-16 points SCalcET8 7.7.019.MI My Notes Ask Your Teacher Given. 27 cos(x2) dx Do

Question

-16 points SCalcET8 7.7.019.MI My Notes Ask Your Teacher Given. 27 cos(x2) dx Do the following (a) Find the approximations Te and Mg for the given integral. (Round your answer to six decimal places.) (b) Estimate the errors in the approximations Ts and Ms in part (a). (Use the fact that the range of the sine and cosine functions is bounded by +1 to estimate the maximum error. Round your answer to seven decimal places.) (c) How large do we have to choose n so that the approximations Th and Mn to the integral are accurate to within 0.0001? (Use the fact that the range of the sine and cosine functions is bounded by 1 to estimate the maximum error.) for Tn for Mn n 2 Need Help?ReatWatch Iaailk to a Tutor Talk to a Tutor

Explanation / Answer

a)

?x = (1 - 0)/8
?x = 1/8
?x = 0.125

T8 =(1/2) * (0.125) * ( f(0) + 2*f(0.125) + 2*f(0.25) + 2*f(0.375) + 2*f(0.5) + 2*f(0.625) + 2*f(0.75) + 2*f(0.875) + f(1) )

=(1/2) * (0.125) * ( 27 + 2*26.996704168617 + 2*26.947282788903 + 2*26.733471885993 + 2*26.160635386187 + 2*24.96612405961 + 2*22.839961479239 + 2*19.465633285534 + 14.58816225844 )

=24.362987

Next

M8 =(0.125) * ( f((0+0.125)/2)) + f((0.125+0.25)/2)) + f((0.25+0.375)/2)) + f((0.375+0.5)/2)) + f((0.5+0.625)/2)) + f((0.625+0.75)/2)) + f((0.75+0.875)/2)) + f((0.875+1)/2)) )

=(0.125) * ( f(0.0625) + f(0.1875) + f(0.3125) + f(0.4375) + f(0.5625) + f(0.6875) + f(0.8125) + f(0.9375) )

=(0.125) * ( 26.99979400661 + 26.983316232641 + 26.871356252919 + 26.506917398017 + 25.65971350722 + 24.039778458054 + 21.327203403347 + 17.225831481413 )

= 24.451739

b)

|ET| <=162/(12*8^2) = 0.2109375

|EM| <=162/(24*8^2) = 0.10546875 =0.1054687 (or 0.1054688)

c)

162/(12*n^2) <= 0.0001

n>= 368

-----------

162/(24*n^2) <= 0.0001

n>= 260

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