1242_Fall2006 Calculus exam - MATH 1242 COMMON FINAL...

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Unformatted text preview: MATH 1242 COMMON FINAL EXAMINATION PART I Fall 2006 This exam is divided into three parts. You have three hours for the entire test. Part I consists of 13 multiple choice questions to be done without using any calculator. Part I is collected after one hour (if exam has started at 8:00 8.111., Part I is collected at 9:00 am). Calculators are allowed only on Part II and III. Part II contains 12 multiple choice questions. A special answer sheet is provided so that your answers can be machine graded. Part III contains 6 free response questions, for which your work on the exam will be graded. These pages contain Part I which consists of 13 multiple choice questions. It must be done without using any calculator. a You must use a pencil with a soft black lead (#2 or HB) to enter your answers on the answer sheet. a For each question choose the response which best fits the question. a If you wish to change an answer, make sure that you completely erase your old answer and any other extraneous marks. 9 There is no penalty for guessing. e If you mark more than one answer to a question, that question will be scored as incorrect. e You may perform your calculations on the test itself or on scratch paper, but do not make any stray marks on the answer sheet. a Make sure that your name appears on the answer sheet and that you fill in the circles corresponding to your name. At the end of the examination you MUST hand in this test booklet, your answer sheet and all scratch paper. MATH 1242 , FINAL EXAM jail 2006 Part I (MULTIPLE CHOICE, NO CALCULATORS). 3 1, / 3$2dso l 934 (a)( +$2~5x> 111cc+C 11:3 (b) E+2xv51na§+0 4 L+932—5m ~4-—————~+C .1332 (C) 2 (d) :132 + 2:6 ~ 5w‘1 + C “4 2‘2w20 (e)u1,+5¢4 93+C 3v /(Sin$+$2+2)d1‘ $3 cosx—*§—+2m+0 GOSH?" 23U+C (a) (b) (c) (30830 + 43; + C’ (d) wcosa: + 2:1: + C » ,5 (e) ~COS$+~E~+2QS+C MATH 1242 2 (d) ~§(1—az2)3/2~C (‘3) ”31;“— 2W30 FINAL EXAM Spring 2006 MATH 1242 FINAL EXAM Spring 2006 7, The MacLaurin Series (Taylor series centered at a z 0) for f (at) : sinzL' is 2 4 6 CC 3: I 1.3 m5 (L7 (b) IE*§+BT“7!‘+ 1 1 3:2 333 1:4 (C) *1+*2-!~~‘§{+ZT+ 1:3 :35 337 (d) l+§+§+7fl+ 1 51:2 ([73 r4 (G) +$+§T+§T+ZT+ (e) :5 — cos (9:) 9. The average value of f (at) 2 3x2 + 2 on the interval [0, 3] is (803 (W2 (C) 8 ) 1 ) CL ( (e ©l—-‘P—-‘ MATH 1242 FINAL EXAM Spring 2006 00 n 10. The series 2 my + 4 (a) Converges by ratio test (b) Diverges by ratio test . 1 (c) Converges by comparison to Z - n n21 (d) Diverges by integral test (8) Converges by integral fiest 11 What is the fjf a:)d:c if the graph of f($ )is a C5“ ( ( ( (d 3 6 9 O 12 i5 ) ) ) ) () 6 6 3 12. If/ f(:c)d:c=17 and/ f(a:)dm=8,then/ f(ac)d:cis 3 0 (a) 9 (b) 25 (c) 8 (d) ~9 (6) ~25 13. Find the value of the improper integral /00 £33m . 1 (a Converges to —6 (b Converges to ~3 ) ) (C) Converges to 3 (d) Converges to 6 > (e Divergent MATH 1242 4 Fall 2006 COMMON FINAL EXAMINATION PART II Name: Instructor: Student ID #: Section/Time: This exam is divided into three parts. You have three hours for the entire test. Part I consists of 13 multiple choice questions to be done without using any calculator. Part I is collected after one hour (if exam has started at 8:00 am, Part I is collected at 9:00 am). Calculators are allowed only on Part II and III. Part II contains 12 multiple choice questions. A special answer sheet is provided so that your answers can be machine graded. Part III contains 6 free response questions, for which your work on the exam will be graded. These pages contain Part II which consists of 12 multiple choice questions. 0 You must use a pencil with a soft black lead (#2 or HB) to enter your answers on the answer sheet. a For each question choose the response which best fits the question. a If you wish to change an answer, make sure that you completely erase your old answer and any other extraneous marks. 4 0 There is no penalty for guessing. e If you mark more than one answer to a question, that question will be scored as incorrect. 0 You may perform your calculations on the test itself or on scratch paper, but do not make any stray marks on the answer sheet. a Make sure that your name appears on the answer sheet and that you fill in the circles corresponding to your name. At the end of the examination you MUST hand in this test booklet, your answer sheet and all scratch paper. MATH 1242 FINAL EXAM Fall 2006 Part II (MULTIPLE CHOICE, CALCULATORS ALLOWED). 1, Find the area of the region bounded by the curves y = fl and y : :13. 2. Which of the following is the sum of the Geometric Series E 3 (~1/2)”‘1 71:1 ivergent 3. A region in the first quadrant is bounded by the curves y = $2, y 2: O and a: = 2. Find the volume obtained by rotating the region about the az—axis. MATH 1242 FINAL EXAM Sprung 2006 3 2 1 4. Let anzw—j—Mfornzl,2,3,~- 8—412 Which of the following is true? (a) The sequence {can} converges to 3 (b) The sequence {an} converges to 3/8 (c) The sequence {on} converges to w3/8 (d) The sequence {an} converges to —3 ) (e The sequence {on} diverges. 5. Let f (x) 2 V4 — 332. Find the Riemann Sum for f (2:) on the interval [0, 2] using n z 4 subdivisions using the midpoint rule. (Round to 3 decimal places) (a) 1/6 (b) 1/3 (C) 1/2 (d) 2 (e) /2 fidm is a divergent improper integral .‘Ifi‘ 9"? 1 (5,“; I; J , (Jr . r1 MATH 1242 FINAL EXAM . @2006 v «A; 7. Hooke’s Law states that the force required to maintain a spring stretched :0 units beyond its natural length is proportional to at: F 2 km, where k is the spring constant. Suppose that a force of 20N is required a spring from its natural length to a length of .5m beyond its natural length. How much work is required to stretch a spring from its natural length to 1m beyond its natural length? 5 1 03 223—2 :E 9. The integral / (a) is a common definite integral and equals 1n 3 (b) is an improper definite integral and equals In 3 ) ) (c (d is a common definite integral, but can only be approximately calculated nu- merically ’ is a divergent improper definite integral (9) none of the above is true. FM MATH 1242 FINAL EXAM Spring 2006 . . — 3)” . 10. The 1ntegral of convergence for the power serles 2 4n IS 71:21 (a) (7, 00) (b) (~00,~1) (C) (—00,—1)U(7,00> (d) ["130 (e) (—177) 1 ‘/ 2 w 2 11. One of the Table Formulas 18 dd/ ln (1 + a u + c. m/ (12“ — 1L2: a U Use this to find /——E—)————-— arx/16— $9332 1 4 + V 16 — 31:2 (a) —~ln ——————~—- +0 4 a: 5 4 V - 2 (13) .mm Lew—:1 +0 4 :c 5 4 + V 16 — 9:232 , ((1) -ln W c 4 :23 1 4 V1 ~ 2 (d) ._1H 1M +6 4 3:0 5 4 + V 16 — 9962 (e) 1n : c 4 3:1: . °° (—1)” 12. Th e senes Z n _ 7121 (a) is divergent (b) is absolutely convergent (c) is convergent, but not absolutely convergent ((1) equals 0 (e) cannot be determined MATH 1242 COMMON FINAL EXAMINATION FREE RESPONSE SECTION FALL 2006 This exam is divided into three parts. These pages contain Part III which consists of 6 free response questions. Please show all of your work on the problem sheet provided. We will not grade loose paper. a If you are basing your answer on a graph on your calculator, sketch a picture of your graph on your sheet and be sure to label your window. 0 Make sure that your name appears on each page. At the end of the examination you MUST hand in this test booklet and all scratch paper. PROBLEMII 12 ‘3 l4 hs 6 GRADE j l ' FREE RESPONSE SCORE: Name: Student No: Instructor: Section No: MATH 1242 FINAL EXAM FALL 2006 PART III (Free Response, ealculators allowed) 1‘ Evaluate the integrals and you must Show all work to receive credit! m+l a 3 oints Find / Weir. ()( p ) 3V$2+2$+5 4523+?) (c) (3 points) Find / :ccos(3x)da:. 2. Consider the finite region in the plane bounded by the curves y z 51:2 and y = a3. (a) (5 points) Set up and evaluate the integral that finds the volume generated by revolving the area about the as—axis. (b) (5 points) Set up and evaluate the integral that finds the volume generated by revolving the area about the line :13 2 2. 3. (a) (3 points) State whether the series is convergent or divergent, and if it converges find the sum (do not forget to state type or test of the series and, show your work!) (1)) (3 points) State whether the series converges or diverges and state the test used. 00 _2 Ens". 11:1 (c) (4 points) Find the interval of converges of the power series 00 a: 3n 2( 7;”). 71:1 4. (a) (4 points) Use the Trapezoid Rule with n : 6 to approximate f0?) 63320333 Note: Tn : égbtbso) +2f(031) +2f($2) +---+2f(93n*1) +f(.7:n)]. (b) (4 points) Use Simpson’s Rule with n z 6 to approximate fog 6332de Note: Sn = 9;- [f ($0) + 4f (1131) + 2f (062) + - ' - + 4f(xn~1) + f (5%)], c 2 point Use your calculator to find 3 @2026. Round to 4 decimal places. 0 2 5. (a) (3 points) Find a poWer series representation for the function f (1:) z 1 + , . cc (b) (1 point) Determine the interval of convergence for the power series represenv 2 tat‘m of :1: r: . l 1 f( ) 1 + a: . . . . . 2 (C) (3 pomts) End a power semes representatron for the functlon f (x) 2 1 + 4 x (Hint: use part (21)). . . . 2 (d) (3 pomt) Use Part (c) to find a power serles representatlon for f 1 + 4033:. m 6. An inverted triangular shaped trough (see picture) which has a height of 10m, a base of 10m, and a width of 18m. If the trough is filled with water, find the work done in pumping all the water out tenangliqsrignngexniyZMIneLersnhnyethe.trengh. (The acceleration due to gravity is 9.8m / 52, and the density of water is 1000/59 / m3). To obtain full credit you must show all details of your calculations. ...
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