Copyright Andrew Chambers 2020. Licensed for non-commercial use only. Visit ibmathsresources.com to download the full worked mark-scheme and for 300 exploration ideas. 22 Exploring Riemann Sums [31 marks] 1. [Maximum marks: 19] We have the function !(!) = ! !!! ! and draw 5 rectangles as shown below. The width of each rectangle is 0.2, and the height of the rectangles are given by !(0), !(0.2), !(0.4) etc. (a) Use the rectangles above to find an upper bound estimate for the area under the curve !(!). Give your answer to 5 significant figures. [3] (b) It is also possible to draw rectangles from the right hand side. The width of each rectangle is 0.2, and the height of the rectangles are given by !(0.2), !(0.4), !(0.6) etc. Use the rectangles above to find a lower bound estimate for the area under the curve !(!). Give your answer to 5 significant figures. [2] (c) We can refine our approximation for the are under the curve by using the trapezium rule. The trapezium rule provides an underestimation of the area when the curve is concave down and an overestimation when the curve is concave up. Use calculus to find the ! coordinate when !(!) changes concavity. [5]

22 Exploring Riemann Sums [31 marks] 1. [Maximum marks: 19]

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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.

[2]

(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.

[5]

Page 2: 22 Exploring Riemann Sums [31 marks] 1. [Maximum marks: 19]

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!! −!

[6]

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

[6]