MATH H53 : Mid-Term-1
|
|
- Jennifer Short
- 6 years ago
- Views:
Transcription
1 MATH H53 : Mid-Term-1 22nd September, 215 Name: You have 8 minutes to answer the questions. Use of calculators or study materials including textbooks, notes etc. is not permitted. Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Question Points Score Total: 5 1
2 1. (1 points) Sketch the curve r = cos θ, θ 2π and find the area of the inner loop. Solution: It is clear that r = when θ = 2π/3 and 4π/3. Hence the equation for the inner loop is r = cos θ, 2π/3 θ 4π/3. So the area is given by A = 1 2 4π/3 2π/3 (1 + 2 cos θ) 2 dθ = 1 2 = 1 2 = = = π/3 π/3 (1 + 2 cos (π + ϕ)) 2 dϕ (where θ = π + ϕ) (1 2 cos ϕ) 2 dϕ (since cos(π + ϕ) = cos ϕ) (1 2 cos ϕ) 2 dϕ (since cos ϕ is even) (1 4 cos ϕ + 4 cos 2 ϕ) dϕ (3 4 cos ϕ + 2 cos 2ϕ) dϕ = 3ϕ 4 sin ϕ + sin 2ϕ = π π/3 2
3 2. (a) (5 points) Find the point of intersection of the lines r 1 (t) = (1,, 2) + t(2, 1, 2) and r 2 (s) = (1, 1, 1) + s( 1, 1, 1) or show that they are skew lines. Solution: If the lines intersect, then we can simultaneously solve 1 + 2t = 1 s t = 1 + s 2 2t = 1 + s From the first two equations we see that t = 1, s = 2. Substituting in the third equation we see that the left hand side is but the right hand side is 1 which is impossible and hence the lines are skew. (b) (5 points) If the lines intersect, find the unique plane containing both the lines. If the lines are skew, find the distance between them. Solution: The common normal to the two lines (which is unique since the lines are not parallel) is given by i j k n = = i + k. Next, points P 1 (1,, 2) and P 2 (1, 1, 1) lie on the two lines, and P 1 P 2 = (, 1, 1). We saw in class that the distance between skew lines is given by the magnitude of the component of P 1 P 2 along n. So, d = comp n P 1 P 2 = n P 1P 2 n =
4 3. (a) (5 points) Show that the points P (1, 2, 2), Q(1, 1, 1), R(, 2, 4) and S(, 2, ) are co-planar. Solution: Let a = PQ = (, 1, 1) b = PR = ( 1,, 2) c = PS = ( 1, 4, 2) Then 1 1 a (b c) = =. Hence the points are co-planar. (b) (5 points) Find the equation of the plane containing all four points. Solution: We can let i j k n = a b = 1 1 = 2i + j k. 1 2 So the equation of the plane is 2x + y z = 2. 4
5 4. (a) (3 points) Find the parametric equation for the curve of intersection of the plane 2z x = 6 with the cone z 2 = x 2 + 3y 2, and identify (circle, ellipse, hyperbola or parabola) the curve. Solution: From the equation of the plane, z = x/ Substituting in the equation on the cone and simplifying we get 3 = x2 4 x + y2 Completing the squares we see that (x/2 1) 2 + y 2 = 4. That is the projection of the curve to the xy-plane is an ellipse which can be parametrized as x = cos t and y = 2 sin t. But then from the equation of the plane z = cos t. So the parametric equation of the curve is r(t) = (2 + 4 cos t, 2 sin t, cos t). Since the projection is bounded, it is clear that the curve will be a circle or an ellipse. Note that r() = (6,, 6) and r(π) = ( 2,, 2). Hence if the curve is a circle then the center will have position vector r c = (2,, 4), and the radius will be R = 2 5. On the other hand r(π/2) = (2, 2, 4), and so r(π/2) r c = 2. If the curve was a circle then this distance should also have been R = 2 5. Hence the curve is not a circle, and is in fact an ellipse! (b) (3 points) Find the curvature at ( 2,, 2). Solution: From he parametric equation above, clearly r(π) = ( 2,, 2). Differentiating the parametric equation, we see that r (π) = (, 2, ), r (π) = (4,, 2) and hence r (π) r (π) = 4( i + 2k). So the curvature κ(π) = r r r 3 = =
6 (c) (4 points) Find the equation for the osculating plane at ( 2,, 2). Solution: A normal to the osculating plane at ( 2,, 2) (or t = π) is given by r (π) r (π) = 4( i + 2k). So we just take n = i + 2k. Then the equation of the plane is x + 2z = 6. Note: This was a silly question. Thanks to Alex for pointing it out. If an entire curve lies in a plane, then the osculating plane of course better be that same plane! A better question might have been to ask for the equation of the osculating circle. Ah...maybe next time! 6
7 5. A particle moves in R 3 so that its position vector r(t) and velocity vector are related by r (t) = b r(t) for some fixed vector b R 3. Suppose also, that r() = 1. (a) (3 points) Show that the particle moves with a constant speed, and that the entire trajectory lies in the unit sphere centered at the origin. Solution: We We compute that d dt r(t) 2 = 2r(t) r (t). But from the equation, r (t) = b r(t), and hence the velocity is orthogonal to the position vector, which in turn implies that the derivative above is zero. Hence r(t) is a constant and the trajectory lies on a sphere centered at the origin. Since r() is a unit vector, the sphere has to be of radius one. To show that the speed is constant we notice that differentiating the equation for velocity, we obtain that r (t) = b r (t). And then the same argument as above shows that the speed is also constant. (b) (4 points) Show that r(t) b is a constant. Using this, show that either the particle is stationary, or it s trajectory is a circle. Solution: Again differentiating d dt r(t) b = r (t) b = (b r(t)) b =. This shows that the trajectory also lies on a plane. Since the (non-empty) intersection of a plane and a sphere is either a point or a circle, the particle is either stationary or the trajectory is a circle. (c) (3 points) If the angle between r() and b is θ, < θ < π show that the particle is not stationary, and calculate the radius of it s circular trajectory in terms of θ. Solution: Since θ, π, the initial velocity is not zero, and the particle is not stationary. The radius can be calculated purely from elementary geometry, or by using our formula for curvature. Method-1 From part (b), b is the normal to the plane containing the trajectory. let the foot of a perpendicular from the origin O to the plane be B (so this will be the center of our trajectory) and denote the tip of r() by P. Then OBP is a right angled triangle with the hypotenuse as OP = 1, and BOP = θ. The radius of the circle is then BP = sin θ. Method-2 Note that, from the above discussion, r (t) = b r (t), r r and r b. Hence κ(t) = r (t) r (t) r (t) 3 = r (t) r (t) r (t) 3 = r (t) b r (t) 2 = b r (t) = b b r(t) sin θ = 1 sin θ. Hence the radius of curvature (which is the radius that we are interested in) is the reciprocal of curvature i.e. R = sin θ. 7
MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane
MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
More informationSOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003
SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationVECTORS AND THE GEOMETRY OF SPACE
VECTORS AND THE GEOMETRY OF SPACE VECTORS AND THE GEOMETRY OF SPACE A line in the xy-plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given.
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 1 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH SOME SOLUTIONS TO EXAM 1 Fall 018 Version A refers to the regular exam and Version B to the make-up 1. Version A. Find the center
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationMath 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:
Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationMath 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.
Math c Solutions: Exam 1 Fall 16 1. Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter. x tan t x tan t y sec t y sec t t π 4 To eliminate the parameter,
More informationIn general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute
alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In
More informationArc Length and Curvature
Arc Length and Curvature. Last time, we saw that r(t) = cos t, sin t, t parameteried the pictured curve. (a) Find the arc length of the curve between (, 0, 0) and (, 0, π). (b) Find the unit tangent vector
More information12.5 Equations of Lines and Planes
12.5 Equations of Lines and Planes Equation of Lines Vector Equation of Lines Parametric Equation of Lines Symmetric Equation of Lines Relation Between Two Lines Equations of Planes Vector Equation of
More informationCalculus III (MAC )
Calculus III (MAC2-) Test (25/9/7) Name (PRINT): Please show your work. An answer with no work receives no credit. You may use the back of a page if you need more space for a problem. You may not use any
More informationPractice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
More information6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary
6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More informationMTHE 227 Problem Set 2 Solutions
MTHE 7 Problem Set Solutions 1 (Great Circles). The intersection of a sphere with a plane passing through its center is called a great circle. Let Γ be the great circle that is the intersection of the
More information(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
More information3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.
Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
More informationChapter 12 Review Vector. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30
Chapter 12 Review Vector MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30 iclicker 1: Let v = PQ where P = ( 2, 5) and Q = (1, 2). Which of the following vectors with the given
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationMath 32A Discussion Session Week 5 Notes November 7 and 9, 2017
Math 32A Discussion Session Week 5 Notes November 7 and 9, 2017 This week we want to talk about curvature and osculating circles. You might notice that these notes contain a lot of the same theory or proofs
More informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence
More informationAP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
More informationMAT 272 Test 1 Review. 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same
11.1 Vectors in the Plane 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same direction as. QP a. u =< 1, 2 > b. u =< 1 5, 2 5 > c. u =< 1, 2 > d. u =< 1 5, 2 5 > 2. If u has magnitude
More information49. Green s Theorem. The following table will help you plan your calculation accordingly. C is a simple closed loop 0 Use Green s Theorem
49. Green s Theorem Let F(x, y) = M(x, y), N(x, y) be a vector field in, and suppose is a path that starts and ends at the same point such that it does not cross itself. Such a path is called a simple
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationMultiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
Multiple Choice.(6 pts) Find smmetric equations of the line L passing through the point (, 5, ) and perpendicular to the plane x + 3 z = 9. (a) x = + 5 3 = z (c) (x ) + 3( 3) (z ) = 9 (d) (e) x = 3 5 =
More informationFinal Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
More informationMATH 2433 Homework 1
MATH 433 Homework 1 1. The sequence (a i ) is defined recursively by a 1 = 4 a i+1 = 3a i find a closed formula for a i in terms of i.. In class we showed that the Fibonacci sequence (a i ) defined by
More informationMath 114, Section 003 Fall 2011 Practice Exam 1 with Solutions
Math 11, Section 003 Fall 2011 Practice Exam 1 with Solutions Contents 1 Problems 2 2 Solution key 8 3 Solutions 9 1 1 Problems Question 1: Let L be the line tangent to the curve r (t) t 2 + 3t + 2, e
More informationSenior Math Circles February 11, 2009 Conics II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 11, 2009 Conics II Locus Problems The word locus is sometimes synonymous with
More informationMTH 234 Exam 1 February 20th, Without fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More information1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.
. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r
More informationMATH UN1201, Section 3 (11:40am 12:55pm) - Midterm 1 February 14, 2018 (75 minutes)
Name: Instructor: Shrenik Shah MATH UN1201, Section 3 (11:40am 12:55pm) - Midterm 1 February 14, 2018 (75 minutes) This examination booklet contains 6 problems plus an additional extra credit problem.
More informationVectors, dot product, and cross product
MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationProjects in Geometry for High School Students
Projects in Geometry for High School Students Goal: Our goal in more detail will be expressed on the next page. Our journey will force us to understand plane and three-dimensional geometry. We will take
More informationMath 11 Fall 2018 Midterm 1
Math 11 Fall 2018 Midterm 1 October 3, 2018 NAME: SECTION (check one box): Section 1 (I. Petkova 10:10) Section 2 (M. Kobayashi 11:30) Section 3 (W. Lord 12:50) Section 4 (M. Kobayashi 1:10) Instructions:
More informationEdexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
More informationAPPM 2350 Section Exam points Wednesday September 26, 6:00pm 7:30pm, 2018
APPM 2350 Section Exam 1 140 points Wednesday September 26, 6:00pm 7:30pm, 2018 ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3) lecture section/time (4) your instructor
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More information2) ( 8 points) The point 1/4 of the way from (1, 3, 1) and (7, 9, 9) is
MATH 6 FALL 6 FIRST EXAM SEPTEMBER 8, 6 SOLUTIONS ) ( points) The center and the radius of the sphere given by x + y + z = x + 3y are A) Center (, 3/, ) and radius 3/ B) Center (, 3/, ) and radius 3/ C)
More informationPRACTICE TEST 1 Math Level IC
SOLID VOLUME OTHER REFERENCE DATA Right circular cone L = cl V = volume L = lateral area r = radius c = circumference of base h = height l = slant height Sphere S = 4 r 2 V = volume r = radius S = surface
More informationThe Distance Formula. The Midpoint Formula
Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x
More informationMathematics 2203, Test 1 - Solutions
Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain
More informationMATH 32A: MIDTERM 1 REVIEW. 1. Vectors. v v = 1 22
MATH 3A: MIDTERM 1 REVIEW JOE HUGHES 1. Let v = 3,, 3. a. Find e v. Solution: v = 9 + 4 + 9 =, so 1. Vectors e v = 1 v v = 1 3,, 3 b. Find the vectors parallel to v which lie on the sphere of radius two
More informationTime : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
More informationMath 210, Final Exam, Spring 2012 Problem 1 Solution. (a) Find an equation of the plane passing through the tips of u, v, and w.
Math, Final Exam, Spring Problem Solution. Consider three position vectors (tails are the origin): u,, v 4,, w,, (a) Find an equation of the plane passing through the tips of u, v, and w. (b) Find an equation
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationIYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical
More informationMath Level 2. Mathematics Level 2
Math Reference Information THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE SAMPLE QUESTIONS. THIS INFORMATION IS ALSO PROVIDED ON THE ACTUAL SUBJECT TEST IN MATHEMATICS LEVEL.
More informationMATH 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2
MATH 3A: MIDTERM REVIEW JOE HUGHES 1. Curvature 1. Consider the curve r(t) = x(t), y(t), z(t), where x(t) = t Find the curvature κ(t). 0 cos(u) sin(u) du y(t) = Solution: The formula for curvature is t
More informationMATHEMATICS 317 April 2017 Final Exam Solutions
MATHEMATI 7 April 7 Final Eam olutions. Let r be the vector field r = îı + ĵj + z ˆk and let r be the function r = r. Let a be the constant vector a = a îı + a ĵj + a ˆk. ompute and simplif the following
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationMATHEMATICS 317 December 2010 Final Exam Solutions
MATHEMATI 317 December 1 Final Eam olutions 1. Let r(t) = ( 3 cos t, 3 sin t, 4t ) be the position vector of a particle as a function of time t. (a) Find the velocity of the particle as a function of time
More informationLook out for typos! Homework 1: Review of Calc 1 and 2. Problem 1. Sketch the graphs of the following functions:
Math 226 homeworks, Fall 2016 General Info All homeworks are due mostly on Tuesdays, occasionally on Thursdays, at the discussion section. No late submissions will be accepted. If you need to miss the
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 informationMATH Final Review
MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric
More informationRotation of Axes. By: OpenStaxCollege
Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationMath 212-Lecture 20. P dx + Qdy = (Q x P y )da. C
15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationA. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
More informationFigure 10: Tangent vectors approximating a path.
3 Curvature 3.1 Curvature Now that we re parametrizing curves, it makes sense to wonder how we might measure the extent to which a curve actually curves. That is, how much does our path deviate from being
More informationAPPM 2350, Summer 2018: Exam 1 June 15, 2018
APPM 2350, Summer 2018: Exam 1 June 15, 2018 Instructions: Please show all of your work and make your methods and reasoning clear. Answers out of the blue with no supporting work will receive no credit
More informationFinal Exam Math 317 April 18th, 2015
Math 317 Final Exam April 18th, 2015 Final Exam Math 317 April 18th, 2015 Last Name: First Name: Student # : Instructor s Name : Instructions: No memory aids allowed. No calculators allowed. No communication
More informationFind c. Show that. is an equation of a sphere, and find its center and radius. This n That. 3D Space is like, far out
D Space is like, far out Introspective Intersections Inverses Gone wild Transcendental Computations This n That 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 c Find
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 information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More information4.Let A be a matrix such that A. is a scalar matrix and Then equals :
1.Consider the following two binary relations on the set A={a, b, c} : R1={(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2={(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then : both R1
More informationHow can we find the distance between a point and a plane in R 3? Between two lines in R 3? Between two planes? Between a plane and a line?
Overview Yesterday we introduced equations to describe lines and planes in R 3 : r = r 0 + tv The vector equation for a line describes arbitrary points r in terms of a specific point r 0 and the direction
More informationMATH 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }
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 information2013/2014 SEMESTER 1 MID-TERM TEST. 1 October :30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
2013/2014 SEMESTER 1 MID-TERM TEST MA1505 MATHEMATICS I 1 October 2013 8:30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY: 1. This test paper consists of TEN (10) multiple choice questions
More 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 informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationworked out from first principles by parameterizing the path, etc. If however C is a A path C is a simple closed path if and only if the starting point
III.c Green s Theorem As mentioned repeatedly, if F is not a gradient field then F dr must be worked out from first principles by parameterizing the path, etc. If however is a simple closed path in the
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationGreen s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem
Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double
More informationConic Sections and Polar Graphing Lab Part 1 - Circles
MAC 1114 Name Conic Sections and Polar Graphing Lab Part 1 - Circles 1. What is the standard equation for a circle with center at the origin and a radius of k? 3. Consider the circle x + y = 9. a. What
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
More informationSenior Math Summer Packet. A. f (x) = x B. f (x) = 2 x. C. f (x) = x 2 D. f (x) = sin x
Name: Date: 1. The set of scores on a mathematics test is 72, 80, 80, 82, 87, 89, and 91. The mean score is A. 8 B. 83 C. 82 D. 80 5. Which function is one-to-one? A. f (x) = x B. f (x) = 2 x C. f (x)
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationCALCULUS 3 February 6, st TEST
MATH 400 (CALCULUS 3) Spring 008 1st TEST 1 CALCULUS 3 February, 008 1st TEST YOUR NAME: 001 A. Spina...(9am) 00 E. Wittenbn... (10am) 003 T. Dent...(11am) 004 J. Wiscons... (1pm) 005 A. Spina...(1pm)
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Approved scientific calculators and templates
More informationExam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus III June, 06 Name: Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work!
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1
OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete
More informationSMT Power Round Solutions : Poles and Polars
SMT Power Round Solutions : Poles and Polars February 18, 011 1 Definition and Basic Properties 1 Note that the unit circles are not necessary in the solutions. They just make the graphs look nicer. (1).0
More information