How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for 2 1 x dx 1 are accurate to within 0.00002? SOLUTION If f(x) = 1 x , then fâ'(x) = , and fâ''(x) = . Since 1 ⤠x ⤠2, we have 1 x ⤠1, so |fâ''(x)| = ⤠2 13 = We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore we chose n so that 2 3 12n2 ⤠0.00002 Solving the inequality for n, we get n2 > 2 3 12(0.00002) or n > 1 0.00012 â 91.29 Thus n = (rounded up to the nearest integer) will ensure the desired accuracy. For the same accuracy with the Midpoint Rule we choose n so that 2 3 24n2 < 0.00002 which gives the following. (Round your answer up to the nearest integer.) n > 1 0.00024 â
How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for 2 1 x dx 1 are accurate to within 0.00002? SOLUTION If f(x) = 1 x , then fâ'(x) = , and fâ''(x) = . Since 1 ⤠x ⤠2, we have 1 x ⤠1, so |fâ''(x)| = ⤠2 13 = We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore we chose n so that 2 3 12n2 ⤠0.00002 Solving the inequality for n, we get n2 > 2 3 12(0.00002) or n > 1 0.00012 â 91.29 Thus n = (rounded up to the nearest integer) will ensure the desired accuracy. For the same accuracy with the Midpoint Rule we choose n so that 2 3 24n2 < 0.00002 which gives the following. (Round your answer up to the nearest integer.) n > 1 0.00024 â
