oldmidterms - Math 152 Sp. 10 Z. Fiedorowicz Name: EA 170...

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Math 152 Sp. ’10 – Z. Fiedorowicz Name: EA 170 Signature: April 14, 2010 Name.nnn: Exam 1 TA: Suh Yeong Park Xiaoyue Xia Form A, 7 pages Rec. Time: 12:30 1:30 General Instructions: Do all the problems. Answer each part thoroughly. Show all of your work! Incorrect answers with work shown may receive partial credit but unsubstantiated answers will receive NO credit. You do not need to simplify numerical answers, e.g. 3 + 2 / 5 - 7 / 6. Calculators are permitted EXCEPT those calculators that have symbolic algebra or calculus capabilities. In particular , the following calculators and their upgrades are not permitted: TI- 89, TI-92,and HP-49. In addition, neither PDAs, laptops nor cell phones are permitted. Special Instructions: The Exam Duration is 48 minutes This exam consists of 5 problems on 6 pages (excluding this cover sheet). Make sure that your exam paper is not missing any pages before you start. Answers without supporting work will receive no credit. A random sample of graded exams will be xeroxed before being returned. Good Luck ! Problem # Points Score 1 15 2 18 3 36 4 15 5 16 Total 100 1
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1) (15 points) In what follows below C and D are constants of integration. (a) If Z f ( x ) dx = 6 x x 2 + 3 + C , find f ( x ). d dx ± 6 x x 2 + 3 ² = 6( x 2 + 3) - 6 x (2 x ) ( x 2 + 3) 2 = 18 - 6 x 2 x 4 + 6 x 2 + 9 (b) If Z x 1 g ( t ) dt = 4 x x 2 + 5 + D , find g ( t ) and D By the fundamental theorem of calculus g ( x ) = d dx ³ 4 x x 2 + 5 ´ = 4 x 2 + 5 + 4 x x x 2 + 5 = 4( x 2 + 5) + 4 x 2 x 2 + 5 = 8 x 2 + 20 x 2 + 5 Thus g ( t ) = 8 t 2 + 20 t 2 + 5 Also plugging in x = 1 we get 0 = Z 1 1 g ( t ) dt = 4(1) 1 2 + 5 + D Thus D = - 4 6 2
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2) (18 points) (a) Express the following sum sin ± - 6 π 3 + 5 π n ² 5 π n + sin ± - 6 π 3 + 10 π n ² 5 π n + sin ± - 6 π 3 + 15 π n ² 5 π n + ... + sin ± - 6 π 3 + 5 n ² 5 π n in the form Σ n i =1 a i . n X i =1 sin ± - 6 π 3 + 5 n ² 5 π n (b) Find a definite integral R b a f ( x ) dx for which the above sum is a right Riemann sum. Z b a sin( x + c ) dx where b - a = 5 π and c = - 6 π 3 - a (c) Use (b) to find the limit of the above sum as n -→ ∞ . 2 3
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calculator will receive no credit.) (a) Z ( e 4 t - 4) 2 e 2 t dt Z ( e 4 t - 4) 2 e 2 t dt = Z e 8 t - 8 e 4 t + 16 e 2 t dt = Z ( e 6 t - 8 e 2 t + 16 e - 2 t ) dt = e 6 t 6 - 8 2 e 2 t - 16 2 e - 2 t + C (b) Z - 4 P + 2 1 - P 2 dP Z - 4 P + 2 1 - P 2 dP = - 4 Z P 1 - P 2 dP + 2 Z 1 1 - P 2 dP = - 4 Z P u du ( - 2 P ) + 2 sin - 1 ( P ) = +4 u + 2 sin - 1 ( P ) + C = +4 1 - P 2 + 2 sin - 1 ( P ) + C where we substituted u = 1 - P 2 in the first integral on the right. 4
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oldmidterms - Math 152 Sp. 10 Z. Fiedorowicz Name: EA 170...

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