12:25 X CAL 2 FINAL EX... V MATH 104 CALCULUS 2 FINAL EXAM Topics: Basic Integration Formulas, Integration by Substitution, Integration Techniques, Sequences, Infinite Series, Tests for Convergence, Power Series A. Multiple Choice. Choose the letter that corresponds to your answer. In a particular item, if the correct answer is not among the choices or there are more than one of the choices that correctly answer that question, choose E. In some items (with blanks). you will have to provide the values(s), word(s) of expression (s) that will make the given statement true. The +C in the given or choices refers to an arbitrary constant. Input your answers in the MS Form provided. 1. Which of the following is true? (B) [ In(r+ 1)d = In'(a + Due 2 (D) 2r VI + 4r de = tanto 2. The slope of the line tangent at any point (a, y) on a curve is given by sinr. Find an equation of the curve if the point (7is on the curve. (A) = cost + 1 (C) y = cost (B) = - cost + 1 (D) y= =cost 3. Determine / (!) d +0 (D) - 4. Which of the following is equal to the expression 21 + Ar - 127 (A) (r + 2)3 - 25 (C) 25 - (x - 2)- (B) 25 - (2 + 2)3 (D) (+ + 2)2 + 25 5. Determine V21 + 4r_pdr (A) sin '(r + 2) +C (C) - tan-1 +2) +C (B) sin (r - 2) + C (D) - tan-1 (- 2) 5 Page 1 of 5 MATH 104 CALCULUS 2 FINAL EXAM12:25 39 X CAL 2 FINAL EX... V Page 1 of 5 8 2 of 5 MATH 104 CALCULUS 2 FINAL EXAM 6. Determine ( 2 57 da. (A) 2ring to (C) Jhis + C (B) (D) 3.r In 5 +C 7. Consider / arceos(2x) dir. Using integration by parts, if we use the substitution u = arecos(2). then du = (A) VI - 42 (C) VI- 47 (B) 2 der 2 dr VI -12 (D) 8. Consider / arecos(2) dir. Using integration by parts, if we use the substitution u = arecos(27), and do = dor, then which of the following is equivalent to / arecost2) da. (A) raresin(2) - ;I-4+ C (C) raresin(2) - VI (B) rarewin(20) + vi -4 + 0 (D) caresin (20) + ; V14 18 + 0 9. Using trigonometric substitution to determine de, we can use the substitution, Inr = 2tand. Then da = (A) 2 soc30 do (C) roc 9 de (B) 2r sec# 10 (D) 2 soc 0 do 10. Using trigonometric substitution to determine Vln't+ 4 In de, we can use the substitution, Ine = 2tand. In terms of e, Vin'r + 4=. (A) 2seed (C) 2v secu (B) 2 sec' 9 (D) 2v/sec-9 - 1 11. Using, trigonometric substitution to determine de, we can use the substitution, Inr = 2 tan 0. Which of the following is equivalent to Vin T+ 1 (A) 2 ( see di (C) 2 ( seed + esco) do (B) 2 / see- 9 di (D) 2 (sectand + cased) de Page 2 of 512:25 .. .I .. . . 9 X CAL 2 FINAL EX... V Page 2 of 5 8 3 of 5 MAIn1 104 VALVULON 4 FINAL EXAM 12. Using trigonometric substitution to determine VIn' x + 4 dx, we can use the substitution, c In x In x = 2 tan 0. Which of the following is equivalent to / VIn x + 4 dx? (A) sec x tan x + In | secx + tan x] + C (C) 2(In |sec 0 + tan 0| + In [csc0 - cot #)) + C (B) tane + C (D) 2(sec @ In [csc 0 - cot #1) + C 13. Evaluate (A) 2 290 29 (B) , (C) 290 (D) 2 14. Which of the following is an improper integral? (A) J (x + 1) 213 dx dx ( 1 + 1 ) 2/3 15. Evaluate " Itsect at . (A) 2.x2 sec x (B) -2x2x sec x (C) 2x sec x (D) -2x sec x 16. Which of the following is true about the sequence 5, -25, 125, -625, ...? (A) The sequence is convergent with limit -1. (C) The sequence is convergent with limit 1. (B) The sequence is convergent with limit 0. (D) The sequence is divergent. 17. Which of the following is true about the sequence {2" - n},? (A) The given sequence is decreasing. (C) The given sequence is increasing. B) The given sequence is strictly decreasing. (D) The given sequence is strictly increasing. 18. Which of the following is true about the sequence (A) The given sequence is decreasing. (C) The given sequence is increasing. (B) The given sequence is strictly decreasing. (D) The given sequence is strictly increasing. Page 3 of 512:25 .. .I .. . . 9 X CAL 2 FINAL EX... V MA 8 4 of 5 FINAL EXAM 19. Which of the following is true about the sequence {n3 - 8},? (A) The given sequence is convergent and the limit is 8. (B) The given sequence is convergent and the limit is 0. (C) The given sequence is convergent and the limit is -8. (D) The given sequence is divergent. 20. Which of the following is true about the sequence ? - -(1 -2") } ? (A) It converges to -1. (C) It converges to -2. (B) It converges to 0. (D) It is divergent. 21. Determine if )?= + + 3 2n2 + 1 is convergent or divergent. n= ] (A) divergent (B) convergent 22. Which of the following series is convergent? 8 (A) 2 (B) n5 1 (D) n= 1 n=1 23. Which of the following series is divergent? n-1 (A) (C) > n=1 (n + 2) (n + 3) 1= 1 2-1 (B) (D) 2 3n-1 n= 1 n= 1 24. Which of the following is convergent? (A) 8 (B) (C) 1 n3 (D) M : 1 25. Determine if the series > sinna is convergent or divergent. n=1 (A) divergent (B) convergent 26. Determine if the series _ ( 1 + ;)" is convergent or divergent. (A) divergent (B) convergent Page 4 of 512:25 .. .I .. . . 9 X CAL 2 FINAL EX... V 80 5 of 5 MA FINAL EXAM 18 27. Determine if the series _ n=1 5n2 - n is convergent or divergent. (A) divergent (B) convergent 28. Which of the following is true given the series (-1 )+1 n! n=1 (2n - 1)!' (A) It is absolutely convergent. (C) It is divergent. (B) It is conditionally convergent. 29. Determine if the series > (-1)"the" is convergent or divergent. n=1 (A) divergent (B) convergent 30. Use sigma notation to write the Maclaurin series for the function f(x) = x2 sin x. (A) (-1)" 2 2n+ 2 (C) (-1)" n=0 (n + 1)! n=0 (2n + 2)! 2 2n+2 (B) (2n + 1)! (D) (-1) In n =0 7= 0 n ! 31. Determine the solution set of the inequality |x - 5|