MAS113 CALCULUS II SPRING 2008, QUIZ 4 SOLUTIONS
|
|
- Patrick Carroll
- 6 years ago
- Views:
Transcription
1 MAS113 CALCULUS II SPRING 8, QUIZ 4 SOLUTIONS Quiz 4a Solutions 1 Find the area of the surface obtained by rotating the curve y = x 3 /6 + 1/x, 1/ x 1 about the x-axis. We have y = x / 1/x. Therefore, x 4 = dx x = 4 Therefore, the surface area is x 3 π x x 1/ + 1 x x + 1 x = x + 1 x. x 6 dx = π 7 + x = 63 x 1/ 56 π. Solve the differential equation y = y + 5 with the initial condition y = y. T 1 = e t. e t y = 5e t = y = 5 + ce t. y = y gives that c = y 5. Consequently, the solution is yt = 5 + y 5e t. 3 Determine the values of r for which the differential equation y + y = has solutions of the form y = e rt. y = e rt gives that y = re rt. We must have re rt + e rt =, or r + =. r =. Quiz 4b Solutions 1 Find the length of the curve y = 4x, y. We have dx/ = y/. Therefore, the length of the curve is y. Making the change of variables u = y/1, we get that the above integral is u u du = u + 1 lnu + 1 u 1 = + 1 ln.
2 Solve the differential equation y = y + 5 with the initial condition y = y. T = e t. e t y = 5e t = y = 5 + ce t. y = y gives that c = y 5/. Consequently, the solution is yt = 5 + y 5 e t. 3 Determine the values of r for which the differential equation y y = has solutions of the form y = e rt. y = e rt gives that y = re rt and y = r e rt. We must have r e rt e rt =, or r 1 =. r = ±1. Quiz 4c Solutions 1 Find the area of the surface obtained by rotating the curve x = a coshy/a, a y a about the y-axis. We have dx Therefore, the surface area is π a a a cosh y a cosh y a = 4πa a = sinh y a = cosh y a. = πa cosh y = πa a y + a sinh y a a a cosh y a = πa 1 sinh. Solve the differential equation y = y + 1 with the initial condition y = y. T = e t. e t y = 1e t = y = 5 + ce t. y = y gives that c = y 5. Consequently, the solution is yt = 5 + y 5e t.
3 3 Determine the values of r for which the differential equation t y + 4ty + y = has solutions of the form y = t r for t >. y = t r gives that y = rt r 1 and y = rr 1t r. t rr 1t r + 4trt r t r = or = rr 4r + = r 3r + = r 1r. r = 1 or r =. Quiz 4d Solutions 1 Find the length of the curve y = lncos x, x π/3. In the original print out of the quiz the formula of the curve was y = lncosx which is a mis-print. The intended problem is as given here. You will be given full credit so long as you set up the integral correctly. We have = sin x = sec x. dx cos x Therefore, the arc length is π/3 1 sec xdx = ln sec x + tanx π/3 = ln + 3. Solve the differential equation y =.5y 45 with the initial condition y = 85. T 1 = e t/. e t/ y = 45e t/ = y = 45 + ce t/. y = 85 gives that c = 13. Consequently, the solution is yt = e t/. 3 Determine the values of r for which the differential equation y + y 3y = has solutions of the form y = e rt. y = e rt gives that y = re rt and y = r e rt. We must have r e rt + re rt 3e rt =, or r + r 3 = r + 3r 1 =. r = 1 or r = 3. Quiz 4e Solutions
4 1 Find the area of the surface obtained byg rotating the curve x = y, 1 y about the x-axis. We have Hence the surface area is π 1 y 16y = π 16 dx = 16y. 16y + 1 3/ = π Solve the differential equation y = y 5 with the initial condition y = y. 1 = e t. e t y = 5e t = y = 5 + ce t. y = y gives that c = y + 5. Consequently, the solution is yt = 5 + y + 5e t/. 3 Determine the values of r for which the differential equation y + y 6y = has solutions of the form y = e rt. y = e rt gives that y = re rt and y = r e rt. We must have r e rt + re rt 6e rt =, or r + r 6 = r + 3r =. r = or r = 3. Quiz 4f Solutions 1 Find the length of the curve y = x / lnx/4, x 4. We have y = x 1/4x and we see that dx = x x = x x Therefore, the arc length is 4 L = 1 x dx = 4x + ln x 4 = 6 + ln 4 4. Solve the differential equation y = y 5 with the initial condition y = y. = e t.
5 e t y = 5e t = y = 5 + ce t. y = y gives that c = y + 5. Consequently, the solution is yt = 5 + y + 5e t/. 3 Determine the values of r for which the differential equation y 13y + 1y = has solutions of the form y = e rt. y = e rt gives that y = re rt and y = r e rt. We must have r e rt 13re rt + 1e rt =, or r 13r + 1 = r 1r 1 =. r = 1 or r = 1. Quiz 4g Solutions 1 Find the area of the surface obtained by rotating the curve y = 1 x, x 1 about the y-axis. We have y = x. Therefore, the surface area is πx 4x dx = π 4x 1 = π 5 1. Solve the differential equation y = y 1 with the initial condition y = y. = e t. e t y = 1e t = y = ce t. y = y gives that c = y + 1. Consequently, the solution is yt = y + 1e t/. 3 Determine the values of r for which the differential equation t y 4ty + 4y = has solutions of the form y = t r for t >. y = t r gives that y = rt r 1 and y = rr 1t r. t rr 1t r 4trt r 4t r = or = rr 1 4r + 4 = r 5r + 4 = r 1r 4. r = 1 or r = 4. Quiz 4h Solutions
6 1 Find the length of the curve y = e x, x 1. We have y = e x. Therefore, the arc length is ex dx = 1+e u u 1 du. The above arrives from the change of variables u = e x. Now we have u u 1 du = 1 1 u + 1 du = x + 1 u 1 ln u 1 u c. Therefore, the arc length in question is e + 1 ln e e Solve the differential equation y = 9.8 y/5 with the initial condition y =. 1 5 = e t/5. e t/5 y = 9.8e t/5 = y = ce t/5. y = gives that c = 9.8. Consequently, the solution is yt = e t/. 3 Determine the values of r for which the differential equation y 3y + y = has solutions of the form y = e rt. y = e rt gives that y = re rt, y = r e rt and y = r e rt. We must have r 3 e rt 3r e rt + re rt =, or r 3 3r + r = rr 1r =. r =, r = 1 or r =.
Introduction to Differential Equations
Chapter 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
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 informationSolution: APPM 1350 Final Exam Spring 2014
APPM 135 Final Exam Spring 214 1. (a) (5 pts. each) Find the following derivatives, f (x), for the f given: (a) f(x) = x 2 sin 1 (x 2 ) (b) f(x) = 1 1 + x 2 (c) f(x) = x ln x (d) f(x) = x x d (b) (15 pts)
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 informationChapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More informationMATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
More informationMath 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationJUST THE MATHS UNIT NUMBER INTEGRATION 1 (Elementary indefinite integrals) A.J.Hobson
JUST THE MATHS UNIT NUMBER 2. INTEGRATION (Elementary indefinite integrals) by A.J.Hobson 2.. The definition of an integral 2..2 Elementary techniques of integration 2..3 Exercises 2..4 Answers to exercises
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 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 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 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 informationAB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2
AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6
More information1. Solve the boundary-value problems or else show that no solutions exist. y (x) = c 1 e 2x + c 2 e 3x. (3)
Diff. Eqns. Problem Set 6 Solutions. Solve the boundary-value problems or else show that no solutions exist. a y + y 6y, y, y 4 b y + 9y x + e x, y, yπ a Assuming y e rx is a solution, we get the characteristic
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 informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More informationHyperbolics. Scott Morgan. Further Mathematics Support Programme - WJEC A-Level Further Mathematics 31st March scott3142.
Hyperbolics Scott Morgan Further Mathematics Support Programme - WJEC A-Level Further Mathematics 3st March 208 scott342.com @Scott342 Topics Hyperbolic Identities Calculus with Hyperbolics - Differentiation
More informationPRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
More informationStudy 5.5, # 1 5, 9, 13 27, 35, 39, 49 59, 63, 69, 71, 81. Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework.
Goals: 1. Recognize an integrand that is the derivative of a composite function. 2. Generalize the Basic Integration Rules to include composite functions. 3. Use substitution to simplify the process of
More informationIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy. du du. If y = f (u) then y = f (u) u
Section 3 4B The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then f (u) u The Chain Rule with the Power
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) A.J.Hobson
JUST THE MATHS UNIT NUMBER 104 DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) by AJHobson 1041 Products 1042 Quotients 1043 Logarithmic differentiation 1044 Exercises 1045 Answers
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 informationIntegration Techniques
Review for the Final Exam - Part - Solution Math Name Quiz Section The following problems should help you review for the final exam. Don t hesitate to ask for hints if you get stuck. Integration Techniques.
More informationIf y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy. du du. If y = f (u) then y = f (u) u
Section 3 4B Lecture The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then y = f (u) u The Chain Rule
More informationCalculus & Analytic Geometry I
TQS 124 Autumn 2008 Quinn Calculus & Analytic Geometry I The Derivative: Analytic Viewpoint Derivative of a Constant Function. For c a constant, the derivative of f(x) = c equals f (x) = Derivative of
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 informationMath 307 E - Summer 2011 Pactice Mid-Term Exam June 18, Total 60
Math 307 E - Summer 011 Pactice Mid-Term Exam June 18, 011 Name: Student number: 1 10 10 3 10 4 10 5 10 6 10 Total 60 Complete all questions. You may use a scientific calculator during this examination.
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Exam 4 Review 1. Trig substitution
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationMath Exam I - Spring 2008
Math 13 - Exam I - Spring 8 This exam contains 15 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 informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationMore Final Practice Problems
8.0 Calculus Jason Starr Final Exam at 9:00am sharp Fall 005 Tuesday, December 0, 005 More 8.0 Final Practice Problems Here are some further practice problems with solutions for the 8.0 Final Exam. Many
More informationThe Free Intuitive Calculus Course Integrals
Intuitive-Calculus.com Presents The Free Intuitive Calculus Course Integrals Day 19: Trigonometric Integrals By Pablo Antuna 013 All Rights Reserved. The Intuitive Calculus Course - By Pablo Antuna Contents
More informationMath 122 Test 2. October 15, 2013
SI: Math 1 Test October 15, 013 EF: 1 3 4 5 6 7 Total Name Directions: 1. No books, notes or Government shut-downs. You may use a calculator to do routine arithmetic computations. You may not use your
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 information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationWeBWorK, Problems 2 and 3
WeBWorK, Problems 2 and 3 7 dx 2. Evaluate x ln(6x) This can be done using integration by parts or substitution. (Most can not). However, it is much more easily done using substitution. This can be written
More informationUNIVERSITY OF SOUTHAMPTON
UNIVERSITY OF SOUTHAMPTON MATH03W SEMESTER EXAMINATION 0/ MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min This paper has two parts, part A and part B. Answer all questions from part
More informationMath 266 Midterm Exam 2
Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6
More informationPractice Problems: Integration by Parts
Practice Problems: Integration by Parts Answers. (a) Neither term will get simpler through differentiation, so let s try some choice for u and dv, and see how it works out (we can always go back and try
More information2. Using the graph of f(x) below, to find the following limits. Write DNE if the limit does not exist:
1. [10 pts.] State each of the following theorems. (a) [2 pts.] The Intermediate Value Theorem (b) [2 pts.] The Mean Value Theorem. (c) [2 pts.] The Mean Value Theorem for Integrals. (d) [4 pts.] Both
More informationCalculus for Engineers II - Sample Problems on Integrals Manuela Kulaxizi
Calculus for Engineers II - Sample Problems on Integrals Manuela Kulaxizi Question : Solve the following integrals:. π sin x. x 4 3. 4. sinh 8 x cosh x sin x cos 7 x 5. x 5 ln x 6. 8x + 6 3x + x 7. 8..
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More informationChapter 3: Transcendental Functions
Chapter 3: Transcendental Functions Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 32 Except for the power functions, the other basic elementary functions are also called the transcendental
More informationIntegration by Substitution
Integration by Substitution Dr. Philippe B. Laval Kennesaw State University Abstract This handout contains material on a very important integration method called integration by substitution. Substitution
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 information11.5. The Chain Rule. Introduction. Prerequisites. Learning Outcomes
The Chain Rule 11.5 Introduction In this Section we will see how to obtain the derivative of a composite function (often referred to as a function of a function ). To do this we use the chain rule. This
More informationMA26600 FINAL EXAM INSTRUCTIONS December 13, You must use a #2 pencil on the mark sense sheet (answer sheet).
MA266 FINAL EXAM INSTRUCTIONS December 3, 2 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,
More informationMAT 275 Test 1 SOLUTIONS, FORM A
MAT 75 Test SOLUTIONS, FORM A The differential equation xy e x y + y 3 = x 7 is D neither linear nor homogeneous Solution: Linearity is ruinied by the y 3 term; homogeneity is ruined by the x 7 on the
More informationMA 242 Review Exponential and Log Functions Notes for today s class can be found at
MA 242 Review Exponential and Log Functions Notes for today s class can be found at www.xecu.net/jacobs/index242.htm Example: If y = x n If y = x 2 then then dy dx = nxn 1 dy dx = 2x1 = 2x Power Function
More informationFinal Examination Solutions
Math. 5, Sections 5 53 (Fulling) 7 December Final Examination Solutions Test Forms A and B were the same except for the order of the multiple-choice responses. This key is based on Form A. Name: Section:
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
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 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 informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More information1.1. BASIC ANTI-DIFFERENTIATION 21 + C.
.. BASIC ANTI-DIFFERENTIATION and so e x cos xdx = ex sin x + e x cos x + C. We end this section with a possibly surprising complication that exists for anti-di erentiation; a type of complication which
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 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 information(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
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 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 information3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then
3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationcosh 2 x sinh 2 x = 1 sin 2 x = 1 2 cos 2 x = 1 2 dx = dt r 2 = x 2 + y 2 L =
Integrals Volume: Suppose A(x) is the cross-sectional area of the solid S perpendicular to the x-axis, then the volume of S is given by V = b a A(x) dx Work: Suppose f(x) is a force function. The work
More informationQuiz Solutions, Math 136, Sec 6, 7, 16. Spring, Warm-up Problems - Review
Quiz Solutions, Math 36, Sec 6, 7, 6. Spring, Warm-up Problems - Review. Find appropriate substitutions to evaluate the following indefinite integrals: a) b) t + 4) 3 dt. Set u t + 4. We have du dt t.
More informationMath 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).
Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y).
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 1231 MATHEMATICS 1B Calculus Section 1: - Integration.
MATH 1231 MATHEMATICS 1B 2007. For use in Dr Chris Tisdell s lectures: Tues 11 + Thur 10 in KBT Calculus Section 1: - Integration. 1. Motivation 2. What you should already know 3. Useful integrals 4. Integrals
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
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 informationPuzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 /10 /10 /10 /10 /10
MATH-65 Puzzle Collection 6 Nov 8 :pm-:pm Name:... 3 :pm Wumaier :pm Njus 5 :pm Wumaier 6 :pm Njus 7 :pm Wumaier 8 :pm Njus This puzzle collection is closed book and closed notes. NO calculators are allowed
More informationMath 102 Spring 2008: Solutions: HW #3 Instructor: Fei Xu
Math Spring 8: Solutions: HW #3 Instructor: Fei Xu. section 7., #8 Evaluate + 3 d. + We ll solve using partial fractions. If we assume 3 A + B + C, clearing denominators gives us A A + B B + C +. Then
More informationCalculus II Practice Test Problems for Chapter 7 Page 1 of 6
Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
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 informationMath 113 Fall 2005 key Departmental Final Exam
Math 3 Fall 5 key Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.. Fill in the blanks with the correct answer. (a) The integral
More informationMath 190 (Calculus II) Final Review
Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the
More informationRevision Checklist. Unit FP3: Further Pure Mathematics 3. Assessment information
Revision Checklist Unit FP3: Further Pure Mathematics 3 Unit description Further matrix algebra; vectors, hyperbolic functions; differentiation; integration, further coordinate systems Assessment information
More informationCalculus II Practice Test Questions for Chapter , 9.6, Page 1 of 9
Calculus II Practice Test Questions for Chapter 9.1 9.4, 9.6, 10.1 10.4 Page 1 of 9 This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for
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 informationMAS113 CALCULUS II SPRING 2008, QUIZ 5 SOLUTIONS. x 2 dx = 3y + y 3 = x 3 + c. It can be easily verified that the differential equation is exact, as
MAS113 CALCULUS II SPRING 008, QUIZ 5 SOLUTIONS Quiz 5a Solutions (1) Solve the differential equation y = x 1 + y. (1 + y )y = x = (1 + y ) = x = 3y + y 3 = x 3 + c. () Solve the differential equation
More information18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then
8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =
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 informationfor any C, including C = 0, because y = 0 is also a solution: dy
Math 3200-001 Fall 2014 Practice exam 1 solutions 2/16/2014 Each problem is worth 0 to 4 points: 4=correct, 3=small error, 2=good progress, 1=some progress 0=nothing relevant. If the result is correct,
More informationMath Spring 2014 Solutions to Assignment # 6 Completion Date: Friday May 23, 2014
Math 11 - Spring 014 Solutions to Assignment # 6 Completion Date: Friday May, 014 Question 1. [p 109, #9] With the aid of expressions 15) 16) in Sec. 4 for sin z cos z, namely, sin z = sin x + sinh y cos
More informationdx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
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 informationIntegration 1/10. Integration. Student Guidance Centre Learning Development Service
Integration / Integration Student Guidance Centre Learning Development Service lds@qub.ac.uk Integration / Contents Introduction. Indefinite Integration....................... Definite Integration.......................
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 information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationMath Dr. Melahat Almus. OFFICE HOURS (610 PGH) MWF 9-9:45 am, 11-11:45am, OR by appointment.
Math 43 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus OFFICE HOURS (60 PGH) MWF 9-9:45 am, -:45am, OR by appointment. COURSE WEBSITE: http://www.math.uh.edu/~almus/43_fall5.html Visit
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More informationThe integral test and estimates of sums
The integral test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then the series n= a n is convergent if and only if the improper integral f (x)dx is convergent.
More informationMath 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3
Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some
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 information6.2 Trigonometric Integrals and Substitutions
Arkansas Tech University MATH 9: Calculus II Dr. Marcel B. Finan 6. Trigonometric Integrals and Substitutions In this section, we discuss integrals with trigonometric integrands and integrals that can
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 information