C3 Revision and Exam Answers: Simpson s Rule

Transcription

1 C3 Revision and Exam Answers: Simpson s Rule Simpson s Rule is a way of accurately finding the area under a curve It is more accurate than the Trapezium Rule which we have seen before You start it the same way as you would start the trapezium rule questions which you have seen in C2 Reminder: Trapezium Rule WJEC C2, 27 May Use the Trapezium Rule with five ordinates to find an approximate value for the integral d Show your working and give your answer correct to three decimal places [4] First draw five dots (because there are five ordinates) and label the first dot 1, and the last dot 2 (because they are the integral numbers) Then find the numbers of the three dots in the middle so that there is an equal difference between each of the numbers In the case of this question, the values of will be 1, 125, 15, 175 and 2 Then substitute the values of into the formula given in the question, to get values for,,, and where gives the value for, gives the value for, etc Finally, substitute all these values into the Trapezium Rule formula: d, where is the difference between the ordinates, Then round this value to 3dp When, When, When, When, When, d dp

2 The Simpson s Rule is very similar but the formula is slightly different: d Where is the difference between ordinates Put simply add the first and the last Add the odd s etc and multiply by Add the even s etc and multiply by Use each ONLY ONCE Add these answers and multiply by where is again the difference between the ordinates Exam Questions: Simpson s Rule WJEC C3, 1 June Use Simpson s Rule with five ordinates to find an approximate value for Show your working and give your answer correct to four decimal places [4] Again, draw the five dots (because there are five ordinates) and label the first dot 1, and the last dot 18 (because they are the integral numbers) Next find the numbers of the three middle ordinates so that there is an equal difference between each of the numbers In the case of this question, the values of will be 1, 12, 14, 16 and 18 Then substitute the values of into the formula given in the question, to get values for,,, and where gives the value for, gives the value for, etc Finally, substitute all these values into the Simpson's Rule formula: d where is the difference between the ordinates, Then round this value to 4dp d When, When, When, When, When, d ( ) dp

3 WJEC C3, 19 January Use Simpson s Rule with five ordinates to find an approximate value for the integral d Show your working and give your answer to three decimal places [4] When, When, When, d {( When, When, ) ( ) ( )} dp Harder Exam Questions: Simpson s Rule Some exam questions have a part which require you to use your answer to part to deduce an approximate value for a similar integral WJEC C3, 12 January Use Simpson s Rule with five ordinates to find an approximate value for the integral ln Show your working and give your answer to three decimal places [4] We can use the method previously explained to get the answer to this part of the question, dp d Use your answer to part to deduce an approximate value for ln d We know that ln is the same as log, where is the natural logarithm, and therefore all the usual logarithm rules apply Therefore, using the logarithm rule log log, which we know from C2, ln ln ln So all we have to do to answer part is multiply the part answer by dp [1]

4 WJEC C3, 20 Jan Use Simpson s Rule with five ordinates to find an approximate value for the integral cos d Show your working and give your answer to four decimal places [4] Although this question seems harder because it has an upper limit of, the same method still applies However because we are dealing with cos and, you have to make sure to work out the answer with the calculator set to radians instead of degrees The final answer for this part should come to dp Use your answer to part to deduce an approximate value for the integral sin d The first thing we should do is find a formula linking cos and sin Since we are trying to work out the sin integral, we will use the formula sin cos, which we again should be familiar with from C2 As sin cos, sin d d cos d [2] By integrating normally, it is clear that d = This can be proven by the following; the difference between each ordinate will be same as which was used in part ), this is because we must again use five ordinates with the same limits All the values will be 1, as there is no part to the integral, therefore nothing changes When we substitute these values into the Simpson s Rule formula we get: (the d Now simply substitute all the values we know into the sin d d cos d formula to get an answer sin d

5 Exam Paper Answers WJEC C3, Specimen Paper 2005/2006, Question 2 d dp WJEC C3, 16 June 2005, Question 1 d dp WJEC C3, 12 January 2006, Question 1 d dp WJEC C3, 24 May 2006, Question 1 ln d dp WJEC C3, 12 January 2007, Question 1 ln d dp ln d dp WJEC C3, 5 June 2007, Question 1 ln d dp WJEC C3, 11 Jan 2008, Question 1 d dp WJEC C3, 23 May 2008, Question 1 d dp WJEC C3, 15 Jan 2009, Question 1 ln cos d dp ln cos d dp WJEC C3, 1 June 2009, Question 1 d dp

6 WJEC C3, 20 January 2010, Question 1 ln d dp WJEC C3, 9 June 2010, Question 1 d dp WJEC C3, 19 January 2011, Question 1 d dp WJEC C3, 26 May 2011, Question 1 ln d dp ln ( ) d dp WJEC C3, 20 January 2012, Question 1 cos d dp sin d dp WJEC C3, 1 June 2012, Question 1 d dp d dp WJEC C3, 23 January 2013, Question 1 d dp WJEC C3, 24 May 2013, Question 1 ln d dp ln d dp WJEC C3, 22 January 2014, Question 1 tan d dp sec d dp