Math 1B Section 112 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 13 3 −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 ≈ 6.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)Z10dxx√x=Z10dxx3/2p = 3/2 ≥ 1 1 ptSo integral diverges by the p-Test. 2 pt3. (4 pts)Z∞1x − 4x3+ 3x2+ 2xdxx − 4x3+ 3x2+ 2x≤xx3=1x2, which converges by p-Test.So integral converges by Comparison Test. 2 ptx − 4x3+ 3x2+ 2x=−2x+5x + 1+−3x + 21 ptZ∞1(x − 4) dxx3+ 3x2+ 2x= limt→∞Zt1(x − 4) dxx3+ 3x2+ 2x= limt→∞[−2 ln(x) + 5 ln(x + 1) − 3 ln(x + 2)]t1.5 pt= limt→∞ln(x + 1)5x2(x + 2)3t1= limt→∞ln(t + 1)5t2(t + 2)3− ln(1 + 1)512(1 + 2)3= ln(1) − ln2533= ln(27/32) .5
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