midpoint(and(trapezoid(rule( example :&...

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Lesson #79 and 80 Midpoint and Trapezoid Rule Example #1: Use a midpoint Riemann sum to approximate the Area under the curve of = ! using 4 subintervals from x=1 to x=3. Ø Graph the function. Ø Find the midpoint of each subinterval. Ø Then solve similar to Left and Right-hand endpoints. Review: Set up the following: LRAM: RRAM: Solve for MRAM:

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Page 1: Midpoint(and(Trapezoid(Rule( Example :& …mrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/37… · · 2017-03-08Solvefor&MRAM:& & & & & & & & & & Lesson&#79&and80&! Sometimes&on&theexam,&theywill&not&specify&use&“left”&or&use

Lesson#79and80

MidpointandTrapezoidRuleExample#1:UseamidpointRiemannsumtoapproximatetheAreaunderthecurveof𝑦 = 𝑒! using4subintervalsfromx=1tox=3.

Ø Graphthefunction.Ø Findthemidpointofeachsubinterval.Ø ThensolvesimilartoLeftandRight-handendpoints.

Review:Setupthefollowing:LRAM:RRAM:SolveforMRAM:

Page 2: Midpoint(and(Trapezoid(Rule( Example :& …mrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/37… · · 2017-03-08Solvefor&MRAM:& & & & & & & & & & Lesson&#79&and80&! Sometimes&on&theexam,&theywill&not&specify&use&“left”&or&use

Lesson#79and80

Ø Sometimesontheexam,theywillnotspecifyuse“left”oruse“right”oruse“midpoint”

Ø Ifaquestionsaystofindtheareausingcircumscribedrectangles,thenALLrectanglesmustbeoutsideofthecurve.

Ø Ifaquestionsaystofindtheareausinginscribedrectangles,thenALLrectanglesmustbeinsideofthecurve.

TrapezoidRule:AreaofTrapezoid:IfIhaveseveraltrapezoidsinarowwiththesamewidth:FormulaforTrapezoidRule:Area≈ !

!!!!!

𝑦! + 2𝑦! + 2𝑦! +⋯+ 2𝑦!!! + 𝑦! wherenisthenumberofsubintervals.

Example#2:Usetrapezoidruletofindtheareafromx=-3tox=0of𝑦 = 5𝑥!sin (𝑒!)using3subintervals.

Page 3: Midpoint(and(Trapezoid(Rule( Example :& …mrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/37… · · 2017-03-08Solvefor&MRAM:& & & & & & & & & & Lesson&#79&and80&! Sometimes&on&theexam,&theywill&not&specify&use&“left”&or&use

Lesson#79and80

L’Hopital’sRuleØ Amethodforfindinglimits.Ø Whenlim ! !

! != !

! 𝑜𝑟 !

!,then

Example#1:

lim!→!

𝑥! − 4𝑥 − 2

PreviousMethod: L’Hopital’s:Example#2:

lim!→!

𝑠𝑖𝑛2𝑥𝑥

PreviousMethod: L’Hopital’s:

lim𝑓(𝑥)𝑔(𝑥)

= lim𝑓!(𝑥)𝑔!(𝑥)

Page 4: Midpoint(and(Trapezoid(Rule( Example :& …mrsyoungsmathclasses.weebly.com/uploads/3/7/5/6/37… · · 2017-03-08Solvefor&MRAM:& & & & & & & & & & Lesson&#79&and80&! Sometimes&on&theexam,&theywill&not&specify&use&“left”&or&use

Lesson#79and80

Example#3:

lim!→!!

𝑠𝑖𝑛2𝑥

4𝑥! − 𝜋!

Example#4:

lim!→!

2 − 𝑥! − 2𝑐𝑜𝑠𝑥

𝑥!

Example#5:

lim !→!

𝑥!

𝑒!

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