prac2 - QUESTION 3. (6 + 6 = 12 pts) (i) Determine whether...

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MAC 3473 - PRACTICE TEST 2 Instructions: Answer all questions. Show all necessary working and reasoning. Write in such a way that any student in the class can follow your work. A Table of Integrals is supplied. Only scientific calculators are allowed. 50 points total. QUESTION 1. (6 + 6 = 12 pts) Evaluate the following integrals. Z 5 x - 7 x 2 - 3 x + 2 dx (a) Z (tan( t ) + 1)sec 2 ( t ) (tan( t ) + 3) 2 dt (b) QUESTION 2. (5 + 5 + 2 = 12 pts) (a) Find an approximation to the definite integral Z 1 0 dx 1 + x 3 using Simpson’s rule with n = 4. As a check on your work you are given the following approximation found by MAPLE: Z 1 0 dx 1 + x 3 0 . 8356488485 (b) Find an upper bound for the error in your approximation. You may assume that | f (4) ( x ) | ≤ 35 for 0 x 1. (c) What do your answers to (a) and (b) imply about the exact value of Z 1 0 dx 1 + x 3 ? Is this consistent with the approximation found using MAPLE ?
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Unformatted text preview: QUESTION 3. (6 + 6 = 12 pts) (i) Determine whether the improper integral Z e-2 x dx converges. Evaluate the integral if it is convergent. (ii) Giving reasons for your answer discuss the convergence of the improper integral Z 1 sin 2 x x 2 + 1 dx QUESTION 4. (2 + 4 + 4 + 4 = 14 pts) (i) Dene what it means for the number L to be a limit of the sequence { a n } n =1 . (ii) Prove the following limit using the formal denition. lim n 1 4 n + 3 = 0 . (iii) Determine whether the sequence a n = n sin(1 /n ) converges. If it converges, nd the limit. (iv) Determine whether the sequence a n = ( n +1) ( n +3) is increasing, decreasing or not monotonic. Is the sequence bounded? 1...
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