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mt1-sample-ans - Math 1B Sample Answers to Sample First...

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Sheet1 Page 1 Math 1B. Sample Answers to Sample First Midterm . p 1 (5 points) Estimate the integral sin x dx using Simpson's rule with n Err:520 6 0 x Err:520 =6 , so the answer is p p 2p 5p S6 Err:520 sin0+4sin
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Sheet1 Page 2 +2sin +4sin +2sin +4sin +sin + 18 6323 6 pv p 13 31 4 h 2 h +4+2 h 4 h 1822 22 pp p Err:520 (8+2 3)= (4+ 3) : 18 9 . 4
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Sheet1 Page 3 ln x 2 (6 points) Find v dx x 0 This is an improper integral (the integrand is discontinuous at x Err:520 0), so we do the indenite integral rst: v 2 Use the rationalizing substitution w Err:520 x , x
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Sheet1 Page 4 Err:520 w, dx #NAME? dw to get ln x ln(w2) v dx Err:520 h 2w dw 4 ln wdw . xw Now do integration by parts, with u Err:520 ln w , dv Err:520 dw , du Err:520 dw=w , v Err:520 w :
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Sheet1 Page 5 4 ln w dw #NAME? ln w - dw Err:520 4(w ln w - w)+ C ppv pv Err:520 4( x ln x - x)+ C 2 x ln x - 4 x + C. Therefore . 4 ln x pv
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Sheet1 Page 6 v dx Err:520 lim (4ln4 - 8 - 2 t ln t 4 t) x t!0+ 0 By l'Hospital's rule, v ln t 1=t v lim t ln t #NAME? #NAME? Err:520 ..2 lim t 0 . t!0+ t!0+ t..1=2 t!0+
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Sheet1 Page 7 ..1=2t3=2 t!0+ . 4 ln x This gives a nal answer of v dx Err:520 8ln2 - 8 x 0 1 2 .
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