22 exploring riemann sums [31 marks] 1. [maximum marks: 19]

CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visitibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.

22ExploringRiemannSums[31marks]

1.[Maximummarks:19]Wehavethefunction!(!) = !

!!!! anddraw5rectanglesasshownbelow.Thewidth of each rectangle is0.2, and the height of the rectangles are given by!(0),!(0.2),!(0.4)etc.

(a) Use therectanglesabove to findanupperboundestimate for theareaunder

thecurve!(!).Giveyouranswerto5significantfigures.[3]

(b) Itisalsopossibletodrawrectanglesfromtherighthandside.Thewidthofeachrectangleis0.2,andtheheightoftherectanglesaregivenby

!(0.2),!(0.4),!(0.6)etc.

Usetherectanglesabovetofindalowerboundestimatefortheareaunderthecurve!(!).Giveyouranswerto5significantfigures.

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(c) We can refine our approximation for the are under the curve by using thetrapezium rule. The trapezium rule provides an underestimation of the areawhen the curve is concave down and an overestimation when the curve isconcaveup.Usecalculustofindthe!coordinatewhen!(!)changesconcavity.

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CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visitibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.

23(d) Inthiscasethetrapeziumruleprovidesanunderestimationoftheareaunder

!(!).Byreferringtoyourpreviousanswergivea justificationwhythismightbe.

[2](e) Forthetrapeziumruleherewesplitourareainto5trapeziaeachwithwidth

0.2andwithtopverticesonthefunction!(!).

Usethe5trapeziaabovetofindabetterapproximationfortheareaunder !(!).Giveyouranswerto5significantfigures.

[3](f) By considering !

!!!! !",use your previous results to find a lower and upperboundfor!.Giveyourboundsto3significantfigures.

[4]2.[Maximummarks:12]Below we have the function ! ! = !! , ! > 0 and two possible Riemannapproximations.(a) Usethediagramabovetoshowthat:

2! + 1!! +! < !" ! + 1

! − 1 < 2! − 1!! −!

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(b) Hencefindalowerandupperboundapproximationto5significantfiguresfor!" 1.1 . Find the maximum percentage error when using the upper bound.Explainwhy!" 1.1 wouldhaveamoreaccurateboundthan!" 3 .

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