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Definite IntegralsRiemann SumsClass WorkConsider the region between y=9x-x3and the x-axis for 0≤x≤3.Sketch the graph of the region.Partition the region into 6 and show the rectangles for LRAMCalculate the area using LRAMUsing the same region as question 1, follow the same step to find RRAM.Using the same region as question 1, follow the same step to find MRAM.Using the same region as question 1, find LRAM, RRAM, & MRAM but with 10 partitions.Using the same region as question 1, find LRAM, RRAM, & MRAM but with 50 partitions.Make a conjecture about the area of the region in question 1.Find LRAM, RRAM, and MRAM between f(x) and the x-axis. Given are the bounds [a,b] and the number of partitions n.fx=x, 0,10, n=5fx=x3, 1,3, n=4fx=cosx, 0,2π, n=8The table shows the rate of fuel consumption of a car at given times on a 2 hour trip.Time10am10:1010:2010:4011am11:1511:3011:45Noongal/hour233432234Using 4 partitions and MRAM, estimate the area.What does this area represent?What are the appropriate units for the area?HomeworkConsider the region between y=3x-x2and the x-axis for 0≤x≤3.Sketch the graph of the region.Partition the region into 4 and show the rectangles for LRAMCalculate the area using LRAMUsing the same region as question 13, follow the same step to find RRAM.Using the same region as question 13, follow the same step to find MRAM.Using the same region as question 13, find LRAM, RRAM, & MRAM but with 10 partitions.Using the same region as question 13, find LRAM, RRAM, & MRAM but with 50 partitions.Make a conjecture about the area of the region in question 1.Find LRAM, RRAM, and MRAM between f(x) and the x-axis. Given are the bounds [a,b] and the number of partitions n.fx=x2, 1,9, n=4fx=3x0,8, n=4fx=sinx, 0,π, n=4The table shows the rate of downloads of a new song in the first 6 hours it was available.Time12am12:301233:153:304:456amdownloads/min20010090805020253524Using 4 partitions and MRAM, estimate the area.What does this area represent?What are the appropriate units for the area?Trapezoid RuleClass WorkConsider the region between y=4x-x3and the x-axis for 0≤x≤2.Sketch the graph of the region.Partition the region into 6 and show the trapezoids for using the trapezoid rule.Calculate the area.Using the same region as question 25, apply the trapezoid rule but with 10 partitions.Using the same region as question 25, apply the trapezoid rule but with 50 partitions.Make a conjecture about the area of the region in question 1.Find the area using the trapezoid rule between f(x) and the x-axis. Given are the bounds [a,b] and the number of partitions n.fx=1x, .5,2, n=4fx=x-x3, 1,3, n=4fx=sinx, 0,2π, n=6The table shows the speed of a car at given times on a 2 hour trip.Time10am10:1010:2010:4011am11:1511:3011:45Noonmiles/hour65506045706055600Using 4 partitions and trapezoid rule to estimate the area.What does this area represent?What are the appropriate units for the area?HomeworkConsider the region between y=8-4xand the x-axis for 0≤x≤2.Sketch the graph of the region.Partition the region into 4. Show the trapezoids for using the trapezoid rule.Calculate the area using LRAMUsing the same region as question 35, apply the trapezoid rule but with 10 partitions.Using the same region as question 35, apply the trapezoid rule but with 50 partitions.Make a conjecture about the area of the region in question 1.Find the area using the trapezoid rule between f(x) and the x-axis. Given are the bounds [a,b] and the number of partitions n.fx=x2-4, 1,3, n=6fx=4-x0,4, n=4fx=cosx, 0,π, n=6The table shows the rate of typists typing a manuscript over a 6 hour period.TimeNoon12:301233:153:304:456amwords/min20010090805020253524Using 4 partitions and trapezoid rule to estimate the area.What does this area represent?What are the appropriate units for the area?2523490163830Accumulation FunctionsClass WorkUse the graph of f ‘(x) to answer the following questions. f(0)=203f'xdx05f'xdx-40f'xdx30f'xdx5-4f'xdxf ’(2)f(2)f ”(2)When is f “(x)>0?251115625400HomeworkUse the graph of f ‘(x), a semi-circle and two lines to answer the following questions. f(1)=003f'xdx 05f'xdx-40f'xdx0-4f'xdx5-4f'xdxf ’(-2)f(-2)f ”(0)When is f(x) increasing?Anti DerivativesClass Work-22fxdx=4, -22gxdx=-3,25fxdx=8,02fxdx=3,85fxdx=2-22fx+gxdx-22fx-gxdx-223fx+gxdx-25fxdx284fxdx-28fxdx-20fxdxFind the value of following definite integrals.143dx25xdx-234x3dx151xdx06exdx-213x2+6x-5dx121x2dx0π4sec2x dx02πsinx dx0111+x2dxHomework-22fxdx=5, -22gxdx=9,25fxdx=-6,02fxdx=-1,85fxdx=-7-22fx+2gxdx-222fx-2gxdx-223fx+gxdx25(fx+1)dx283fxdx-28fxdx-20fxdxFind the value of following definite integrals.244dx252xdx-16x3dx162xdx05(4ex+1)dx-213x2+6x-5dx126x3dx0π4secxtanx dx02πcosx dx01211-x2dxMean Value Theorem & Average ValueClass WorkFind the average value of the function on the given interval.y=5x-2, -2,3y=cosx, 0,π21014730125095y=1x2, 1,52879090218440y=1x, [1,e]a. b. HomeworkFind the average value of the function on the given interval.y=6x2+4x, -1,5y=sec2x, 0,π4669290224790y=1x3, -3,-1303657064135y=e2x,[0,ln4]a. b. Fundamental Theorem of CalculusClass WorkFind dy/dxy=1x4t-2dty=22x3u2-4uduy=x4lnvdvy=x204t3-2tdty=3x22x7udu320421078105y=lnxxlnxevdvLet Fx=0xftdt, where f(t) is defined by the graph.f(2) F(2)F’(2)f ‘(2)-2x5u-6du+K=4x(5u-6)du, find KHomeworkFind dy/dxy=2xeuduy=3xt2dty=xπsinv dvy=4-x55-t dty=2x7xu2-4u+2duy=1xx2v dv219329050165Let Fx=0xftdt, where f(t) is defined by the graph.f(2) F(2) F’(2)f ‘(2)1x3u2+2u+1du+K=3x(3u2+2u+1)du, find KMultiple ChoiceNo Calculator Permitted144x+24xdx=ln412ln43+12ln45+12ln42-22x2-4x+7dx0163831003-1003The area under y=1x from x=1 to x=e4 is split into two equal area regions by x=k. Find k.22.5e2ln4eThe average value of y= x over the interval 1 to 4 is149143679Fx=x2xt2-1dt, F ‘(x)=2x2-x4x2-x8x2-x3x2-17x2-1Calculator Permitted13x3dx is approximated using right rectangular approximation method, with 4 equal partitions. Find the approximate area and state whether it is under or over estimate.17.641 u2; under estimate17.641 u2; over estimate20 u2; over estimate20 u2; under estimate27 u2; over estimateUsing the trapezoid rule and n=8, approximate 24x2-6dx6.6883.8397.6476.66713.366What is the average value of y= sin x on the [0,π2]?π22π1-1-π236fxdx=4, 610fx=-8, and 810fxdx=-5, then which of the following statements is true?36xf(x)dx=4x3102f(x)dx=-6810(f(x)-3)dx=-8385f(x)dx=5310f(x)dx=-901ex+1exdx=e-1–e-12-e-11-e-12+e-1Extended Response2167890114935No Calculator PermittedUse the accumulation function to answer the following: 05fxdx0-4fxdxf(0)f'(0)The table represents the fuel consumption of a car at given times.time(min)0.51234.5678gal/min234235342Approximate the fuel consumption using MRAM for 0≤t≤8.What is the approximate rate of change in the fuel consumption at t=3?If the maximum rate of fuel consumption occurs at t=4.5 min, what is the rate of change in the fuel consumption at t=4.5? Explain.Calculator PermittedGiven fx=4x2+6x on the interval [0,6]4276090109855Find the average value of f(x).x=k is a vertical line that divides the area of f(x) in half, find k.The graph of a velocity function, f ‘(x), is shownHow far does the particle travel for 3<x<7?If f(0)= 3, what is f(9)=?What is the particles acceleration at x=5?Is particle speed increasing or decreasing at x=6? Explain.Answer19.638 u219.638u220.5314520.048, 20.048, 20.35120.241, 20.241, 20.75420.2517.384, 23.709, 21.27814, 27, 19.50,0,017/3App fuel used during 2 hr tripGal4.2194.2194.6414.455, 4.455, 4.5294.498, 4.498, 4.5014.50168, 328, 2409.329, 13.329, 13.1301.896, 1.896, 2.05221450Number of downloads in first 6hrs after releaseDownloads3.8898.9603.998 Approximately 4.001.428-16.50109.167Miles traveled in 2 hoursMiles888Approximately 8.7045.146013,640Total words typed in 6 hrsWords32.50-302.5140-2 <x<-1; 4<x<52.54.5-2π2 π2 π-4.5-2π -1/2undefined-2<x<117151224101910.5651.609402.429-15.510.78523-824-3366821323.753.584594.653-152.25.4140.524.5.637.21/(e-1)A) 2.5 B) 2- 1/2 π501.273-.2225.410A) 16 B) 4- π4x-224x2-16x–lnx-8x7+4x314x-42x2(1+lnx)e xlnxA) 4 B) 9 C) 4 D) -1/26Ex? x–sinx1+x335x2-180x+102x2 + 1x2xA) 4 B) 6 C) 4 D) undefinedK= -36MULTIPLE CHOICECDAAEAABDCEXTENDED RESPONSEA) 9 B) 7 C) 0 D) 2A) 30 B) 1≤c’ (+) ≤2 C) 0, max rate of change =0A) 66 B) 4.642A) 8+2π B) 15.5+2π C) 0 D) decreasing, slope of tangent is negative, velocity is positive, acceleration and velocity have opposite signs ................
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