Question

Task 3 - Simpson's Rule (Maximum Mark 15) Find the equation the quadratic y = ax? + bx + c which passes through the points AC-h, Y.), B(0,y), and C(h.yc). Use your quadratic to find an expression for the area under the quadratic between the points A and C. Explaining carefully how your result can be used to prove the general formula for Simpson's Rule. Notes: . You will need to research independently to find the formula for Simpson's Rule. You must show all your working, in particular the algebraic steps to derive the formula. • This part of the assignment is testing that you can do independent research, solve simultaneous equations, do simple integration and use your result to derive further formulae. Marking Criteria Topic Finding Quadratic Poor 0 - 1 mark Just substitutes values. Finding Area Basic attempt with minimally correct integration e.g. one power. Good Excellent 2 - 3 marks 4-5 marks Substitutes values Substitutes values, and attempts to find solves simultaneous coefficients. equations, with very clear working and explanation Good attempt at Correct solution, integration, minor with clear working mistakes - follow and fully simplified. through an incorrect quadratic Attempt at Clear and fully explanation to correct explanation connect result connecting result above to the rule above to formula and explanation of terms. Deriving Simpson's Merely quotes rule, Rule limited explanation of terms.

Answer 1

Ginent The quadratic equation ax² bxto es points Al-hiya), B(o, yo) and c. Chiye) → we have the find the equation which passes through these pointe A B come c gi ax? +bx+c passes through a (thiya) , B (0,40) Use the paint B (O, GB) and put it in go 2 (0)2+ → Mlow - and & Chiyo) ax²t but ga o +670) +C [c = you] • Polad hos ta entry yo ax?abe. He also * YA = abh + b(h)+c ahl - bh tyo • Also paint c has to satisfy y - ga .(4 B aa'+buto - a (4²) + b Chlte ge ah + bh tyB B Ye CSScanned with Camscunner (w)

•Adding (4) and ful) , we Ya+Ye = (u), we get +2 (422 each?) +2 π a- YA+Yc - 2ys 2 h2 -putting the value a im eq W) or fil) ah? 6 h YA ya (y wyr ya. Ja-th ago YA - Yo 2 -bh ) (ye - 9A) 2h Hence the equation ya = ax'+bx+c will become 2. y = (1)-(13-)a yu +YB Any CS$canned with CamScanner

Now, to find the area under the quadrate between the points A and C. Vehiye) () Chiga) ch ak da h .. » Area = / ydx por J[Cho ten284) 2? + dela eta az yo] de 2 th 7 We W h w + ch 2h² ch + yo+h+n] 2h A 3 get Area = (44, 44-243.) CD ( ) ), - your] A = 440-) ** 4X4) () a 24 0) (2 : ) + 2nya fontye-zgoj+ Rhyo A = helga.ty] - 2hgo + 2hys h ga tych + hly I gatua ya + yo + A = Az Ans- 3 banned with CamScanner

canada Cascanner Now, using the forevious result to from the general formula for simpson's Rule - Now if we add two more points - dimilarly, from We get the area as E D c to E ACHE? h (ye tho + YE) 3 • Now add two areas arom (Arc) + ( CE) Area (total) - A[A-99 + Acc76) - 1 fyn rhys + 2ge+ Ayo nge) curie limiderly by indeling two more area by aluding the qurther, we U get the Area ( total cos Area (tatal )Igat layo + 2gc +440 tay 7+ 4ys + Wor] "Hence for n-points , the total area will be An (total) = 4, you by 242 thyst + 2yn-2 + 4yn-1 1 h .+2 + where n must be even. This is called simpoon's rule. Pleau Uprate Ans.

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