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SP10_MAC2311_E3ab - MAC 2311 Spring 2010 Exam 3A A Sign...

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Unformatted text preview: MAC 2311 Spring 2010 Exam 3A A. Sign your scantron sheet in the white area on the back in ink. B. Write and code in the spaces indicated: 1) Name (last name, first initial, middle initial) 2) UF ID number 3) Discussion Section number C. Under “special codes”, code in the test number 3, 1. 1 2 s 4 5 6 7 8 9 0 a 2 3 4 5 6 7 8 9 0 D. At the t0p right of your answer sheet, for “Test Form Code” encode A. o B C D E E. This test consists of 10 five—point, one three—point and two one—point multiple choice questions, bonus questions worth 4 points and two sheets (4 pages) of partial credit questions worth 25 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before turning in your test check for transcribing errors. Any mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion leader or proctor. Be prepared to show your picture ID with a legible signature. 3) The answers will be posted on the MAC 2311 homepage after the exam. Your discussion leader will return yOur tearofi' sheet with your exam score in discussion. Your score will also be posted on Elearning within one week. z; / NOTE: Be sure to bubble the answers to questions 1— 17 on your scantrgyh Questions 1 - 10 are worth 5 points each. 1. Find the absolute maximum and minimum values of f (:13) = e 3 on [0,2]. a. 64, 1 b. 82, 1 c. 64, 0 d. 62, e e. 64, 82 2. Find each value of :1: at which f (:13) i .732 +1110?) has an inflection point. a.:1;=0,:1:=:t1 b.:1:=0,:1:=:t2 c.:1:=:t20nly d. :1: = :l:1 only e. f has no inflection points 3. A particle moves along the graph of y = tan(:1:) sec(:1:) so that y is in— creasing at a rate of 3 units per second. Find the rate at which :1: is . . . 7r chang1ng w1th respect to t1me when :1: = — . . . 3 . a. :1: IS 1ncreas1ng by —— un1ts per second. \/§ b. :1: is decreasing by \/§ units per second. c. :1: is increasing by units per second. 33/5 \/§ d. :1: is decreasing by —3— units per second. . . . 1 . e. :1: 1s 1ncreas1ng by —— un1ts per second. fl 2A 4. Find all values of c guaranteed by the Mean Value Theorem for fix) 2 11:3 — 41: on [0,3]. a. c 2 \/§ only d.c=— 3andc=\/§ 5. Find the critical numbers of f (51:) = sin a _i 31 51 ”"3’ 2’ 3 d :I:=O,5—7r,7r,7—7r 6 6 2 :1: — cosw on [0, 27r). 7r 571' 271' 4% 0 — — =0 _ _ 33771-3 3 C m 7 3771-) 3 27r 7r 47r 3 ’ 2’ 3 6. A particle moves in a straight line so that its position (in feet from a starting point) after t minutes is given by 5(t) = 6752 — 753. Find the displacement of the particle and the total distance traveled on the time interval 0 g t g 5. displacement a. 32 ft b. 25 ft c. 57 ft d. 25 ft e. 32 ft total distance 3A 57ft 39ft 89ft 57ft 39ft (w — 4>2 7. The local extrema of f (:11) = occur at which of the following a: :c—values? local minimum local maximum at :1: 2 at :1: = a 4 none b 0 —4, 4 c 4 —4 d —4, 4 0 e —4 4 8. The radius of a sphere is measured to be 7' =4 inches. If the mea- surement of 7' has a possible error of :l:0.04 inch, use differentials to approximate the percentage error that can occur in using this mea— sure to compute the surface area of the sphere. Note: the surface area S of a sphere is given by S : 47r 7'2. a. 1% b. 2.4% c. 1.02% d. 2.01% e. 2% 2 — 3 9 — 2 2 2 — 36 9. If f(:z:) = ”3 ,then f’(a:) = $4": and f”(a:) = if. Find each interval on which f is both increasing and concave up. a. (—3, 0) o (3&, 00) b. (—3\/'2’, —3) c. ' (—3, 3) d. (—3,0) only e. (—00, —3\/§) U (3,00) 4A 10. Which of the following statements is / are true of f (as) = (932 — 1)1/3? 293 Note that f,($) = m. A. f (as) has exactly one critical number. B. f(:1:) has vertical tangent lines at :1: = —1 and a: = 1. C. f(:1:) is increasing on (0,1) and (1, 00). a. A only . b. B and C only c. A and C only (:1. B only e. A, B, and C 11. (3 points) The position function of a snail moving along a straight path - t — 3 . is 5(t) = —, where 5(t) is the distance in centimeters from the starting 25 + 2 point after t minutes. Find the acceleration of the snail after 1 minute. 5 10 a. — cm/min2 b. —1 cm/min2 c. —— cm/min2 , 9 27 d 1 m/ ' 2 e A 2 / ' 2 . — c min . —— cm mm 9 27 The following true/ false questions are worth one point each. 12. The position function for an object moving in a straight line is given by s : f(t). lf f'(b) > 0 and f”(b) < 0, then the object is slowing down at time t = b. a. True b. False 5A 13. If y = f(a:) is a continuous function so that f'(3) = 0 and f”(3) = —4, then f has a relative minimum at a: = 3. a. True b. False ‘ Bonus” (one point each) The graph of the derivative of a contin- uous function f is sketched below. ' Use the graph to determine if the following statements about f are true or false. 14. f(:c) has a local minimum at a: = 0. a. True b. False 15. f is increasing on the interval (—1,1) only. a. True b. False 16. f is concave down on intervals (—00, —1) and (1,00). a. True b. False 17. f has inflection points at m = —1 and m = 1. a. True b. False' 6A MAC 2311 TEST 3A PART II SPRING 2010 Sect # Name UFID Signature SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. Let f(a:) 2 mm” a. Use logarithmic differentiation to find f '(33). f’($) = b. Find the slope of the tangent line to f (3:) at a: = 7r. 7A 2. To which of the functions below can we apply Rolle’s Theorem on the interval [0, 8]? Write “can apply” or “cannot apply”. If you can apply Rolle’s Theorem, find each value of c guaranteed by the theorem. If not, state a condition not satisfied by the function. 332—83: am): 35—1 b. f(a:) = 332/3 — 2331/3 3. A boy standing at the end of a pier which is 12 feet above the water is pulling on a rope that is attached to a rowboat in the water. If he is pulling the rope in at 6 feet per minute, how fast is the boat docking (moving towards the pier) when it is 16 feet from the pier? Pier boar The boat is docking at feet per minute. 8A Sect # Name 4. Let fix) = \/1 + 2m. A. Find the linearization L(:c) of f at a :4. LOB) = B. Use L(m) to approximate V9.06. Hint: \/9-0 = f(4.03). Write your answer as a single fraction or a decimal. 9A 5. Let f(a:) = 41:5 — 5334. A. Determine the following for f (if none, write “none”): (a) w—intercept (s) (b) Show the signs of f’ and f ” on the number lines below: fl ‘——_~—_f” (c) local maximum at a: 2 local maximum value: ((1) local minimum at a: 2 local minimum value: (e) inflection point(s) at a: = B. Sketch the graph of y = 42:5 ~— 5234, using the information above. Label 3 all important features of the graph. Note that f (1) m —O.6. 10B MAC 2311 Spring 2010 Exam 3B A. Sign your scantron sheet in the white area on the back in ink. B. Write and code in the spaces indicated: 1) Name (last name, first initial, middle initial) 2) UF ID number - 3) Discussion Section number C. Under “special codes”, code in the test number 3, 2. 1 1 2 e 4 5 6 7 8 9 0 ' 1 a 3 4 5 6 7 8 9 0 D. At the top right of your answer sheet, for “Test Form Code” encode B. A e C D E E. This test consists of 10 five—point, one three—point and two one—point multiple choice questions, bonus questions worth 4 points and two sheets (4 pages) of partial credit questions worth 25 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before turning in your test check for transcribing errors. Any mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion leader or proctor. Be prepared to Show your picture ID with a legible signature. 3) The answers will be posted on the MAC 2311 homepage after the exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted on Elearning within one week. Questions 1 - 10 are worth 5 points each. 1. Find each value of as at which f (cc) 2 11:2 + ln(:c2) has an inflection point. a.:c=0,:c=:l:1 b.:c=:l:10nly c.:c=0,:c=:l:2 d. cc 2 :l:2 only e. f has no inflection points. 2. Find the absolute maximum and minimum values of f(:c) I= ems‘3m+2 on [0, 2]. a. 62, e b. 64, 62 c. 64, 1 d. 64, 0 e. 62, 1 3. A particle moves in a straight line so that its position (in feet from a starting point) after t minutes is given by 5(t) = 6752 — t3. Find the displacement of the particle and the total distance traveled on the time interval 0 S t S 5. displacement total distance a. 25 ft ' 57 ft b. 32 ft 39 a c. 32 ft 57 ft d. 25 ft . 39 ft e. 57 ft . 89 ft 2B xi” .,I 4. The 1adius of a sphere is measured to be 7' = 3 inches. If tlie/‘inear surement of 7‘ has a possible error of i0. 03 inch, use dlfferentlalséjt 3% approximate the percentage error that can occur in using this mea—‘Vr 95/ sure to compute the surface area of the sphere. . Note: the surface area S of a sphere is given by S = 471' T2. a. 2.4% b. 1% c. 2.01% d. 1.02% e. 2% 5. A particle moves along the graph of y = tan(:c) sec(:c) so that y is in— creasing at a rate of 3 units per second. Find the rate at which :1: is . . , 7r changmg Wlth respect to time when a: = — . . . 1 . . a. :5 1s increasmg by — un1ts per second. \/§ b cc is increasing by— units per second. 3—35 V? c. :5 is decreasing by ? units per second. 3 d. :L' is increasing by # units per second. \/§ e. :L' is decreasing by \/§ units per second. 6. Find all values of c guaranteed by the Mean Value Theorem for f(:c) = :33 — 451: on [0,3]. 1 a.c=1onl b.c=— 3andc=\/3 c.c=—onl y \/3 y dc—x/30nl ec——iandc——1~ ' y ' \f5— x/E 3B 7 . 47.51% a! s‘ 2 a: — cos a: on [0, 277’)” 7. Find the critical numbers of f (as) = sin 57r 77r 27r 47r 7r 3 ax=0,F,7r,F bm—O,?,7r,? cos—3,3— 27T 7T 47T 7T 57r d.$—?,§,§' e.$—0,§,7T,? (a: — 1)2 . . 8. The local extrema of f (3:) = 3: occur at Wthll of the followmg as—values? local minimum local maximum at a: 2 at a: = a. 0 —1, 1 b. —1 1 c 1 —1 d 1 none e —1, 1 0 9. Which of the following statements is / are true of f (as) = (3:2 — 4)1/3? , 23: Note that f (33) = m. A. f (as) has exactly one critical number. B. f(3:) is increasing on (0,2) and (2,00). C. f (as) has vertical tangent lines at a: = —2 and a: = 2. a. B and C only b. C only c. A, B and C d. A only e. A and B only 4B 532—3 9—932 _2:1:2—36 $3 ,then f’(:1:)= 11:4 and f”’(:1:) 10. If f(:1:) = :35 Find each interval on which f is both increasing and concave up. a. (—3,3) b. (—00,—3\/§)U(3,oo) , c. (—3\/§,—3) d. (—3,0)U(3\/§,oo) e. (—3,0) only 11. (3 points) The position function of a snail moving along a straight path t — 3 . . . . is 3(t) = m, where 3(t) is the d1stance in centimeters from the start1ng point after t minutes. Find the acceleration of the snail after 1 minute. 1 2 5 a. Q cm/min2 b. —2—7 cm/min2 c. 6 cm/min2 10 . 2 . 2 (1. —§% cm/mm ‘e. —1 cm/m1n The following true/ false questions are worth one point each. 12. If y = f(:1:) is a continuous function so that f’(6) = 0 and f”(6) = 2, then f has a relative maximum at :1: = 3. a. True b. False 13. The position function for an object moving in a straight line is given by .9 = fft). If f’(b) > 0 and f”(b) < 0, then the object is slowing down at time t : b. a. True b. False 5B BOIIUS” (one point each) The graph of the derivative of7 . ' uous function f is sketched below. e Use the graph to determine if the following statements about f are true or false. 14. f is concave up on the interval (0,00) only. a. True b. False 15. f is increasing on intervals (—2, 0) and (2,00). a. True b. False 16. f(a:) has a local maximum at a: = 0. a. True b. False 17. f has inflection points at a: = —1 and a: = 1. a. True b. False 6B MAC 2311 TEST 3B PART 11 SPRING 2010 Sect # a Name UFID Signature SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. Let‘f(:c) : $51”. a. Use logarithmic differentiation to find f ’(cc). f’(rc) = 3 1). Find the slope of the tangent line to f(:c) at cc = g. 7Tb 7B 2. A boy standing at the end of a pier which is 12 feet above the water is pulling on a rope that is attached to a rowboat in the water. If he is pulling the rope in at 8 feet per minute, how fast is the boat docking (moving towards the pier) when it is 16 feet from the pier? The boat is docking at feet per minute. 3. To which of the functions below can we apply Rolle’s Theorem on the interval [0,8]? Write “can apply” or “cannot apply”. If you can apply Rolle’s Theorem, find each value of c guaranteed by the theorem. If not, state a condition not satisfied by the function. a. f(:1:) = 1E2/3 — 2331/3 8:13—33 10- flit) (13—1 8B' Sect # Name 4. Let f(:c) : \/2IE — 1. A. Find the linearization L(a:) of f at a = 5. L(a:) = B. Use L(:c) to approximate V9.18. Hint: v9.1 : f(5.09). Write your answer as a decimal. 9B 5. Let f(:c) : 5:36 — 6:35. A. Determine the following for f (if none1 write “none”): (a) m-intercept(s) (b) Show the signs of f’ and f ” on the number lines below: fl mf/l (c) local maximum at :6 i local maximum value: ((1) local minimum at :c 2 local minimum value: (e) inflection point(s) at :c = B. Sketch the graph of y = 5x6—6x5, using the information above. Label 4 all important features of the graph. Note that f (—5—) % —O.65. 10B ...
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