Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
|
|
- Conrad Sherman
- 5 years ago
- Views:
Transcription
1 MATH 222 (Lectures,2, and 4) Fall 205 Midterm Solutions Student ID#: Circle your TA s name from the following list. Carolyn Abbott Tejas Bhojraj achary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian Tao Ju Ahmet Kabakulak Dima Kuzmenko Ethan McCarthy Tung Nguyen Jaeun Park Adrian Tovar Loez Polly Yu Please inform your TA if you find any errors in the solutions. Score Problem Problem 2 Problem Problem 4 Problem 5 Problem 6 Problem 7 Instructions Write neatly on this exam. If you need extra aer, let us know. On Problems, 2, and, only the answer will be graded. On Problems 4, 5, 6, and 7 you must show your work and we will grade the work and your justification, and not just the final answer. Each roblem worth either 4 or 5 oints. No calculators, books, or notes (excet for those notes on your inch by 5 inch notecard.) Please simlify any formula involving a trigonometric function and an inverse trigonometric function. For examle, lease write cos(arcsin x) x 2. Note that we have rovided some formulas on the next age to hel with this.
2 Formulas You may freely quote any algebraic or trigonometric identity, as well as well as any of the following formulas or minor variants of those formulas. cos(arcsin x) x 2 sec(arctan x) +x 2. tan(arcsec x) x 2. R x n dx ( x n+ n+ R e x dx e x + C R cos xdx sinx + C + C when n 6 ln x + C when n R sin xdx R tan xdx cos x + C ln cos x + C R cot xdx ln sin x + C R sec xdx ln sec x + tan x + C. R csc xdx ln csc x + cot x + C. R +x 2 dx arctan(x)+c.
3 . For each statement below, CIRCLE true or false. (a) (b) (c) (d) (e) True False True False True False True False True False (a) If x 7 cos then tan 49 x 2 x. (b) R sin 2 ( )d sin cos + C (c) +sin(x) x x for all x. (d) R 2 x 2 9dx is a finite number. (e) R x x x + dx is a finite number. (a) True. (b) False. (c) False. (d) False. (e) True.
4 2. On this age, only the answer will be graded. (a) Comute R sin 2 (x) cos 2 (x)dx. sin 2 (x) cos 2 (x)x sin 2 (x) ( sin 2 (x))dx (2 sin 2 (x) )dx ( cos(2x)) )dx cos(2x)dx 2 sin(2x)+c (b) Comute R 4 (x )(x+) dx. We rewrite this in the form: 4 (x )(x + ) dx x x + Solving using the method of equating coe cients yields A and B. (c) Comute R x 2 +6x+0 dx. b x 2 +6x + 0 lim b! +(x + ) 2 dx lim b! [arctan(x + )] b lim b! (arctan(b + ) arctan(0)) /2.
5 . On this age, only the answer will be graded. (a) Find a ositive number A such that R 00 x 2 +7x 5dx < A. Any A bigger than.0075 will work. (b) Comute R xe 7x+ dx. Let f x and g 0 e 7x+ so that f 0 and g 7 e7x+.then xe 7x+ dx fg 0 fg f 0 g x 7 e7x+ 7 e 7x+ dx x 7 e7x+ 49 e7x+ + C (c) Comute R 2x x 2dx. Comlete the square to get 2x x 2 (x ) 2. Then we get: 2x x 2 dx (x ) 2 dx Using x sin and dx cos d this yields: cos d sin 2 d + C arcsin(x ) + C.
6 4x + 4. Comute dx or exlain why the integral does x(2x +)(2x +) 4x+ not exist. (You may freely use the formula 2x+.) We comute: 4x + x(2x + )(2x + ) dx There are other equivalent answers. lim b! lim b! lim b! x ale ln x " ln ale ln 2x + x(2x+)(2x+) x 2x + dx b 2 ln 2x + 2 ln 2x + x (2x + )(2x + ) # b x 4x 2 +8x + b lim ln b! 4b 2 +8b + ln( 4 ) ln( 5 )ln( b ln ) 2x+
7 5. Comute R (z + e z )sin(z)dz. We comute this as the sum of two integrals: (z + e z )sin(z)dz z sin(z)dz + e z sin(z)dz For R z sin(z)dz we double back. First do integration by arts with f z so f 0 and g 0 sin(z) sog cos(z). And we get: z sin(z)dz fg f 0 g z cos(z)+ cos(z)dz z cos(z)+ 9 sin(z)+c Then we let I R e z sin(z)dz. We first integrate by arts with f sin(z) and g 0 e z. Then f 0 cos(z) and g e z, yielding: I e z sin(z)dz fg f 0 g sin(z)e z cos(z)e z dz We integrate by arts again, with h cos(z) and k 0 e z so h 0 sin(z) and k e z : sin(z)e z hk 0 sin(z)e z hk h 0 k sin(z)e z cos(z)e z ( sin(z))e z dz sin(z)e z cos(z)e z 9 sin(z))e z dz sin(z)e z cos(z)e z 9I We thus have the equation: I sin(z)e z cos(z)e z 9I which, after moving all of the I terms to the left side, yields: We thus obtain: ( + 9) I sin(z)e z cos(z)e z + C. I 0 (sin(z)ez cos(z)e z )+C Putting this together yields: (z + e z )sin(z)dz z cos(z)+ 9 sin(z)+ 0 (sin(z)ez cos(z)e z )
8 6. Comute R e x 4 e 2x dx. Set z e x so that dz e x dx and dx dz e x e x 4 e 2x dx z 2 4 z 2 dz dz z.thenwehave: Now let z 2sin so that dz 2 cos d and we get: (2 sin ) 2 4 4sin 2 2 cos d 4 cos 2 4sin 2 d cos 2 sin 2 d (cot 2 )d csc 2 d cot + C Since sin( ) z 2 we get arcsin( z 2 ) and cot( ) 4 z 2 z yielding: 4 z 2 z 4 e 2x e x arcsin( z 2 )+C arcsin( ex 2 )+C
9 7. (a) For n 0,,... let I n R x n e x+2 dx. Deriveareductionformula for I n. (b) Let J n R x 5 (ln x) n dx for n 0. This satisfies the reduction formula J n (lnx) n x6 n 6 6 J n for n. Comute J 2. (a): Let f x n so f 0 nx n and let g 0 e x+2 so that g I n x n e x+2 dx fg 0 fg f 0 g xn e x+2 n xn e x+2 n I n x n ex+2 Then: e x+2 dx (b): J 0 R x 5 dx x6 6 Then + C. Then J 2 (lnx) 2 x6 6 J (lnx) x J (lnx) 2 x6 6 6 J 0 6 x ln x x C. If you simlify, you might get other, equivalent, answers. 6 x ln x x 6 + C 6
Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
Practice Final a MATH 222 (Lectures 1,2, and 4) Fall 2015. Name: Student ID#: Circle your TAs name: Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler Di Fang Bingyang Hu
More informationCarolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
Final a MATH 222 (Lectures,2, and 4) Fall 205. Name: Student ID#: Circle your TAs name: Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi
More informationCarolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
MATH 222 (Lectures 1,2, and 4) Fall 2015 Midterm 2a Name: Student ID#: Circle your TA s name from the following list. Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler
More informationCarolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
Practice Final b MATH 222 (Lectures,2, and 4) Fall 205. Name: Student ID#: Circle your TAs name: Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler Di Fang Bingyang Hu Canberk
More informationCarolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey. Jale Dinler Di Fang Bingyang Hu Canberk Irimagzi Chris Janjigian
Practice Final a Solutions (/7 Version) MATH (Lectures,, and 4) Fall 05. Name: Student ID#: Circle your TAs name: Carolyn Abbott Tejas Bhojraj achary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler Di Fang
More informationMATH 222 (Lectures 1,2, and 4) Fall 2015 Practice Midterm 2.1 Solutions. Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey
MATH (Lectures,, and 4) Fall 05 Practice Midterm. Solutions Student ID#: Circle your TA s name from the following list. Carolyn Abbott Tejas Bhojraj Zachary Carter Mohamed Abou Dbai Ed Dewey Jale Dinler
More informationAllen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He
MATH 222 (Lectures 2,3, and 4) Fall 2017 Practice Midterm 2.1 Name: Circle your TA s name from the following list. Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park James Hanson Julia Lindberg
More informationAllen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He
MATH 222 (Lectures 2,3, and 4) Fall 2017 Midterm 2 Name: Circle your TA s name from the following list. Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park James Hanson Julia Lindberg Mao
More informationAllen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He
MATH 222 (Lectures 2,3, and 4) Fall 207 Practice Midterm 2.2 Solutions Circle your TA s name from the following list. Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park James Hanson Julia
More informationEd Dewey Stephen Neal Dae Han Kang Sharath Prased Alisha Zachariah. Chris Janjigian Animesh Anand Reese Johnston Jeremy Schwend Alex Troesch
MATH 222 (2 and 4) Fall 213 Practice Final Solutions Circle your TA s name from the following list. Ed Dewey Stephen Neal Dae Han Kang Sharath Prased Alisha Zachariah Chris Janjigian Animesh Anand Reese
More informationMATH 101: PRACTICE MIDTERM 2
MATH : PRACTICE MIDTERM INSTRUCTOR: PROF. DRAGOS GHIOCA March 7, Duration of examination: 7 minutes This examination includes pages and 6 questions. You are responsible for ensuring that your copy of the
More informationMath 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
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 information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationMath 181, Exam 1, Spring 2013 Problem 1 Solution. arctan xdx.
Math, Exam, Sring 03 Problem Solution. Comute the integrals xe 4x and arctan x. Solution: We comute the first integral using Integration by Parts. The following table summarizes the elements that make
More informationTest one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
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 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 informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
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 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 informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
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 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 informationFinal Exam. V Spring: Calculus I. May 12, 2011
Name: ID#: Final Exam V.63.0121.2011Spring: Calculus I May 12, 2011 PLEASE READ THE FOLLOWING INFORMATION. This is a 90-minute exam. Calculators, books, notes, and other aids are not allowed. You may use
More informationThere are some trigonometric identities given on the last page.
MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during
More informationMath 181, Exam 1, Study Guide Problem 1 Solution. xe x2 dx = e x2 xdx. = e u 1 2 du = 1. e u du. = 1 2 eu + C. = 1 2 ex2 + C
Math 8, Exam, Study Guide Problem Solution. Evaluate xe x dx. Solution: We evaluate the integral using the u-substitution method. Let u x. Then du xdx du xdx and we get: xe x dx e x xdx e u du e u du eu
More informationMath 121. Exam II. November 28 th, 2018
Math 121 Exam II November 28 th, 2018 Name: Section: The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all
More informationMATH 127 SAMPLE FINAL EXAM I II III TOTAL
MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer
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 122 Test 3. April 17, 2018
SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator
More informationHave a Safe and Happy Break
Math 121 Final EF: December 10, 2013 Name Directions: 1 /15 2 /15 3 /15 4 /15 5 /10 6 /10 7 /20 8 /15 9 /15 10 /10 11 /15 12 /20 13 /15 14 /10 Total /200 1. No book, notes, or ouiji boards. You may use
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationUniversity of Regina Department of Mathematics and Statistics Math 111 All Sections (Winter 2013) Final Exam April 25, 2013
University of Regina Department of Mathematics and Statistics Math 111 All Sections (Winter 013) Final Exam April 5, 013 Name: Student Number: Please Check Off Your Instructor: Dr. R. McIntosh (001) Dr.
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 informationSpring 2017 Midterm 1 04/26/2017
Math 2B Spring 2017 Midterm 1 04/26/2017 Time Limit: 50 Minutes Name (Print): Student ID This exam contains 10 pages (including this cover page) and 5 problems. Check to see if any pages are missing. Enter
More informationMAT 132 Midterm 1 Spring 2017
MAT Midterm Spring 7 Name: ID: Problem 5 6 7 8 Total ( pts) ( pts) ( pts) ( pts) ( pts) ( pts) (5 pts) (5 pts) ( pts) Score Instructions: () Fill in your name and Stony Brook ID number at the top of this
More informationMath 132 Exam 3 Fall 2016
Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.6: Sequences, Series, Integral, Comparison, Alternating, Absolute
More information4. Theory of the Integral
4. Theory of the Integral 4.1 Antidifferentiation 4.2 The Definite Integral 4.3 Riemann Sums 4.4 The Fundamental Theorem of Calculus 4.5 Fundamental Integration Rules 4.6 U-Substitutions 4.1 Antidifferentiation
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 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 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 informationLast/Family Name First/Given Name Seat #
Math 2, Fall 27 Schaeffer/Kemeny Final Exam (December th, 27) Last/Family Name First/Given Name Seat # Failure to follow the instructions below will constitute a breach of the Stanford Honor Code: You
More informationSt. Augustine, De Genesi ad Litteram, Book II, xviii, 37. (1) Note, however, that mathematici was most likely used to refer to astrologers.
Quote: [...] Beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians (1) have made a covenant with the devil to darken the spirit and to confine
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationInverse Trigonometric Functions. September 5, 2018
Inverse Trigonometric Functions September 5, 08 / 7 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions..0 0.5 Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what
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 informationTHE UNIVERSITY OF WESTERN ONTARIO
Instructor s Name (Print) Student s Name (Print) Student s Signature THE UNIVERSITY OF WESTERN ONTARIO LONDON CANADA DEPARTMENTS OF APPLIED MATHEMATICS AND MATHEMATICS Calculus 1000A Midterm Examination
More informationMATH 151, FALL SEMESTER 2011 COMMON EXAMINATION 3 - VERSION B - SOLUTIONS
Name (print): Signature: MATH 5, FALL SEMESTER 0 COMMON EXAMINATION - VERSION B - SOLUTIONS Instructor s name: Section No: Part Multiple Choice ( questions, points each, No Calculators) Write your name,
More informationMath 132 Exam 3 Fall 2016
Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.6: Sequences, Series, Integral, Comparison, Alternating, Absolute
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More informationFinal Exam. Math 3 December 7, 2010
Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.
More informationName: Instructor: 1. a b c d e. 15. a b c d e. 2. a b c d e a b c d e. 16. a b c d e a b c d e. 4. a b c d e... 5.
Name: Instructor: Math 155, Practice Final Exam, December The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for 2 hours. Be sure that your name
More informationMath 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 informationCh 5 and 6 Exam Review
Ch 5 and 6 Exam Review Note: These are only a sample of the type of exerices that may appear on the exam. Anything covered in class or in homework may appear on the exam. Use the fundamental identities
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 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 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 informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
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 informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More informationMath 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts
Introduction Math : Calculus - Fall 0/0 Review of Precalculus Concets Welcome to Math - Calculus, Fall 0/0! This roblems in this acket are designed to hel you review the toics from Algebra and Precalculus
More informationGrade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please
More informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More informationTrigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
More informationMath 1: Calculus with Algebra Midterm 2 Thursday, October 29. Circle your section number: 1 Freund 2 DeFord
Math 1: Calculus with Algebra Midterm 2 Thursday, October 29 Name: Circle your section number: 1 Freund 2 DeFord Please read the following instructions before starting the exam: This exam is closed book,
More informationUniversity Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.
MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,
More informationMATH 152 Spring 2018 COMMON EXAM I - VERSION A
MATH 52 Spring 28 COMMON EXAM I - VERSION A LAST NAME: FIRST NAME: INSTRUCTOR: SECTION NUMBER: UIN: DIRECTIONS:. The use of a calculator, laptop or cell phone is prohibited. 2. TURN OFF cell phones and
More informationMATH 162. Midterm Exam 1 - Solutions February 22, 2007
MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationThe Substitution Rule
The Sbstittion Rle Kiryl Tsishchanka THEOREM The Fndamental Theorem Of Calcls, Part II): If f is continos on [a,b], then where F is any antiderivative of f, that is F f. b a ] b fx)dx Fb) Fa) Fx) a NOTATION:
More informationPractice Differentiation Math 120 Calculus I Fall 2015
. x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although
More informationMAT Calculus for Engineers I EXAM #3
MAT 65 - Calculus for Engineers I EXAM #3 Instructor: Liu, Hao Honor Statement By signing below you conrm that you have neither given nor received any unauthorized assistance on this exam. This includes
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 information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationMath 131 Exam 2 November 13, :00-9:00 p.m.
Name (Last, First) ID # Signature Lecturer Section (01, 02, 03, etc.) university of massachusetts amherst department of mathematics and statistics Math 131 Exam 2 November 13, 2017 7:00-9:00 p.m. Instructions
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 information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
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 informationPractice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental
More informationFinal exam for MATH 1272: Calculus II, Spring 2015
Final exam for MATH 1272: Calculus II, Spring 2015 Name: ID #: Signature: Section Number: Teaching Assistant: General Instructions: Please don t turn over this page until you are directed to begin. There
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 1)
Chapter 5-6 Review Math 116 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the fundamental identities to find the value of the trigonometric
More informationMA 110 Algebra and Trigonometry for Calculus Spring 2017 Exam 3 Tuesday, 11 April Multiple Choice Answers EXAMPLE A B C D E.
MA 110 Algebra and Trigonometry for Calculus Spring 017 Exam 3 Tuesday, 11 April 017 Multiple Choice Answers EXAMPLE A B C D E Question Name: Section: Last digits of student ID #: This exam has twelve
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: 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 be used. You may use a graphing
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
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 informationTaylor Series. Math114. March 1, Department of Mathematics, University of Kentucky. Math114 Lecture 18 1/ 13
Taylor Series Math114 Department of Mathematics, University of Kentucky March 1, 2017 Math114 Lecture 18 1/ 13 Given a function, can we find a power series representation? Math114 Lecture 18 2/ 13 Given
More informationMath 1 Lecture 22. Dartmouth College. Monday
Math 1 Lecture 22 Dartmouth College Monday 10-31-16 Contents Reminders/Announcements Last Time Implicit Differentiation Derivatives of Inverse Functions Derivatives of Inverse Trigonometric Functions Examish
More informationMath 122 Test 3. April 15, 2014
SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not
More informationTechniques of Integration
Chapter 8 Techniques of Integration 8. Trigonometric Integrals Summary (a) Integrals of the form sin m x cos n x. () sin k+ x cos n x = ( cos x) k cos n x (sin x ), then apply the substitution u = cos
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 informationSET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
More information1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
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 informationf(g(x)) g (x) dx = f(u) du.
1. Techniques of Integration Section 8-IT 1.1. Basic integration formulas. Integration is more difficult than derivation. The derivative of every rational function or trigonometric function is another
More informationTurn off all noise-making devices and all devices with an internet connection and put them away. Put away all headphones, earbuds, etc.
CRN: NAME: INSTRUCTIONS: This exam is a closed book exam. You may not use your text, homework, or other aids except for a 3 5-inch notecard. You may use an allowable calculator, TI-83 or TI-84 to perform
More information