Math Exam I - Spring 2008
|
|
- Job Holmes
- 5 years ago
- Views:
Transcription
1 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 of 5 points. The latter questions will be evaluated not only for having the correct solutions but also for clarity. Points may be taken for confusing and disorganized writing, even when the answer is correct. 1. Evaluate the sum 3 j= (j + j ). A) 1374 B) 1375 C) 1376 D) 1377 E) 1378 F) 1379 G) 138 H) 1381 I) 138 J) j= (j + j ) = 3 + = j=1 3 j=1 (j + j ) j + 3 j j=1 ( ) = = =
2 . Find a formula for the Riemann sum R N for the function f(x) = x over the interval [1, ]. A) R N = 1 N B) R N = 1 N C) R N = N D) R N = 1 N E) R N = 1 N F) R N = N G) R N = N H) R N = 1 N I) R N = 1 N J) R N = 1 N N 1 j= N j=1 N j=1 N j=1 N 1 j= N 1 ( + j+1 N ) ( + j+1 N ) (1 + j+1 N ) (1 + j+1 N ) (1 + ( j+1 N ) ) j= (1 + j N ) N 1 j= N 1 j= ( j+1 N ) N 1 (1 + j+1 N ) j= (1 + j N ) N 1 j+1 j= (1 + N ) x = 1 N, a j = 1 + j N N 1 R N = x f(a j+1 ) = 1 N j= N 1 j= ( 1 + j + 1 ). N
3 3. The limit lim N 3 N N j=1 ( + 3j ) 4 N represents a definite integral. What is this integral? A) 3 x4 dx B) 5 x4 dx C) 3 ( + 3x)4 dx D) 3 ( + 3x/N)4 dx E) 5 (x/n)4 dx F) 5 ( + x)4 dx G) 5 ( + 3x)4 dx H) N 1 ( + 3x/N)4 dx I) 1 ( + x)4 dx J) ( + x)4 dx The Riemann sum can be written in the form x N f(a j ) j=1 where f(x) = x 4 and the set of points a j = + 3j/N, j = 1,..., 3, approximates the interval [, 5]. Notice that x = (5 )/N = 3/N. This corresponds to the integral 5 x 4 dx. 3
4 4. Evaluate the sum of integrals f(x)dx + 6 f(x) dx where f(x) is the function shown in the graph A) π B) π/4 C) π/4 D) π/ E) π/ F) π G) π H) 3π I) 3π/ J) 4π Notice that f(x)dx + 6 f(x) dx = 6 f(x)dx is the area of the larger half-disc (of radius ) which is π. 4
5 5. Calculate the integral 5 (3f(x) 5g(x))dx assuming that 5 f(x)dx = 7 and 5 g(x)dx = 5. A) 3 B) 3 C) 4 D) 4 E) 7 F) 7 G) H) I) 5 J) 5 5 (3f(x) 5g(x))dx = 3 5 f(x)dx 5 5 = = 4. g(x)dx 5
6 6. Calculate the integral x 1 dx. A) 4 B) 3 C) D) 1 E) F) 1 G) H) 3 I) 4 J) 5 We have x 1 < over the interval ( 1, 1), and positive for x > 1 and x < 1. Thus x 1 dx = 1 (1 x )dx + 1 (x 1)dx = (x x 3 /3) 1 + (x 3 /3 x) 1 = 1 1/3 + 8/3 1/3 + 1 =. 6
7 7. Calculate the integral (3x e x ) dx. A) 4e B) e + 8 C) e 8 D) e + 8 E) e 8 F) e 4 G) e 8 H) e 4 I) e + 8 J) 4e 4 (3x e x ) dx = (3x / e x ) = (6 e ) = e 8. 7
8 8. Calculate the integral 3a a dt t. A) (ln )/a B) (ln 3)/a C) a ln 3 D) a ln E) ln 3/ ln a F) ln(3a)/ ln(a) G) ln(a) H) ln(3/a) I) ln(3a) J) ln 3 3a a dt t = ln t 3a a = ln(3a) ln a = ln 3a a = ln 3. 8
9 9. A function G(s) is defined by the integral Find G (s). G(s) = cos(s) 6 A) G (s) = cos(s)(cos 4 (s) 3 cos(s)) B) G (s) = cos(s)(cos 4 (s) 3 cos(s)) + 6 C) G (s) = sin(s)(cos 4 (s) 3 cos(s)) D) G (s) = sin(s)(cos 4 (s) 3 cos(s)) + 6 ( u 4 3u ). E) G (s) = sin(s)(cos 4 (s) 3 cos(s))(4 cos 3 (s) sin(s) 3 sin(s)) F) G (s) = sin(s)(cos 4 (s) 3 cos(s) ) G) G (s) = cos(s)(cos 4 (s) 3 cos(s) ) H) G (s) = sin(s)(cos 4 (s) 3 cos(s) ) I) G (s) = cos(s)(sin 4 (s) 3 sin(s)) J) G (s) = sin(s)(cos 4 (s) 3 cos(s)) 6 Writing F (x) = x 6 (u4 3u) and g(s) = cos s, we have G(s) = F (g(s)). The chain rule gives G (s) = F (g(s))g (s). Therefore, G (s) = (cos 4 s 3 cos s)( sin s). 9
10 1. Calculate the derivative d x tan tdt. dx x A) x sec (x ) sec ( x)/( x) B) sec (x )/x + sec ( x) x C) tan(x )/x tan( x)/ x D) sec (x )/x + tan( x) x E) tan(x )/x tan( x) x F) x tan(x ) tan( x) x G) x tan(x ) tan( x)/( x) H) x tan(x ) tan( x)/( x) I) x tan(x ) tan( x)/( x) J) x tan(x ) + tan( x)/( x) d x tan tdt = d x tan tdt d dx x dx dx x = x tan(x ) tan( x)/ x. tan tdt 1
11 11. Water flows into an empty reservoir at a rate of 6 + 5t gallons per hour. What is the quantity of water in the reservoir after hours? A) 17 B) 18 C) 19 D) 11 E) 111 F) 11 G) 113 H) 114 I) 115 J) 116 This quantity is given by the integral (6 + 5t)dt = / = =
12 1. The traffic flow rate past a certain point on a highway is q(t) = 3 + t 3t, where t is in hours and t = is 8 AM. How many cars pass by ring the time interval from 8 to 1 AM? A) 84 B) 86 C) 88 D) 9 E) 9 F) 94 G) 96 H) 98 I) 1 J) 1 The number of cars is (3 + t 3t )dt = /3 = 9. 1
13 13. Express the integral 1 x + x (4x 3 + 3x ) dx as an integral in the variable u, using the substitution u = 4x 3 + 3x. A) 1 B) 1 C) 6 1 u u u D) u E) 7 F) 6 7 G) H) 1 6 I) 3 J) 3 1 u u u u u u u If u = 4x 3 + 3x, then = 6(x + x)dx, u() = and u(1) = = 7. Therefore, 1 x + x (4x 3 + 3x ) dx = u. 13
14 14. Find the definite integral ln e t e t + e t + 1 dt. A) B) 6 C) 1/6 D) ln E) 1/ ln F) 6 G) e 1 H) 1 e I) 6 e J) e e We use the substitution u = e t. Then = e t dt, u() = 1, u(ln ) =, and ln e t e t + e t + 1 dt = u + u + 1 = = (1 + u) dv v = ( v 1 ) 3 = 1/3 + 1/ = 1/6. 14
15 15. Find the definite integral 1 θ tan(θ )dθ. A) ln(cos 1) 1 B) 1 ln(cos 1) + 1/ C) ln(cos 1) D) 1 ln(sin 1) E) 1 ln(cos 1) F) ln(cos 1) G) ln(cos 1) H) ln(tan 1) 1 I) 1 ln(tan 1) J) ln(tan 1) tan 1 We use the substitution u = cos θ. Thus = θ sin(θ )dθ and 1 θ tan(θ )dθ = 1 cos 1 1 u = 1 ln(cos 1). 15
16 16. (16 points)find the following two indefinite integrals. In each case, clearly indicate the substitutions used. (a) xe 4x dx (b) (cos x)3 sin x dx For integral (a) we use the substitution u = 4x. So = 8xdx and xe 4x dx = 1 e u = 1 + C. 8 8 e 4x For integral (b) we use the substitution u = 3 sin x. So = ln 3(cos x)3 sin x dx and (cos x)3 sin x dx = ln 3 = u/ ln 3 + C = 3sin x ln 3 + C. 16
17 17. (9 points) Find all solutions to the differential equation y = 5y. Which solution satisfies the initial condition y() = 3.4? We know that the general solution of this equation is y(t) = y e 5t. The initial condition gives y = 3.4. Under this condition the solution reces to y(t) = 3.4e 5t. 17
Math 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 informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
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 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 informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationSolutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =
Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists
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 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 informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
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 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 information1. Evaluate the integrals. a. (9 pts) x e x/2 dx. Solution: Using integration by parts, let u = x du = dx and dv = e x/2 dx v = 2e x/2.
MATH 8 Test -SOLUTIONS Spring 4. Evaluate the integrals. a. (9 pts) e / Solution: Using integration y parts, let u = du = and dv = e / v = e /. Then e / = e / e / e / = e / + e / = e / 4e / + c MATH 8
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 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 informationLecture 5: Integrals and Applications
Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The
More information1. (13%) Find the orthogonal trajectories of the family of curves y = tan 1 (kx), where k is an arbitrary constant. Solution: For the original curves:
5 微甲 6- 班期末考解答和評分標準. (%) Find the orthogonal trajectories of the family of curves y = tan (kx), where k is an arbitrary constant. For the original curves: dy dx = tan y k = +k x x sin y cos y = +tan y
More informationMath 131 Exam 2 Spring 2016
Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0
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 informationFinal exam practice 4 (solutions) UCLA: Math 3B, Winter 2019
Final exam practice 4 (solutions) Instructor: Noah White Date: UCLA: Math 3B, Winter 2019 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions
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 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 informationApplied Calculus I. Lecture 36
Applied Calculus I Lecture 36 Computing the volume Consider a continuous function over an interval [a, b]. y a b x Computing the volume Consider a continuous function over an interval [a, b]. y y a b x
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More informationf(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx
Chapter 7 is concerned with all the integrals that can t be evaluated with simple antidifferentiation. Chart of Integrals on Page 463 7.1 Integration by Parts Like with the Chain Rule substitutions with
More informationPrelim 2 Math Please show your reasoning and all your work. This is a 90 minute exam. Calculators are not needed or permitted. Good luck!
April 4, Prelim Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Trigonometric Formulas sin x sin x cos x cos (u + v) cos
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 informationFinal exam practice 4 UCLA: Math 3B, Winter 2019
Instructor: Noah White Date: Final exam practice 4 UCLA: Math 3B, Winter 2019 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in
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 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 informationMath 122 Fall Unit Test 1 Review Problems Set A
Math Fall 8 Unit Test Review Problems Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee
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 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.
MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =
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 informationLecture 4: Integrals and applications
Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18
More informationMathematics 1052, Calculus II Exam 1, April 3rd, 2010
Mathematics 5, Calculus II Exam, April 3rd,. (8 points) If an unknown function y satisfies the equation y = x 3 x + 4 with the condition that y()=, then what is y? Solution: We must integrate y against
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 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 informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: -
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 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 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 information1 + x 2 d dx (sec 1 x) =
Page This exam has: 8 multiple choice questions worth 4 points each. hand graded questions worth 4 points each. Important: No graphing calculators! Any non-graphing, non-differentiating, non-integrating
More informationt 2 + 2t dt = (t + 1) dt + 1 = arctan t x + 6 x(x 3)(x + 2) = A x +
MATH 06 0 Practice Exam #. (0 points) Evaluate the following integrals: (a) (0 points). t +t+7 This is an irreducible quadratic; its denominator can thus be rephrased via completion of the square as a
More informationAP Calculus BC Spring Final Part IA. Calculator NOT Allowed. Name:
AP Calculus BC 6-7 Spring Final Part IA Calculator NOT Allowed Name: . Find the derivative if the function if f ( x) = x 5 8 2x a) f b) f c) f d) f ( ) ( x) = x4 40 x 8 2x ( ) ( x) = x4 40 +x 8 2x ( )
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 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 informationOBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
More informationMATH 1242 FINAL EXAM Spring,
MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t
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 informationQuestions from Larson Chapter 4 Topics. 5. Evaluate
Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of
More informationCalculus I Practice Final Exam B
Calculus I Practice Final Exam B This practice exam emphasizes conceptual connections and understanding to a greater degree than the exams that are usually administered in introductory single-variable
More information. Section: Your exam contains 4 problems. The entire exam is worth 60 points.
Math 125 Section D (Pezzoli) Fall 2017 Midterm #1 (60 points) Name TA:. Section: Your exam contains 4 problems. The entire exam is worth 60 points. This exam is closed book. You may use one 8 1 11 sheet
More informationName: Instructor: Exam 3 Solutions. Multiple Choice. 3x + 2 x ) 3x 3 + 2x 2 + 5x + 2 3x 3 3x 2x 2 + 2x + 2 2x 2 2 2x.
. Exam 3 Solutions Multiple Choice.(6 pts.) Find the equation of the slant asymptote to the function We have so the slant asymptote is y = 3x +. f(x) = 3x3 + x + 5x + x + 3x + x + ) 3x 3 + x + 5x + 3x
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 informationSummary: Primer on Integral Calculus:
Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of
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 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 informationMATH 152, Spring 2019 COMMON EXAM I - VERSION A
MATH 15, Spring 19 COMMON EXAM I - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: ROW NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN
More informationMATH 2300 review problems for Exam 1 ANSWERS
MATH review problems for Exam ANSWERS. Evaluate the integral sin x cos x dx in each of the following ways: This one is self-explanatory; we leave it to you. (a) Integrate by parts, with u = sin x and dv
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 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 informationMath 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator
Math Test - Review Use differentials to approximate the following. Compare your answer to that of a calculator.. 99.. 8. 6. Consider the graph of the equation f(x) = x x a. Find f (x) and f (x). b. Find
More informationAB 1: Find lim. x a.
AB 1: Find lim x a f ( x) AB 1 Answer: Step 1: Find f ( a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f ( x). Do any cancellations and go back to Step 1. If you
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 informationDifferential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
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 informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
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 informationMA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.
MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationFinal Exam 2011 Winter Term 2 Solutions
. (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L
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 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 informationApplications of Differentiation
Applications of Differentiation Definitions. A function f has an absolute maximum (or global maximum) at c if for all x in the domain D of f, f(c) f(x). The number f(c) is called the maximum value of f
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
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 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 information1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2
Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,
More informationMath 116 Second Midterm March 20, 2013
Math 6 Second Mierm March, 3 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has 3 pages including this cover. There are 8 problems. Note that the
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More informationMATH 1207 R02 FINAL SOLUTION
MATH 7 R FINAL SOLUTION SPRING 6 - MOON Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () Let f(x) = x cos x. (a)
More informationMath 222 Spring 2013 Exam 3 Review Problem Answers
. (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w
More informationGrade: The remainder of this page has been left blank for your workings. VERSION D. Midterm D: 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 D: Page of 2 Indefinite Integrals. 9 marks Each part is worth marks. Please
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
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 informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationAssignment 9 Mathematics 2(Model Answer)
Assignment 9 Mathematics (Model Answer) The Integral and Comparison Tests Problem: Determine converges or divergence of the series. ) (a) 0 = (b) ) (a) =8 (b) + 3) (a) = (b) 3 + ) (a) e = (b) 5) (a) =0
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t;
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified.
More informationMAS113 CALCULUS II SPRING 2008, QUIZ 4 SOLUTIONS
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,
More informationMath 115 HW #5 Solutions
Math 5 HW #5 Solutions From 29 4 Find the power series representation for the function and determine the interval of convergence Answer: Using the geometric series formula, f(x) = 3 x 4 3 x 4 = 3(x 4 )
More informationBC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true?
BC Exam 1 - Part I 8 questions No Calculator Allowed - Solutions 6x 5 8x 3 1. Find lim x 0 9x 3 6x 5 A. 3 B. 8 9 C. 4 3 D. 8 3 E. nonexistent ( ) f ( 4) f x. Let f be a function such that lim x 4 x 4 I.
More informationTurn off all cell phones, pagers, radios, mp3 players, and other similar devices.
Math 25 B and C Midterm 2 Palmieri, Autumn 26 Your Name Your Signature Student ID # TA s Name and quiz section (circle): Cady Cruz Jacobs BA CB BB BC CA CC Turn off all cell phones, pagers, radios, mp3
More informationMidterm 1 practice UCLA: Math 32B, Winter 2017
Midterm 1 practice UCLA: Math 32B, Winter 2017 Instructor: Noah White Date: Version: practice This exam has 4 questions, for a total of 40 points. Please print your working and answers neatly. Write your
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
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 informationMATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
More informationMIDTERM 2. Section: Signature:
MIDTERM 2 Math 3A 11/17/2010 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. When you use a major theorem (like
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 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 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 information