Math 1B Section 107 Quiz #4Thursday, 20 September 2007Theo [email protected]:1. (3 pts) Let’s say I believe that y is a function of x, and I do an experiment and getthe following values of x and y:x y(x) y00(x)0 0 —1 0 12 1 03 2 14 4 —For the particular measurement I’m making, I want to estimateR40y(x)dx. If I usethe trapezoid approximation with n = 4, I getZ40y(x)dx ≈ 5.How good of an estimate is this? I.e. what is the expected error? You can’t evaluatethe second derivative y00(x) exactly, but you can estimate it: if y(x−1) = a, y(x) = b,and y(x + 1) = c, then y00(x) ≈ a − 2b + c. So estimate y00(1), y00(2), and y00(3), anduse these to estimate the error ET4.ETn.K(b−a)312n21 ptK & max |y00(x)| = 1 1 ptET4.1×4312×42= 1/3 1 pt1Determine whether the following definite integrals are convergent or divergent. Evaluateeach convergent integral.2. (3 pts)Z20dxx√x=Z20dxx3/2p = 3/2 ≥ 1 1 ptSo integral diverges by the p-Test. 2 pt3. (4 pts)Z∞1x − 2x3+ 3x2+ 2xdxx − 2x3+ 3x2+ 2x≤xx3=1x2, which converges by p-Test.So integral converges by Comparison Test. 2 ptx − 2x3+ 3x2+ 2x=−1x+3x + 1+−2x + 21 ptZ∞1(x − 2) dxx3+ 3x2+ 2x= limt→∞Zt1(x − 2) dxx3+ 3x2+ 2x= limt→∞[−ln(x) + 3 ln(x + 1) − 2 ln(x + 2)]t1.5 pt= limt→∞ln(x + 1)3x(x + 2)2t1= limt→∞ln(t + 1)3t(t + 2)2− ln(1 + 1)31(1 + 2)2= ln(1) − ln2332= ln(9/8) .5
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