Math 1B Section 107 Quiz 3 Thursday 13 September 2007 Theo Johnson Freyd theojf math berkeley edu Name 1 3 pts Say we want to find Z 3 1 cos x dx x to within an error of 0 001 using the trapezoid rule What s a reasonable number of intervals into which to divide the domain 1 3 Your number should be big enough to guarantee that the value of the approximation is within the allowed error but fewer intervals means less computation for the computer Error trapezoid K b a 3 12n2 5 pt We need K f 00 x sin x d2 cos x d dx x x dx2 cosx x cos x x 1 2 x x2 1 2 1 12 cos x x2 sin x x x sin 2 cos x2 x2 x3 2 sin x x 2 cos x2 x3 2 x3 2 5 13 1 pt So take K 5 Error trapezoid 5 2 3 12n2 We want n such that Error trapezoid 1 1000 q 5 2 3 This certainly happens if n 12 1000 For instance n 1 70 works 1 pt 5 pt 2 3 pts Evaluate the integral Z 1 ee x x dx 1 u e x du e x dx e x x dx 1 e R1 1 pt e x e x dx 1 e R1 R 1 e eu 1 e u e ee e1 e u u e e du 1 pt 1 pt 3 4 pts Evaluate the integral Z 1 ln x2 1 dx 0 u ln x2 1 dv dx 2 du 2x x 1 v x R R1 2 dx 1 1 2 x ln x2 1 x 0 0 2x 0 ln x 1 dx x2 1 i R1h 2 ln 2 0 2 x2 1 dx R1 ln 2 2 2 0 x2dx 1 ln 2 2 arctan x 1x 0 1 pt 1 pt 1 pt ln 2 2 arctan 1 ln 2 2 4 2 1 pt
View Full Document
Unlocking...