1. (4 % each, total 20 %) Answer each of the following. (No need to show your work for this problem). 3 n. n!? n=1
|
|
- Jonah Adams
- 6 years ago
- Views:
Transcription
1 NAME: EXAM 4 - Math 56 SOlutions Instruction: Circle your answers and show all your work CLEARLY Partial credit will be given only when you present what belongs to part of a correct solution (4 % each, total 0 %) Answer each of the following (No need to show your work for this problem) (a) Write down the Maclaurin series of f(x) = e x x n Answer:, Interval of convergence: (, ) (b) Write down the Maclaurin series of f(x) = x Answer: x n, Interval of convergence: (, ) (c) What is the sum of this convergent series Answer: As n= n = e, then answer is e Discussion: Many get the answer e But e = n? n = + n= n The problem wants to see if students know what e really is in terms of Maclaurin series, and if students know how to compute the first term of the Maclaurin series { x = e t + t + (d) For the curve y = t, find the value(s) t at which the curve has horizontal t + 5 tangent lines Answer: Compute dy dt = (t 4) Therefore, the value(s) t at which the curve has horizontal tangent lines are t = and t = (e) Find a Cartesian equation for the curve r = 4 cos(θ) Answer: Multiplying both sides by r to get r = 4r cos(θ) This leads to x + y = 4x, or, after completing the squares, (x ) + y = 4 Discussion: Common errors are from those students who do not know what to compute Some of us use y = 4x x But y = 4x x represents only part of the original curve, and so it is a wrong answer
2 (0 %) Find the Maclaurin series for the function xe x+ and determine the radius of convergence Solution Since this is a Maclauring series, this means a = 0 Therefore, we first need to single out e x To do that, we use Laws of exponents to obtain the wanted Maclaurin series: xe x+ = xe e x = xe To determine the radius of convergence, we compute lim n Hence the radius of convergence in e (n+)! e x n = e xn+ = lim n n + = 0 Discussion: The purposes of this exercise is two fold: one is to see if students know how to use algebra to correctly compute the Maclaurin series using the known ones (such as e x ); the other is to see if students know the difference between a Taylor series and a Maclaurin series, which is the case when a = 0 The most common wrong solution is: xe x+ (x + ) n = x x(x + ) n = This indicates that the student did not understand Maclauring series must expand the series at a = 0, and the series can only have x n terms, and cannot have (x + ) n terms Only a Taylor series at a = can have (x + ) n terms (5 %) Find the Maclaurin series for the function tan ( x ), and determine the radius of convergence and the interval of convergence Solution If we can remember that tan (u) = we can directly use u = x to get tan ( x ) = ( ) n ( ) x n+ = n + ( ) n n + un+, with IC = [, ], then ( ) n n+ (n + ) xn+ As x, we conclude that the interval of convergence of this Maclaurin series is [, ] Solution Suppose that we do not remember the Maclaurin series of tan (u) We can use + u = ( ) n u n,
3 together with differentiation and integration to get it Let f(x) = tan ( x ) Then Then f (x) = f(x) = + ( ) x = ( ) n n+ ( ) x n ( ) n = x n dx = ( ) n n+ xn ( ) n n+ (n + ) xn+ + C Set x = 0 to get 0 = tan (0) = C, and so tht Maclauring series is tan ( x ) = ( ) n n+ x n dx = ( ) n n+ (n + ) xn+ To determine the interval of convergence, we use ration test For us to conclude that the seriesis convergent, we need lim n x n+ n+ (n+) x n+ n+ (n+) x (n + ) = lim n (n + ) = x < It follows that x < or x < We now need to check the convergency at x = and x = At x =, the series becomes ( ) n n+ (n + ) n+ = By Alternating series test, this is convergent ( ) n n + At x =, the series becomes ( ) n n+ (n + ) ( )n+ = ( ) n ( ) n+ n + = ( ) n+ n + By Alternating series test, this is convergent Therefore, the interval of convergence is [, ] 4 (5 %) Find the Taylor series of and the interval of convergence Solution Write x at a =, and determine the radius of convergence x = (x ) = (x ) n Using Ratio test, the radius of convergence is, and so the interval of convergence contains (, + ) = (, ) At x =, the series becomes ( ) n = ( ) n By the Divergence Test, the series is divergent At x =, the series becomes ( ) n = n By the Divergence Test, the series is divergent Thus the interval of convergence is (, )
4 Discussion: There are only two Taylor series problems in the test This is one of them As we are seeking Taylor series at a =, the answer must have the form f (n) () (x ) n Some of us, mostly those not attending the classes, chose to directly compute f (n) () and end up nowhere, due to the complexity of the higher order of differentiations The uniqueness of Taylor is the way we recommended in class One common error is the following: x = ( x ) = ( ) x n Those who answered the problem this way might be trying to remember what they did in the past, without understanding what we are now computing That was for the Maclaurin series (that is, Taylor series at a = 0) But we are now computing Taylor series at a =, a totally different problem 5 (0 %) Find a function f(x) whose Maclaurin series is ( ) n xn Solution Compare it with the e x Maclaurin series, we note that ( ) n xn Thus the answer is f(x) = e x = ( ) n ( x ) n = e x 6 (0 %) Find the first 4 terms of the Taylor series of f(x) = + x at a = Solution Thus, Use the formula a n = f (n) (a) and compute f(x) = ( + x) f() = a 0 = f (x) = ( + x) f (x) = ( + x) f () = f (x) = 0 7 ( + x) f () = f(x) = + x = + f () = a = (x ) + ( ) a = 0 7 a = 5 8 (x ) (x ) +
5 Discussion: Many chose to apply the formula: ( ) k ( + x) k = x n k(k )(k )(k n + ) = + x n, r = n n= Why this is NOT applicable here? Because the formula above is for the Maclaurin series (that is, Taylor series at a = 0), and in this problem, we are computing the Taylor series at a = As discussed in class, we recommended that in this case, we should use f(x) = f (n) (a) (x a) n, with f(x) = ( + x) and a = Can we use the Maclaurin series in this problem? Yes, we can But we must formulate the problem correctly Here is how Using algebra, firstly we rewrite the function as f(x) = ( + x) (x ) = ( + (x )) = ( + ) Then we can apply the formula as follows: ( f(x) = + = ( + = + n= ) (x ) n= ( )( )( n + ) ( )( )( n + ) ( ) ) x n ( ) x n Then, we compute: (it takes a bit high school algebra to see the two solutions yield the same answers) n = 0, a 0 = n =, a = = 6 n =, a = n =, a =! =! = (5 % each, 0% total) Given a parametric curve x = 4 + t and y = t + t, (i) Find dy dx (ii) Find d y dx (iii) Find the values of t at which the curve has horizontal tangent line and at which the curve has vertical tangent line
6 (iv) Write down the integral that computes the length of the curve with t (No need to simplify your answer Do NOT evaluate the integral ) Solution Compute dy dt = t + t and dx dt = t (i) Thus y = dy dx = t+t t (ii) Compute dy dt (iii) Set y lim t 0 dy dx tangent lines) = ( + t)/ = Hence d y dx = t = 4t = 0 to get t = Thus at t =, we have a horizontal tangent line As =, the curve has no vertical tangent line (Review Calculus I for vertical (iv) The integral computing the Arc length is (t + t ) + (t) dt Grade Distribution of Exam 4: Meaning of the scores: The highest score of exam 4 is 4/00 at least 0 = Very good, familiar with the related materials and skillful, with minimal computational errors Keep o 80-8 = good, familiar with most of the related materials, with a few computations errors Make an effort to do better 70-7 = OK, not so familiar with the related materials, with relatively more computational errors We have room to improve (For this quiz, not familiar with differentiation) 60-6 = Passing, We are ont the borderline of failing It indicates that we are less familiar with the related materials and more computational errors and algebraic errors We have lots of room to improve at most 5 = We failed We need to catch it up It should definitely be the time for us to see the instructor and get assistance to understand the materials and to practice MORE Scores Frequency Percentage 64 64
Math 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: CUID: Section: Fall 27 8., 8.2,. -.4 Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook,
More informationMath156 Review for Exam 4
Math56 Review for Eam 4. What will be covered in this eam: Representing functions as power series, Taylor and Maclaurin series, calculus with parametric curves, calculus with polar coordinates.. Eam Rules:
More informationMA 126 CALCULUS II Wednesday, December 14, 2016 FINAL EXAM. Closed book - Calculators and One Index Card are allowed! PART I
CALCULUS II, FINAL EXAM 1 MA 126 CALCULUS II Wednesday, December 14, 2016 Name (Print last name first):................................................ Student Signature:.........................................................
More informationMA CALCULUS II Friday, December 09, 2011 FINAL EXAM. Closed Book - No calculators! PART I Each question is worth 4 points.
CALCULUS II, FINAL EXAM 1 MA 126 - CALCULUS II Friday, December 09, 2011 Name (Print last name first):...................................................... Signature:........................................................................
More informationMath 113 Winter 2005 Key
Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple
More informationWithout fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
More informationTest 3 - Answer Key Version B
Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop,
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More informationUNIVERSITY OF REGINA Department of Mathematics and Statistics. Calculus I Mathematics 110. Final Exam, Winter 2013 (April 25 th )
UNIVERSITY OF REGINA Department of Mathematics and Statistics Calculus I Mathematics 110 Final Exam, Winter 2013 (April 25 th ) Time: 3 hours Pages: 11 Full Name: Student Number: Instructor: (check one)
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series.
MATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series. The objective of this section is to become familiar with the theory and application of power series and Taylor series. By
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationMA 126 CALCULUS II Wednesday, December 10, 2014 FINAL EXAM. Closed book - Calculators and One Index Card are allowed! PART I
CALCULUS II, FINAL EXAM 1 MA 126 CALCULUS II Wednesday, December 10, 2014 Name (Print last name first):................................................ Student Signature:.........................................................
More informationExam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:
Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may
More informationFriday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
More informationWithout fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response
More informationMATH 152, Fall 2017 COMMON EXAM II - VERSION A
MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones
More informationMathematics 111 (Calculus II) Laboratory Manual
Mathematics (Calculus II) Laboratory Manual Department of Mathematics & Statistics University of Regina nd edition prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry University of Regina Department
More informationMA162 EXAM III SPRING 2017 APRIL 11, 2017 TEST NUMBER 01 INSTRUCTIONS:
MA62 EXAM III SPRING 207 APRIL, 207 TEST NUMBER 0 INSTRUCTIONS:. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and use a #
More informationYou can learn more about the services offered by the teaching center by visiting
MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources
More informationDO NOT WRITE ABOVE THIS LINE!! MATH 181 Final Exam. December 8, 2016
MATH 181 Final Exam December 8, 2016 Directions. Fill in each of the lines below. Circle your instructor s name and write your TA s name. Then read the directions that follow before beginning the exam.
More informationFall 2016, MA 252, Calculus II, Final Exam Preview Solutions
Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and
More informationSection 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series
Section 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series Power Series for Functions We can create a Power Series (or polynomial series) that can approximate a function around
More information5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationSpring 2015, MA 252, Calculus II, Final Exam Preview Solutions
Spring 5, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card,
More informationNORTHEASTERN UNIVERSITY Department of Mathematics
NORTHEASTERN UNIVERSITY Department of Mathematics MATH 1342 (Calculus 2 for Engineering and Science) Final Exam Spring 2010 Do not write in these boxes: pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 Total (100 points)
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationMTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Solutions to Exam 2 April 9, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
More informationJim Lambers MAT 169 Fall Semester Practice Final Exam
Jim Lambers MAT 169 Fall Semester 2010-11 Practice Final Exam 1. A ship is moving northwest at a speed of 50 mi/h. A passenger is walking due southeast on the deck at 4 mi/h. Find the speed of the passenger
More informationMath 113/113H Winter 2006 Departmental Final Exam
Name KEY Instructor Section No. Student Number Math 3/3H Winter 26 Departmental Final Exam Instructions: The time limit is 3 hours. Problems -6 short-answer questions, each worth 2 points. Problems 7 through
More informationMultiple Choice Answers. MA 114 Calculus II Spring 2013 Final Exam 1 May Question
MA 114 Calculus II Spring 2013 Final Exam 1 May 2013 Name: Section: Last 4 digits of student ID #: This exam has six multiple choice questions (six points each) and five free response questions with points
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationMATH 1241 Common Final Exam Fall 2010
MATH 1241 Common Final Exam Fall 2010 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 1241 Final Exam consists of three parts. You have three hours for the
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background We have seen that some power series converge. When they do, we can think of them as
More informationMTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam 2 November 5, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
More informationThis exam is closed book. You may use one sheet of handwritten notes (both sides OK). Do not share notes. No photocopied materials are allowed.
Math 125 Final Examination Winter 2013 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name This exam is closed book. You may use one 8.5 11 sheet of handwritten notes (both sides
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 November 4th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show
More informationWithout fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 April th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all
More informationMath 116 Second Midterm March 19, 2012
Math 6 Second Midterm March 9, 202 Name: Instructor: Section:. Do not open this exam until you are told to do so. 2. This exam has pages including this cover. There are 9 problems. Note that the problems
More informationMath 113 Final Exam Practice
Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 31B, Spring 2017 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in
More informationAP Calculus Testbank (Chapter 9) (Mr. Surowski)
AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series
More informationMath 112 Rahman. Week Taylor Series Suppose the function f has the following power series:
Math Rahman Week 0.8-0.0 Taylor Series Suppose the function f has the following power series: fx) c 0 + c x a) + c x a) + c 3 x a) 3 + c n x a) n. ) Can we figure out what the coefficients are? Yes, yes
More informationMath 31A Differential and Integral Calculus. Final
Math 31A Differential and Integral Calculus Final Instructions: You have 3 hours to complete this exam. There are eight questions, worth a total of??? points. This test is closed book and closed notes.
More informationTest 2 - Answer Key Version A
MATH 8 Student s Printed Name: Instructor: Test - Answer Key Spring 6 8. - 8.3,. -. CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed
More informationRED. Math 113 (Calculus II) Final Exam Form A Fall Name: Student ID: Section: Instructor: Instructions:
Name: Student ID: Section: Instructor: Math 3 (Calculus II) Final Exam Form A Fall 22 RED Instructions: For questions which require a written answer, show all your work. Full credit will be given only
More informationMath 121: Final Exam Review Sheet
Exam Information Math 11: Final Exam Review Sheet The Final Exam will be given on Thursday, March 1 from 10:30 am 1:30 pm. The exam is cumulative and will cover chapters 1.1-1.3, 1.5, 1.6,.1-.6, 3.1-3.6,
More informationMath 112 (Calculus I) Final Exam
Name: Student ID: Section: Instructor: Math 112 (Calculus I) Final Exam Dec 18, 7:00 p.m. Instructions: Work on scratch paper will not be graded. For questions 11 to 19, show all your work in the space
More informationMath 143 Flash Cards. Divergence of a sequence {a n } {a n } diverges to. Sandwich Theorem for Sequences. Continuous Function Theorem for Sequences
Math Flash cards Math 143 Flash Cards Convergence of a sequence {a n } Divergence of a sequence {a n } {a n } diverges to Theorem (10.1) {a n } diverges to Sandwich Theorem for Sequences Theorem (10.1)
More informationMathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
More informationMath 113 Winter 2005 Departmental Final Exam
Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple
More informationMath 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class.
Math 18 Written Homework Assignment #1 Due Tuesday, December 2nd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 18 students, but
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationWithout fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam November th, 08 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
More informationMath 116 Second Midterm March 19, 2012
Math 6 Second Midterm March 9, 22 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so. 2. This exam has pages including this cover. There are 9 problems. Note that
More informationFalse. 1 is a number, the other expressions are invalid.
Ma1023 Calculus III A Term, 2013 Pseudo-Final Exam Print Name: Pancho Bosphorus 1. Mark the following T and F for false, and if it cannot be determined from the given information. 1 = 0 0 = 1. False. 1
More informationMTH 133 Final Exam Dec 8, 2014
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Problem Score Max Score 1 5 3 2 5 3a 5 3b 5 4 4 5 5a 5 5b 5 6 5 5 7a 5 7b 5 6 8 18 7 8 9 10 11 12 9a
More informationFinal Exam Practice Problems Part II: Sequences and Series Math 1C: Calculus III
Name : c Jeffrey A. Anderson Class Number:. Final Exam Practice Problems Part II: Sequences and Series Math C: Calculus III What are the rules of this exam? PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationTest 3 Version A. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test.
Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop,
More informationHomework Problem Answers
Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln
More informationMTH 133 Exam 2 November 16th, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response
More information(to be used later). We have A =2,B = 0, and C =4,soB 2 <AC and A>0 so the critical point is a local minimum
Math 8 Show Your Work! Page of 6. (a) Find and classify any critical points of f(x, y) =x 2 +x+2y 2 in the region x 2 +y 2
More informationFall 2015 Exam 3 PIN: 17
MARK BOX problem points 0 10 1 10 2-6 50 7 15 8a/8b 15 NAME: PIN: 17 KEY-e-poo % 100 INSTRUCTIONS On Problem 0, fill in the blanks and boxes. As you know, if you do not make at least half of the points
More informationFall 2016 Exam 3 NAME: PIN:
MARK BOX problem points 0 18 1 12 2-11 50=10(5) 12 10 13 10 % 100 NAME: PIN: HAND IN PART INSTRUCTIONS This exam comes in two parts. (1) HAND IN PART. Hand in only this part. (2) STATEMENT OF MULTIPLE
More informationUniversity of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 2012 First Year - CHE, CIV, IND, LME, MEC, MSE
University of Toronto FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, JUNE, 212 First Year - CHE, CIV, IND, LME, MEC, MSE MAT187H1F - CALCULUS II Exam Type: A Examiner: D. Burbulla INSTRUCTIONS:
More information2013/2014 SEMESTER 1 MID-TERM TEST. 1 October :30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
2013/2014 SEMESTER 1 MID-TERM TEST MA1505 MATHEMATICS I 1 October 2013 8:30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY: 1. This test paper consists of TEN (10) multiple choice questions
More informationMath Exam III - Spring
Math 3 - Exam III - Spring 8 This exam contains 5 multiple choice questions and hand graded questions. The multiple choice questions are worth 5 points each and the hand graded questions are worth a total
More informationChapter 4 Notes, Calculus I with Precalculus 3e Larson/Edwards
4.1 The Derivative Recall: For the slope of a line we need two points (x 1,y 1 ) and (x 2,y 2 ). Then the slope is given by the formula: m = y x = y 2 y 1 x 2 x 1 On a curve we can find the slope of a
More informationMath 125 Final Examination Spring 2015
Math 125 Final Examination Spring 2015 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name This exam is closed book. You may use one 8.5 11 sheet of handwritten notes (both sides
More informationMath 120 online. Practice Midterm Exam #2 Prof. Kinoshita. Fall (Actual midterm will have 100 pts)
Note: The format of this practice midterm will be similar to the real midterm. However, the actual midterm will have less questions and be worth 100 points. There will also be more room to work on the
More informationMath 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class.
Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180
More informationMath 162: Calculus IIA
Math 162: Calculus IIA Final Exam December 15, 2015 NAME (please print legibly): Your University ID Number: Your University email Indicate your instructor with a check in the box: JJ Lee Doug Ravenel Timur
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationP-CEP. Mathematics. From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet
P-CEP Mathematics From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet Dear future AP Calculus BC student: There are certain math skills that have been
More informationWithout fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard
More informationHAND IN PART. Prof. Girardi Math 142 Spring Exam 3 PIN:
HAND IN PART Prof. Girardi Math 142 Spring 2014 04.17.2014 Exam 3 MARK BOX problem points possible your score 0A 9 0B 8 0C 10 0D 12 NAME: PIN: solution key Total for 0 39 Total for 1 10 61 % 100 INSTRUCTIONS
More informationMAT 146. Semester Exam Part II 100 points (Part II: 50 points) Calculator Used Impact on Course Grade: approximately 30% Score
MAT 146 Semester Exam Part II Name 100 points (Part II: 50 points) Calculator Used Impact on Course Grade: approximately 30% Score Questions (17) through (26) are each worth 5 points. See the grading rubric
More informationPRACTICE FINAL 2 MATH 16b
PRACTICE FINAL MATH 6b Find possible relative maxima and minima of the function f (x, y) = x + y + y + In each case apply the second derivative test if possible Evaluate the integral 0 (x + ) 00dx Describe
More informationSection 8.7. Taylor and MacLaurin Series. (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications.
Section 8.7 Taylor and MacLaurin Series (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications. MATH 126 (Section 8.7) Taylor and MacLaurin Series The University of Kansas
More informationMath Exam 2-11/17/2014
Math 121 - Exam 2-11/17/2014 Name: Section: Section Class,Times Day Instructor Section Class,Times Day Instructor 1 09:00%AM%(%09:50%AM M%T%W%%F% Li,%Huilan 15 11:00%AM%(%11:50%AM M%T%W%%F% Perlstadt,%Marci
More informationMath 114: Make-up Final Exam. Instructions:
Math 114: Make-up Final Exam Instructions: 1. Please sign your name and indicate the name of your instructor and your teaching assistant: A. Your Name: B. Your Instructor: C. Your Teaching Assistant: 2.
More informationMA EXAM 3 INSTRUCTIONS VERSION 01 April 18, Section # and recitation time
MA 16600 EXAM 3 INSTRUCTIONS VERSION 01 April 18, 2018 Your name Student ID # Your TA s name Section # and recitation time 1. You must use a #2 pencil on the scantron sheet (answer sheet). 2. Check that
More informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
More informationMTH 234 Exam 1 February 20th, Without fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
More informationMath 113: Quiz 6 Solutions, Fall 2015 Chapter 9
Math 3: Quiz 6 Solutions, Fall 05 Chapter 9 Keep in mind that more than one test will wor for a given problem. I chose one that wored. In addition, the statement lim a LR b means that L Hôpital s rule
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationMultiple Choice. (c) 1 (d)
Multiple Choice.(5 pts.) Find the sum of the geometric series n=0 ( ) n. (c) (d).(5 pts.) Find the 5 th Maclaurin polynomial for the function f(x) = sin x. (Recall that Maclaurin polynomial is another
More informationHave a Safe Winter Break
SI: Math 122 Final December 8, 2015 EF: Name 1-2 /20 3-4 /20 5-6 /20 7-8 /20 9-10 /20 11-12 /20 13-14 /20 15-16 /20 17-18 /20 19-20 /20 Directions: Total / 200 1. No books, notes or Keshara in any word
More information