Math 1B Section 107 Quiz #3Thursday, 13 September 2007Theo [email protected]:1. (3 pts) Say we want to findZ31cos xxdxto within an error of 0.001 using the trapezoid rule. What’s a reasonable number ofintervals into which to divide the domain [1, 3]? Your number should be big enoughto guarantee that the value of the approximation is within the allowed error, butfewer intervals means less computation for the computer.Error(trapezoid) ≤K(b−a)312n2.5 ptWe need K ≥ |f00(x)|d2dx2cos xx =ddx−sin xx−cos xx2 =−cos xx+sin xx2+sin xx2+2 cos xx3≤cos xx+2 sin xx2+2 cos xx3≤1x+2x2+2x3≤11+212+213= 5 1 ptSo take K = 5Error(trapezoid) ≤5(2)312n2We want n such that Error(trapezoid) ≤ 1/1000This certainly happens if n ≥q5(2)312/10001 ptFor instance, n = 70 works. .5 pt12. (3 pts) Evaluate the integralZ1−1ee−x−xdxu = e−xdu = −e−xdx 1 ptR1−1ee−x−xdx =R1−1ee−xe−xdx=R1/eu=e−eudu 1 pt= −eu|1/eu=e= ee− e1/e1 pt3. (4 pts) Evaluate the integralZ10ln(x2+ 1) dxu = ln(x2+ 1) dv = dxdu = 2x/(x2+ 1) v = x1 ptR10ln(x2+ 1) dx =x ln(x2+ 1) 1x=0−R102x2dxx2+1= ln 2 −R10h2 −2x2+1idx 1 pt= ln 2 − 2 + 2R10dxx2+1= ln 2 − 2 + [arctan(x)]1x=01 pt= ln 2 − 2 + arctan(1)= ln 2 − 2 + π/4 1
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