 2021, Paper 2 · 6 marks

Q13 · 2021 · Technology-activeSimple familiar6 marks

QUESTION 13 (6 marks)

The area under the graph of the function

f(x)=0.2e^{-0.2x}

for

1\le x\le9

is shaded.

Use Simpson’s rule with four intervals to determine an approximation for this area.

[4 marks]

Use a calculus approach to evaluate the reasonableness of your area approximation from a).

[2 marks]

WORKED SOLUTION

6 marks

ANSWER

Simpson approximation

\approx0.6535

; exact area

\approx0.6534

, so the approximation is reasonable.

Worked solution

a) With four intervals,

h=2

. Using

x=1,3,5,7,9

, Simpson’s rule gives approximately

0.6535

b) The exact area is

\int_1^9 0.2e^{-0.2x}dx=[-e^{-0.2x}]_1^9=e^{-0.2}-e^{-1.8}\approx0.6534.

The values are very close, so the Simpson approximation is reasonable.

correctly determines the interval width required for the calculations

[1 mark]

correctly determines all required values of

f(x)

[1 mark]

substitutes values into Simpson’s rule

[1 mark]

approximates the area using Simpson’s rule

[1 mark]

correctly expresses the exact area as a definite integral

[1 mark]

calculates the value of the definite integral and provides a suitable comment on the reasonableness of the response

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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Compare your working with the guide above.