MATH 1242 FALL 2008 COMMON FINAL EXAMINATION PART I. Instructor:

Transcription

1 MATH 14 FALL 008 COMMON FINAL EXAMINATION PART I Name Student ID Instructor: Section/Time This exam is divided into three parts. Calculators are not allowed on Part I. You have three hours for the entire test, but you have only one hour to finish Part I. You may start working on the other two parts of the exam whenever you are done with Part I, but you cannot use your calculator until ALL of the Part I answer sheets are collected. After these answer sheets are collected, your instructor will announce that calculators are allowed on Parts II and III. These pages contain Part I which consists of 1 multiple choice questions. These questions must be answered without the use of a calculator. You must use a pencil with a soft black lead ( or HB) to enter your answers on the answer sheet For each question choose the response which best fits the question If you wish to change an answer, make sure that you completely erase your old answer and any other extraneous marks. There is no penalty for guessing If you mark more than one answer to a question, the question will be scored as incorrect. You may perform your calculations on the test itself or on scratch paper, but do not make any stray marks on the answer sheet. Make sure that your name appears on the answer sheet and that you fill in the circles corresponding to your name. After 1 hour, you MUST hand in the answer sheet for Part I. At the end of the exam, you MUST hand in all remaining test materials including test booklets, the answer sheet for Part II, and scratch paper.

2 MATH 14 FINAL EXAM PART I FALL Given the table of values below, estimate the area under the graph of C œ 0ÐBÑ on the interval Ò"ß&Ó using two approximating rectangles of equal width and right endpoints. B 0ÐBÑ! " " % & % ) & "% (a) 31 (b) 33 (c) 34 (d) 36 (e) 38 &!!. Let ' 0ÐBÑ.B œ &, let ' 0ÐBÑ.B œ ß and let ' 0ÐBÑ.B œ Þ Evaluate ' 0ÐBÑ.BÞ (a) ( (b) & (c) & (d) ( (e) )! "& "% ' "& " ' 3. Evaluate ' " B.B (a) (b) (c) (d) (e) 4. Evaluate ' B.B ) " " (a) G (b) G (c) G (d) G (e) G B B B %B B % % 5. Evaluate ' " B Ð" BÑ.B (a) ˆ " ˆ " " B B B B G (b) % " B " B " " (c) % " B " B (d) B B G " (e) B B G % 6. Evaluate 'Ð B BÑ.B " % " " (a) B B G (b) 3 B+ B G (c) B %68lBl G " " (d) B B %68lBl G (e) B B %68lBl G 1

3 MATH 14 FINAL EXAM PART I FALL Evaluate '! B " B " " " % %.B (a) (b) (c) (d) (e) B! " B ' 8. Use the substitution? œ " B to evaluate.b. Which of the followiing is '! B equal to " B.B? (a) '.? & (b) '.? " (c) '.? " & (d) '.? & (e) '.?.!? "?!? "? "? 9. Evaluate ' BÈB ".B " (a) (b) (c) 1 (d) (e) " "% "' "( "& "& "& "& 10. Evaluate ' =38 B %-9= B.B (a) -9= B %=38ÐBÑ G (b) -9= B %=38 B G (c) -9=ÐBÑ =38 B G (d) -9=ÐBÑ %=38 B G (e) -9=ÐBÑ %=38 B G 11. Evaluate ' B B/.BÞ " " / B " B B B (a) B / G (b) B/ / G (c) B B " G (d) B/ B / B " G (e) B / B Î G 1. Evaluate '! lb "l.b " " (a) 0 (b) (c) (d) (e) 1 % %

4 MATH 14 FALL 008 COMMON FINAL EXAMINATION PART II Name Student ID Instructor: Section/Time These pages contain Part II which consists of 13 multiple choice questions. After the answer sheets for Part I have all been collected, and your intstructor announces that calculators are OK, you are allowed to use a calculator on this part of the exam. You must use a pencil with a soft black lead ( or HB) to enter your answers on the answer sheet For each questions choose the response which best fits the question If you wish to change an answer, make sure that you completely erase your old answer and any other extraneous marks. There is no penalty for guessing If you mark more than one answer to a question, the question will be scored as incorrect. You may perform your calculations on the test itself or on scratch paper, but do not make any stray marks on the answer sheet. Make sure that your name appears on the answer sheet for Part II and that you fill in the circles corresponding to your name. At the end of the exam, you MUST hand in all remaining test materials including test booklets, the answer sheet for Part II, and scratch paper.

5 MATH 14 FINAL EXAM PART II FALL Given that i) ' " " " >+8?.? œ?>+8? 68 "? G "? " ii) '?>+8?.? œ >+8? G Evaluate the integral '/ B >+8 " / B.B. (a) e (b) e " B " B " %B % B " B " %B "? >+8 / 68 " / G >+8 / 68 " / G " B / >+8 / G %B B (c) / " % % (d) / %B " " B /B (e) / %B " B " B / % >+8 / G >+8 / G ' " B ". Use the fundamental theorem of calculus to find the derivative of 0ÐBÑ œ " >.> % " " " (a) " B% (b) " B% (c) " B% " (d) (e) " "'B " "'B % % 3. Find the area enclosed by the curves C œ B and C œ B Þ (a) (b) (c) (d) (e) " " " " " ' & % 4. What is the volume of the solid obtained by rotating the region bounded by C œ B ß C œ ß and, B œ! about the C axis? (Note that the answers below are given in terms of integrals--you do not need to evaluate them.) " (a) 1'! Ð B Ñ.B (b)1' "! Ð B Ñ.B (c)1' Ð B Ñ.B! "!! (d) 1' B B.B (e) 1' B B.B 1

6 MATH 14 FINAL EXAM PART II FALL A particle is moved along the B -axis by a force that measures "! " B pounds at a point B feet from the origin. Find the work done in moving the particle from the origin to a distance of 5 feet. (a) 75 ft-lbs. (b) 300 ft-lbs (c) 35 ft-lbs. (d) 350 ft-lbs. (e) 400 ft-lbs. 6. Consider two infinite sequences: I " 8 8 " II š 8 " È8 Which of the following statements is correct? (a) Neither sequence converges. (b) Both sequences converge. (c) I converges, but II does not converge. (d) II converges, but I does not converge. (e) The limit of sequence I is one Suppose that + " œ and + 8 " œ + for 8 "Þ Evaluate 8 + Þ Round your answer to % decimal places. lim 8Ä 8 (a) The limit does not exist. (b) Þ%%!" (c) Þ%%)( (d) Þ%%*& (e) Þ%&!" + 8. Which of the following is closest to the sum of the series 3? (a) 1.4 (b) 1.45 (c) 1.5 (d) 1.55 (e) 1.6 8œ" 8 "

7 MATH 14 FINAL EXAM PART II FALL Consider three series: 8 8 8œ" 8œ" 8x 8œ" I. 8 ß II. III. 8 Which series coverge(s)? (a) I only (b) II only (c) III only (d) I and III only (e) II and III only 10. Find a power series representation for the function B (a) " B %D )B ÞÞÞ (b) " B B B ÞÞÞ % (c) B B %B )B ÞÞÞ % (d) B B B )B ÞÞÞ % (e)b B %B )B ÞÞÞ " B In the Taylor series for 0ÐBÑ œ =38ÐBÑ centered at, what is the coefficient of 1 ÐB Ñ (a) 1 (b) " (c) " (d) " (e) " '? % 1. Use the Trapezoidal rule with 8 œ to approximate the value of ' "! B ". Round your answer to two decimal places. (a) 1.55 (b) 1.64 (c) 1.73 (d) 1.87 (e) What is the average value of the function 0ÐBÑ œ 3B on the interval Ò"ß 3 Ó? (a) 10 (b) 11 (c) 1 (d) 13 (e) 14 3

8 MATH 14 FALL 008 COMMON FINAL EXAMINATION PART III Name Student ID Instructor: Section/Time These pages contain Part III which consists of 5 free response questions. Please show all your work in this test booklet. Loose paper will not be graded. If you are basing your answer on a graph on your calculator, sketch this graph in the answer booklet. Be sure to label your window by putting a scale on each axis. At the end of the exam, you MUST hand in all remaining test materials including test booklets, answer sheet, and scratch paper. PROBLEM GRADE FREE RESPONSE SCORE:

9 MATH 14 FINAL EXAM PART III FALL In this problem, we calculate the arclength of the parametric curve C œ > ß B œ > ß! Ÿ > Ÿ % (a) Express the arclength of the parametric curve C œ > ß B œ > ß! Ÿ > Ÿ % in terms of an integral. Be sure to indicate the limits of integration, but do not evaluate the integral yet. " " (b) Evaluate the integral from part (a). (Factoring the expression inside the square root and using È+, œ È+ È, will help.) 1

10 MATH 14 FINAL EXAM PART III FALL 008 B. In this problem, we wish to approximate the value of B/.B using a power series representation. B B B B (a) Recall that / œ " B x x ÞÞÞ Find a power series representation for / Þ You need only show the first four non-zero terms. '! " (b) Use the result of part (a) to find a power series representation for only show the first four non-zero terms. B B/ Þ You need B (c) Use the series in part (b) to evaluate B/.B with an error less than '! "

11 MATH 14 FINAL EXAM PART III FALL (a) Find the partial fraction expansion of B ) B B. (b) Use the answer you found in part (a) to evaluate ' B ) B B.B. 3

12 MATH 14 FINAL EXAM PART III FALL 008 B 8 " 8œ! 4. Find the interval of convergence for the power series Be sure to check for convergence or divergence at the endpoints of this interval

13 MATH 14 FINAL EXAM PART III FALL A pool is shaped liked a cone. If we look down at the pool from above, the cross sections are circles. If we look from the side, the cross section is an isosceles triangle as shown below. The pool is filled with water (density 6.5 lb/ft ) to a depth of 8 feet ( feet below the top). How much work does it take to pump the water to a level 3 feet above the top of the pool? 5