x = y = A = (1) (2) = 2.
|
|
- Jordan Webb
- 5 years ago
- Views:
Transcription
1 MATH 3 Final Eam (Version 1) Solutions May, 8 S. F. Ellermeyer Name Instructions. Your work on this eam will be graded according to two criteria: mathematical correctness and clarity of presentation. In other words, you must know what you are doing (mathematically) and you must also epress yourself clearly. In particular, write answers to questions using correct notation and using complete sentences where appropriate. Also, you must supply su cient detail in your solutions (relevant calculations, written eplanations of why you are doing these calculations, etc.). It is not su cient to just write down an answer with no eplanation of how you arrived at that answer. As a rule of thumb, the harder that I have to work to interpret what you are trying to say, the less credit you will get. You may use your calculator but you may not use any books or notes. 1. (a) Estimate the volume of the solid that lies below the surface z = 5 and above the rectangle R = f(; y) j 5; y g. Use a Riemann Sum with m = 5 and n = and take the sample points to be the upper right corners of each subrectangle. (Note: The point of this part of the problem is to estimate the volume by using a Riemann sum. o not compute the eact volume using a double integral.) (b) Use a double integral to nd the eact volume of the solid described above. Solution: Note that and thus The desired estimate is = 5 5 y = = 1 = A = (1) () =. [f (1; ) + f (1; ) + f (; ) + f (; ) + f (3; ) + f (3; ) + f (; ) + f (; ) + f (5; ) + f (5; )] = [ ] =. The eact volume is ZZ R (5 ) da = Z 5 Z (5 ) dy d = 5.. Set up and evaluate a double integral that gives the volume of the solid lying under the elliptic paraboloid + y 9 + z = 1 1
2 and above the rectangle R = [ 1; 1] [ ; ]. Solution: The volume is ZZ 1 R y Z 1 Z da = y dy d = Sketch the region of integration for the iterated integral Z 3 Z p 9 y f (; y) d dy and then write an equivalent double integral with the order of integration changed. (Hint: The equivalent integral is actually a sum of two integrals.) Solution: The region,, of integration is pictured below. The integral written above views as a Type II region. 8 y = 9? 6 y = 3 6,3 = By viewing as a Type I Region, the above integral can also be computed as Z p 6 Z 3 f (; y) dy d + Z 3 p 6 Z 9 f (; y) dy d.. The curve r = cos () (in polar coordinates) is pictured below. This curve is called a rose with eight petals.
3 y Set up and evaluate a double integral that gives the area of one petal of the rose. Solution: One half of a petal is traced out as ranges from to =8. Thus the area of one petal is ZZ Z =8 Z cos() 1 da = r dr d = Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length a if the density function at any point is proportional to the square of the distance from the verte opposite the hypotenuse. (Assume that the verte opposite the hypotenuse is at the point (; ).) Solution: We assume that the verte opposite the hypotenuse is located at (; ). This means that the density function is (; y) = K ( + y ) where K is some positive constant. The total mass of the lamina is ZZ m = K + y Z a Z a da = K + y dy d = 1 6 Ka. The coordinate of the center of mass is = 1 ZZ K + y da m = 6 a Z a = 5 a Z a + y dy d 3
4 and the y coordinate of the center of mass is y = 1 ZZ Ky + y da m = 6 a Z a = 5 a. Z a y + y dy d 6. Match the functions, f, with the plots of their gradient vector elds (which are given on the attached sheet). (a) f (; y) = p + y matches Graph 3 (b) f (; y) = y matches Graph 9 (c) f (; y) = y matches Graph (d) f (; y) = y matches Graph 8 (e) f (; y) = + y matches Graph 1 (f) f (; y) = y matches Graph 1 (g) f (; y) = y matches Graph 6 (h) f (; y) = y matches Graph 7 (i) f (; y) = y matches Graph 5 (j) f (; y) = + y matches Graph 7. Find the work done by the force eld F (; y) = sin (y) i + yj on a particle that moves along the parabola y = from the point ( (; ). Solution: The work done is Z F dr 1; 1) to the point where is the curve = t y = t 1 t. Since d = dt and dy = t dt, the work done is Z Z F dr= P d + Q dy = Z 1 t sin t + t 3 dt = 1 cos (1) 1 15 cos () +.
5 8. Eplain why the line integral Z 3y 3 d + 9y dy is independent of path and evaluate this integral where is any path beginning at the point (; 1) and ending at the point ( 3; ). Solution: The integral is independent of path because P (; y) = 3y 3 and Q (; y) = 9y are both de ned on all of R (which is a simply connected domain) and because A potential = f (; y) = 3y 3 and thus, by the Fundamental Theorem of Line Integrals, Z 3y 3 d + 9y dy = f ( 3; ) f (; 1) = Use Green s Theorem to nd the work done by the force F (; y) = + y i + y j in moving a particle from the origin along the ais to the point (1; ) ; then along the line segment from (1; ) to (; 1), and then back to the origin along the y ais. Solution: By Green s Theorem, the work done is Z + y d + y dy = Z 1 Z 1 = dy d 1. Find the area of the quadrilateral pictured below. 5
6 (-8,9) (-,-7) (,-7) (8,-8) Solution: By a result derived in class, the area of this quadrilateral is 1 (( ) ( 7) () ( 7) + () ( 75. 6
(x + y) ds. 2 (1) dt = p Find the work done by the force eld. yzk
MATH Final Exam (Version 1) Solutions May 4, 11 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.
More informationMATH 2203 Final Exam Solutions December 14, 2005 S. F. Ellermeyer Name
MATH 223 Final Exam Solutions ecember 14, 25 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In
More informationMATH 2203 Exam 3 Version 2 Solutions Instructions mathematical correctness clarity of presentation complete sentences
MATH 2203 Exam 3 (Version 2) Solutions March 6, 2015 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.
More informationMATH 1190 Exam 4 (Version 2) Solutions December 1, 2006 S. F. Ellermeyer Name
MATH 90 Exam 4 (Version ) Solutions December, 006 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation.
More information1. (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. Solution: Such a graph is shown below.
MATH 9 Eam (Version ) Solutions November 7, S. F. Ellermeer Name Instructions. Your work on this eam will be graded according to two criteria: mathematical correctness and clarit of presentation. In other
More informationy 4x 2 4x. The velocity vector is v t i 4 8t j and the speed is
MATH 2203 - Exam 2 (Version 1) Solutions September 23, 2014 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of
More informationMATH Exam 2 (Version 2) Solutions September 23, 2014 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to
MATH 2203 - Exam 2 (Version 2) Solutions September 23, 2014 S. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of
More informationf x,y da 2 9. x 2 y 2 dydx y 2 dy x2 dx 2 9. y x da 4 x
MATH 3 (Calculus III) -Exam 4 (Version ) Solutions March 5, 5 S. F. Ellermeer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarit of
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationx 2 yds where C is the curve given by x cos t y cos t
MATH Final Exam (Version 1) olutions May 6, 15. F. Ellermeyer Name Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In
More informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More informationG r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam Answer Key
G r a d e P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Eam Answer Key G r a d e P r e - C a l c u l u s M a t h e m a t i c s Final Practice Eam Answer Key Name: Student Number:
More informationMath 141: Trigonometry Practice Final Exam: Fall 2012
Name: Math 141: Trigonometry Practice Final Eam: Fall 01 Instructions: Show all work. Answers without work will NOT receive full credit. Clearly indicate your final answers. The maimum possible score is
More information17. Find the moments of inertia I x, I y, I 0 for the lamina of. 4. D x, y 0 x a, 0 y b ; CAS. 20. D is enclosed by the cardioid r 1 cos ; x, y 3
SCTION 2.5 TRIPL INTGRALS 69 2.4 XRCISS. lectric charge is distributed over the rectangle, 2 so that the charge densit at, is, 2 2 (measured in coulombs per square meter). Find the total charge on the
More information4. (6 points) Express the domain of the following function in interval notation:
Eam 1-A L. Ballou Name Math 131 Calculus I September 1, 016 NO Calculator Allowed BOX YOUR ANSWER! Show all work for full credit! 1. (4 points) Write an equation of a line with y-intercept 4 and -intercept
More informationMath 1050 Final Exam Form A College Algebra Fall Semester Student ID ID Verification Section Number
Math 1050 Final Eam Form A College Algebra Fall Semester 2010 Instructor Name Student ID ID Verification Section Number Time Limit: 120 minutes Graphing calculators without CAS are allowed. All problems
More informationSolutions to Practice Exam 2
Solutions to Practice Eam Problem : For each of the following, set up (but do not evaluate) iterated integrals or quotients of iterated integral to give the indicated quantities: Problem a: The average
More informationMath 142: Trigonometry and Analytic Geometry Practice Final Exam: Fall 2012
Name: Math 14: Trigonometry and Analytic Geometry Practice Final Eam: Fall 01 Instructions: Show all work. Answers without work will NOT receive full credit. Clearly indicate your final answers. The maimum
More informationThe Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set
LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using
More information( ) 2 + 2x 3! ( x x ) 2
Review for The Final Math 195 1. Rewrite as a single simplified fraction: 1. Rewrite as a single simplified fraction:. + 1 + + 1! 3. Rewrite as a single simplified fraction:! 4! 4 + 3 3 + + 5! 3 3! 4!
More informationBrief Revision Notes and Strategies
Brief Revision Notes and Strategies Straight Line Distance Formula d = ( ) + ( y y ) d is distance between A(, y ) and B(, y ) Mid-point formula +, y + M y M is midpoint of A(, y ) and B(, y ) y y Equation
More informationPart I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
More informationy x arctan x arctan x x dx 2 ln by first expressing the integral as an iterated integral. f t dt dz dy 1 2 y x x t 2 f t dt F 2 Gm d 1 d 1 sr 2 d 2
CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER A Click here for answers. S Click here for solutions.. If denotes the greatest integer in, evaluate the integral where R, y 3, y 5.. Evaluate the integral
More information1. State the definition of the integral. Be certain to include all of the appropriate words and symbols. tan(t 2 ) dt, find f (x).
Math 26 First Eam Spring 2 Write neat, concise, and accurate solutions to each of the problems below I will not give partial credit for steps I cannot follow Include all relevant steps, use correct notation,
More informationPrecalculus Summer Packet
Precalculus Summer Packet These problems are to be completed to the best of your ability by the first day of school You will be given the opportunity to ask questions about problems you found difficult
More informationMATHEMATICS EXTENSION 2
Sydney Grammar School Mathematics Department Trial Eaminations 008 FORM VI MATHEMATICS EXTENSION Eamination date Tuesday 5th August 008 Time allowed hours (plus 5 minutes reading time) Instructions All
More informationInstructions for Section 2
200 MATHMETH(CAS) EXAM 2 0 SECTION 2 Instructions for Section 2 Answer all questions in the spaces provided. In all questions where a numerical answer is required an eact value must be given unless otherwise
More informationAP Calculus AB Free-Response Scoring Guidelines
Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationUBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007
This eamination has 15 pages including this cover. UBC-SFU-UVic-UNBC Calculus Eam Solutions 7 June 007 Name: School: Signature: Candidate Number: Rules and Instructions 1. Show all your work! Full marks
More information2018 Summer Assignment AP Calculus BC. Due Dates: Part 1 is due no later than Aug. 1 Part 2 is due no later than Aug. 15
08 Summer Assignment AP Calculus BC Due Dates: Part is due no later than Aug. Part is due no later than Aug. 5 The assignment must be postmarked or hand delivered to the school (preferably to me Part :
More informationMath 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.
. Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.
More informationMAC 2311 Final Exam Review Fall Private-Appointment, one-on-one tutoring at Broward Hall
Fall 2016 This review, produced by the CLAS Teaching Center, contains a collection of questions which are representative of the type you may encounter on the eam. Other resources made available by the
More informationMath 180, Exam 2, Spring 2013 Problem 1 Solution
Math 80, Eam, Spring 0 Problem Solution. Find the derivative of each function below. You do not need to simplify your answers. (a) tan ( + cos ) (b) / (logarithmic differentiation may be useful) (c) +
More informationJUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES DIRECTORATE TERM
JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 1 DIRECTORATE TERM 1 017 This document has been compiled by the FET Mathematics Subject Advisors together with Lead Teachers.
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
More informationMATH 52 MIDTERM 1. April 23, 2004
MATH 5 MIDTERM April 3, Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the honor code. During the test, you may not use
More informationDepartment of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO
Department of Mathematical Sciences Math 6 Calculus Spring 6 Eam V DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO NAME (Printed): INSTRUCTOR: SECTION NO.: When instructed, turn over this cover page
More informationAP Calculus 2 Summer Review Packet
AP Calculus Summer Review Packet This review packet is to be completed by all students enrolled in AP Calculus. This packet must be submitted on the Monday of the first full week of class. It will be used
More informationCALCULUS AB SECTION II, Part A
CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank
More informationMath 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationMath Pre-Calc 20 Final Review
Math Pre-Calc 0 Final Review Chp Sequences and Series #. Write the first 4 terms of each sequence: t = d = - t n = n #. Find the value of the term indicated:,, 9,, t 7 7,, 9,, t 5 #. Find the number of
More informationIt s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]
It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)
More informationThe slope, m, compares the change in y-values to the change in x-values. Use the points (2, 4) and (6, 6) to determine the slope.
LESSON Relating Slope and -intercept to Linear Equations UNDERSTAND The slope of a line is the ratio of the line s vertical change, called the rise, to its horizontal change, called the run. You can find
More information( ) ( 4) ( ) ( ) Final Exam: Lessons 1 11 Final Exam solutions ( )
Show all of your work in order to receive full credit. Attach graph paper for your graphs.. Evaluate the following epressions. a) 6 4 6 6 4 8 4 6 6 6 87 9 b) ( 0) if ( ) ( ) ( ) 0 0 ( 8 0) ( 4 0) ( 4)
More informationMidterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
More informationWithout fully opening the exam, check that you have pages 1 through 10.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 10. Show all your work on the standard
More informationFranklin Math Bowl 2007 Group Problem Solving Test 6 th Grade
Group Problem Solving Test 6 th Grade 1. Consecutive integers are integers that increase by one. For eample, 6, 7, and 8 are consecutive integers. If the sum of 9 consecutive integers is 9, what is the
More informationPractice Problems for Test II
Math 117 Practice Problems for Test II 1. Let f() = 1/( + 1) 2, and let g() = 1 + 4 3. (a) Calculate (b) Calculate f ( h) f ( ) h g ( z k) g( z) k. Simplify your answer as much as possible. Simplify your
More informationFor questions 5-8, solve each inequality and graph the solution set. You must show work for full credit. (2 pts each)
Alg Midterm Review Practice Level 1 C 1. Find the opposite and the reciprocal of 0. a. 0, 1 b. 0, 1 0 0 c. 0, 1 0 d. 0, 1 0 For questions -, insert , or = to make the sentence true. (1pt each) A. 5
More informationA.P. Calculus Summer Assignment
A.P. Calculus Summer Assignment This assignment is due the first day of class at the beginning of the class. It will be graded and counts as your first test grade. This packet contains eight sections and
More informationMath 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
More informationSection 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44
Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationPart 1: Integration problems from exams
. Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating
More information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More informationMath 231 Final Exam Review
Math Final Eam Review Find the equation of the line tangent to the curve 4y y at the point (, ) Find the slope of the normal line to y ) ( e at the point (,) dy Find d if cos( y) y 4 y 4 Find the eact
More informationInstructions for Section B
11 2016 MATHMETH EXAM 2 SECTION B Answer all questions in the spaces provided. Instructions for Section B In all questions where a numerical answer is required, an eact value must be given unless otherwise
More informationThis exam contains 9 problems. CHECK THAT YOU HAVE A COMPLETE EXAM.
Math 126 Final Examination Autumn 2011 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name This exam contains 9 problems. CHECK THAT YOU HAVE A COMPLETE EXAM. This exam is closed
More informationALGEBRA 1 CP FINAL EXAM REVIEW
ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8
More informationMATHEMATICS A2/M/P1 A LEVEL PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS A LEVEL PAPER 1 Bronze Set A (Edexcel Version) CM Time allowed: 2 hours Instructions to
More informationCreated by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.
IYGB GCE Mathematics MP Advanced Level Practice Paper N Difficulty Rating: 3.550/.68 Time: hours Candidates may use any calculator allowed by the regulations of this eamination. Information for Candidates
More information= ( 2) = p 5.
MATH 0 Exam (Version ) Solutions Setember, 00 S. F. Ellermeyer Name Instructions. Your work on this exam will be raded accordin to two criteria: mathematical correctness clarity of resentation. In other
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
More informationGeometry Honors Summer Packet
Geometry Honors Summer Packet Hello Student, First off, welcome to Geometry Honors! In the fall, we will embark on an eciting mission together to eplore the area (no pun intended) of geometry. This packet
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationM151B Practice Problems for Final Exam
M5B Practice Problems for Final Eam Calculators will not be allowed on the eam. Unjustified answers will not receive credit. On the eam you will be given the following identities: n k = n(n + ) ; n k =
More information(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.
Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive
More informationMath Double Integrals in Polar Coordinates
Math 213 - Double Integrals in Polar Coordinates Peter A. Perry University of Kentucky October 22, 2018 Homework Re-read section 15.3 Begin work on 1-4, 5-31 (odd), 35, 37 from 15.3 Read section 15.4 for
More informationAP Calculus Review Assignment Answer Sheet 1. Name: Date: Per. Harton Spring Break Packet 2015
AP Calculus Review Assignment Answer Sheet 1 Name: Date: Per. Harton Spring Break Packet 015 This is an AP Calc Review packet. As we get closer to the eam, it is time to start reviewing old concepts. Use
More informationAB Calculus 2013 Summer Assignment. Theme 1: Linear Functions
01 Summer Assignment Theme 1: Linear Functions 1. Write the equation for the line through the point P(, -1) that is perpendicular to the line 5y = 7. (A) + 5y = -1 (B) 5 y = 8 (C) 5 y = 1 (D) 5 + y = 7
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationHKUST. MATH1013 Calculus IB. Directions:
HKUST MATH101 Calculus IB Midterm Eamination (Sample Version) Name: Student ID: Lecture Section: Directions: This is a closed book eamination. No Calculator is allowed in this eamination. DO NOT open the
More informationArkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam
0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker
More informationMath 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper Practice Paper E Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion
More informationG r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Midterm Practice Exam Answer Key
G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Midterm Practice Eam Answer Key G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s Midterm Practice Eam Answer Key Name:
More informationTechnical Calculus I Homework. Instructions
Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the
More informationBC Calculus Diagnostic Test
BC Calculus Diagnostic Test The Eam AP Calculus BC Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour and 5 minutes Number of Questions
More informationUniversity of Waterloo Final Examination MATH 116 Calculus 1 for Engineering
Last name (Print): First name (Print): UW Student ID Number: University of Waterloo Final Eamination MATH 116 Calculus 1 for Engineering Instructor: Matthew Douglas Johnston Section: 001 Date: Monday,
More information11.1 Double Riemann Sums and Double Integrals over Rectangles
Chapter 11 Multiple Integrals 11.1 ouble Riemann Sums and ouble Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated b the following important questions:
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More informationM151B Practice Problems for Exam 1
M151B Practice Problems for Eam 1 Calculators will not be allowed on the eam. Unjustified answers will not receive credit. 1. Compute each of the following its: 1a. 1b. 1c. 1d. 1e. 1 3 4. 3. sin 7 0. +
More informationUC Merced: MATH 21 Final Exam 16 May 2006
UC Merced: MATH 2 Final Eam 6 May 2006 On the front of your bluebook print () your name, (2) your student ID number, (3) your instructor s name (Bianchi) and () a grading table. Show all work in your bluebook
More informationAP Calculus AB Information and Summer Assignment
AP Calculus AB Information and Summer Assignment General Information: Competency in Algebra and Trigonometry is absolutely essential. The calculator will not always be available for you to use. Knowing
More informationMATHEMATICS Higher Grade - Paper I (Non~calculator)
Prelim Eamination 005 / 006 (Assessing Units & ) MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefully. Calculators may not be used in this paper.. Full credit
More informationPart Two. Diagnostic Test
Part Two Diagnostic Test AP Calculus AB and BC Diagnostic Tests Take a moment to gauge your readiness for the AP Calculus eam by taking either the AB diagnostic test or the BC diagnostic test, depending
More information(c) Find the gradient of the graph of f(x) at the point where x = 1. (2) The graph of f(x) has a local maximum point, M, and a local minimum point, N.
Calculus Review Packet 1. Consider the function f() = 3 3 2 24 + 30. Write down f(0). Find f (). Find the gradient of the graph of f() at the point where = 1. The graph of f() has a local maimum point,
More information5.2 Solving Quadratic Equations by Factoring
Name. Solving Quadratic Equations b Factoring MATHPOWER TM, Ontario Edition, pp. 78 8 To solve a quadratic equation b factoring, a) write the equation in the form a + b + c = b) factor a + b + c c) use
More informationSummer Packet Honors PreCalculus
Summer Packet Honors PreCalculus Honors Pre-Calculus is a demanding course that relies heavily upon a student s algebra, geometry, and trigonometry skills. You are epected to know these topics before entering
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 informationMATH 1220 Midterm 1 Thurs., Sept. 20, 2007
MATH 220 Midterm Thurs., Sept. 20, 2007 Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the eam and nothing else. Calculators
More informationWEDNESDAY, 18 MAY 9.00 AM AM. 1 Full credit will be given only where the solution contains appropriate working.
X00/0 NATINAL QUALIFICATINS 0 WEDNESDAY, 8 MAY 9.00 AM 0.0 AM MATHEMATICS HIGHER Paper (Non-calculator) Read carefull Calculators ma NT be used in this paper. Section A Questions 0 (40 marks) Instructions
More informationFind the following limits. For each one, if it does not exist, tell why not. Show all necessary work.
Calculus I Eam File Spring 008 Test #1 Find the following its. For each one, if it does not eist, tell why not. Show all necessary work. 1.) 4.) + 4 0 1.) 0 tan 5.) 1 1 1 1 cos 0 sin 3.) 4 16 3 1 6.) For
More informationReview Test 2. c ) is a local maximum. ) < 0, then the graph of f has a saddle point at ( c,, (, c ) = 0, no conclusion can be reached by this test.
eview Test I. Finding local maima and minima for a function = f, : a) Find the critical points of f b solving simultaneousl the equations f, = and f, =. b) Use the Second Derivative Test for determining
More informationMath 103 Final Exam Review Problems Rockville Campus Fall 2006
Math Final Eam Review Problems Rockville Campus Fall. Define a. relation b. function. For each graph below, eplain why it is or is not a function. a. b. c. d.. Given + y = a. Find the -intercept. b. Find
More information