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Zachary DorffZhuMAT266ONLINEBFall2019Assignment Test1FallB2019 due 10/29/2019 at 03:36pm MSTProblem 1.4.(1 point)Estimate the area under the graph in the figure by using (a)the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’sRule, each withn=4.T4≈M4≈S4≈Solution:SOLUTIONWe havea=0,b=4 andn=4. SoΔx=b-an=44=1, andxk=a+kΔxsox0=0,x1=1 ,x2=2,x3=3, andx4=4.We use the graph of the function to estimate the values of thefunction at the required values ofx:1.T4=Δx2hf(x0)+2f(x1)+2f(x2)+2f(x3)+f(x4)i=12hf(0)+2f(1)+2f(2)+2f(3)+f(4)i≈12h0+2·3+2·5+2·3+1i=11.52.The midpoints are ¯x1=0.5, ¯x2=1.5, ¯x3=2.5 and¯x4=3.5M4=Δxhf(¯x1)+f(¯x2)+f(¯x3)+f(¯x4)i=1·hf(0.5)+f(1.5)+f(2.5)+f(3.5)i≈1·h1+4.5+4.5+2i=123.S4=Δx3hf(x0)+4f(x1)+3f(x2)+4f(x3)+f(x4)i=13hf(0)+4f(1)+2f(2)+4f(3)+f(4)i≈13h0+4·3+2·5+4·3+1i=353Correct Answers:•1/2*(0+2*3+2*5+2*3+1)•1*(1+4.5+4.5+2)•1/3*(0+4*3+2*5+4*3+1)Problem 2.8.(1 point)Consider the four functions shown below. On the first two,an approximation forRbaf(x)dxis shown.126.96.36.199.(Click on any graph to get a larger version.)1.For graph number 1, Which integration method is shown?•A. right rule•B. midpoint rule•C. left rule•D. trapezoid ruleIs this method an over- or underestimate?•A. over•B. under2.For graph number 2, Which integration method is shown?•A. trapezoid rule•B. left rule•C. right rule•D. midpoint ruleIs this method an over- or underestimate?•A. over•B. under3.On a copy of graph number 3, sketch an estimate withn=2 subdivisions using the midpoint rule.Is this method an over- or underestimate?•A. under•B. over4.On a copy of graph number 4, sketch an estimate withn=2 subdivisions using the trapezoid rule.Is this method an over- or underestimate?•A. under•B. over1
Solution:SOLUTIONWe can see by inspection or by looking at how the functionsincrease/decrease or their concavity that graph 1 shows the rightrule.Similarly, graph 2 shows the right rule.The graphs showing the numerical estimates for graphs 3 and4 are shown below.3.4.(Click on any graph to get a larger version.)Then graph 3, showing the midpoint rule.Then graph 4, showing the trapezoid rule.