Practice Problems for Final Exam
|
|
- Denis James
- 5 years ago
- Views:
Transcription
1 Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5, ẏ = 4x + x 3 xy 4 Solution: Denote f(y) = 9y 5y 5, g(y) = 4x + x 3 xy 4. f( y) = 9y + 5y 5 = f(y), f(y) is odd; g( y) = 4x + x 3 x( y) 4 = 4x + x 3 xy 4 = g(y), g(y) is even. Therefore, the system is reversible. It is easy to check that the origin is a FP because both right hand sides vanish when x = 0 and y = 0. [ y Jacobian is A = x 2 y 4 4xy 3 [ 0 9 A(0, 0) = T = 0, = 36 > 0 Hence, there is a center at the origin. 4 0 The system has a center at the origin and it is reversible. Hence it has a nonlinear center at the origin. 2. Show that the given system is reversible but not conservative. ẋ = 4y cos y + 3x sin y, ẏ = 5 cos y + 4 sin x Hint: Show that the point ( 0, π 2 ) is a FP and determine its type. Solution: Denote f(y) = 4y cos y + 3x sin y, g(y) = 5 cos y + 4 sin x. f( y) = 4( y) cos( y) + 3x sin( y) = 4y cos y 3x sin y = f(y), f(y) is odd; g( y) = 5 cos( y) + 4 sin x = 5 cos y + 4 sin x = g(y), g(y) is even. Therefore, the system is reversible. It is easy to check that the point ( 0, π 2 ) is a FP because both right hand sides vanish when x = 0 and y = π 2.
2 [ 3 sin y 4 cos y 4y sin y + 3x cos y Jacobian is A = 4 cos x 5 sin y [ 3 2π A(0, π) = T = 1 < 0, = 15+8π > 0 The FP ( ) 0, π 2 is an attractor. A conservative system can never have an attracting FP. Therefore, the system cannot be conservative. 3. Consider the system ẋ = x y x(x 2 + y 2 ), ẏ = x + y y(x 2 + y 2 ) (a) Write the system in polar coordinates Solution: ṙ = 1 r (xẋ + yẏ) = 1 r (x2 xy x 2 r 2 + xy + y 2 y 2 r 2 ) = 1 r (r2 r 4 ) ṙ = r(1 r 2 ) θ = 1 r (xẏ yẋ) = 1 2 r 2 (x2 + xy xyr 2 xy + y 2 + xyr 2 ) = 1 r 2 (r2 ) = 1 The system in polar coordinates is ṙ = r(1 r 2 ), θ = 1 (b) Show that a limit cycle exists and find it. Solution: If r(1 r 2 ) = 0, then r = 0 or r = 1. The circle r = 1 is the limit cycle. 4. Consider the system ẋ = x + y 2, ẏ = y(2 + 2x y 2 ) (a) Use index theory to show that the system has no closed orbits Solution: x-nullcline: x = y 2, y-nullcline: y = 0, x = 1 2 y2 1. The curves x = y 2 and x = 1 2 y2 1 don t intersect. Indeed, assume they intersect. Then we have y 2 = 1 2 y2 1 or 1 2 y2 = 1 which is impossible. Therefore, the only FP is the intersection of the curve x = y 2 and the line y = 0, Page 2
3 which is the point (0, 0). [ 1 2y Jacobian is A = 2y 2 + 2x 3y 2 [ 1 0 A(0, 0) = 0 2 = 2 < 0 and the FP (0, 0) is saddle. If there is a closed orbit C then it either encloses the saddle FP at the origin or does not. An index of a saddle point is 1 and an index of a closed orbit according to the index theory has to be +1. If C encloses the origin its index is 1. If C doesn t enclose it its index is 0. In either case the index of C is not +1. Therefore, the system cannot have a closed orbit. (b) Use a method different from index theory and Dulac s criterion to show that the system has no closed orbits Solution: Denote f(x, y) = x + y 2, g(x, y) = y(2 + 2x y 2 ). Then f y = 2y = g x and the system is gradient. Gradient systems have no closed orbits. Therefore, the system has no closed orbits. (c) Use Dulac s criterion to show that the system has no closed orbits. You may use a Dulac s function g(x, y) = 1 y. Solution: (g(x, y)(ẋ, ẏ)) = x (gẋ) + y (gẏ) = ( xy ) x + y + y (2 + 2x y2 ) = 1 y 2y. Now let s look at the x-axis. Suppose (x 0, 0) is a point on the axis. Let x(t) be a solution to the equation ẋ = x + y 2 with the initial condition x(0) = x 0, and let y(t) = 0. Then (x(t), y(t)) = (x(t), 0) is a solution to the given system. This means that the x-axis is a trajectory and cannot be crossed by another trajectory by existence and uniqueness theorem. The x-axis divides the xy-plane into two parts: the upper half-plane (y > 0) and the lower half-plane (y < 0). In each halves (g(x, y)(ẋ, ẏ)) preserves its sign ( (g(x, y)(ẋ, ẏ)) < 0 when y > 0 and (g(x, y)(ẋ, ẏ)) > 0 when y < 0 ). Each half is a simply connected region. Hence in each half conditions for Dulac s criterion hold and there are no closed orbits. Closed orbits being trajectories cannot cross the x-axis. Therefore, the system has no closed orbits in the xy-plane. Page 3
4 5. For ẋ = y + 3xy 2, ẏ = x 4x 2 y use Lyapunov function to show that the system has no closed orbits. Specifically, show that V (x, y) = x 2 + y 2 is Lyapunov function for the system and hence no closed orbits are allowed. Solution: Consider a Lyapunov function V (x, y) = x 2 + y V (0, 0) = V (x, y) > 0 for all (x, y) V = Vx ẋ + V y ẏ = 2x( y + 3xy 2 ) + 2y(x 4x 2 y) = 2xy + 6x 2 y 2 + 2yx 8x 2 y 2 = 2x 2 y 2 < 0 for all (x, y), a < b. V (x, y) = x 2 + y 2 and the system has no closed orbits. satisfies all requirements for a Lyapunov function 6. Determine if the given system is gradient. If it is, find its potential function V. ẋ = 2xy 3 + 3y 2 cos x, ẏ = 3(x 2 1)y 2 + 6y sin x Solution: Denote f(x, y) = 2xy 3 + 3y 2 cos x, g(x, y) = 3(x 2 1)y 2 + 6y sin x. f y = 6xy 2 + 6y cos x, g x = 6xy 2 + 6y cos x. f y = g x. Therefore, the system is gradient. V x (x, y) = f(x, y) = 2xy 3 + 3y 2 cos x, V (x, y) = x 2 y 3 + 3y 2 sin x + φ(y). V y (x, y) = g(x, y). 3x 2 y 2 + 6y sin x + φ (y) = 3(x 2 1)y 2 + 6y sin x. φ (y) = 3y 2, φ(y) = ( 3y 2 ) dy = y 3 + C. V (x, y) = x 2 y 3 + 3y 2 sin x y 3 + C. We can put C = 0. V (x, y) = (x 2 1)y 3 + 3y 2 sin x. 7. Consider the system ẋ = 3x 5y x(x 2 + 4y 2 ), ẏ = 5x + 3y y(2x 2 + y 2 ) Page 4
5 (a) Classify the FP at the origin. Solution: Jacobian: A = A(0, 0) = [ (0, 0) is an unstable spiral. [ 3 3x 2 4y 2 5 8xy 5 4xy 3 2x 2 3y 2, T = 3, = 34, > T 2. (b) Switch to polar coordinates and write an equation for ṙ Solution: ṙ = 1 r (xẋ + yẏ) = 1 r (3x2 5xy x 4 4x 2 y 2 + 5xy + 3y 2 2x 2 y 2 y 4 ) = 1 r (3r2 (x 4 + 2x 2 y 2 + y 4 ) 4x 2 y 2 ) = 1 ( 3r 2 (x 2 + y 2 ) 2 4r 4 cos 2 θ sin 2 θ ) = 1 ( 3r 2 r 4 r 4 sin 2 2θ ) r r ṙ = r ( 3 r 2 r 2 sin 2 2θ ) (c) Determine the circle of maximum radius r 1, centered at the origin such that all trajectories have a radially outward component on it. Solution: On this circle ṙ > 0. To find its radius we require 3 r 2 r 2 sin 2 2θ = 3 ( 1 + sin 2 2θ ) r 2 > 0. 0 sin 2 2θ 1, sin 2 2θ 2. Then a sufficient condition is 3 2r 2 > 0 3 Which gives r <. We take r 2 1 = (d) Determine the circle of minimum radius r 2, centered at the origin such that all trajectories have a radially inward component on it. Solution: On this circle ṙ < 0. To find its radius we require 3 ( 1 + sin 2 2θ ) r 2 < 0. As before, sin 2 2θ 2. Then a sufficient condition is 3 r 2 < 0 Page 5
6 Which gives r > 3. We take r 2 = (e) There is no FPs in the region r 1 r r 2. You don t have to show that. Prove that the system has a limit cycle somewhere in the trapping region r 1 r r 2. Solution: There is a source at the origin. All trajectories go inside the trapping region through both circles of radii r 1 and r 2 centered at the origin. There are no FPs inside the region. By Poincare-Bendixson theorem there must be a limit cycle inside the region r 1 r r 2. Page 6
1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationNon-Linear Dynamics Homework Solutions Week 6
Non-Linear Dynamics Homework Solutions Week 6 Chris Small March 6, 2007 Please email me at smachr09@evergreen.edu with any questions or concerns reguarding these solutions. 6.8.3 Locate annd find the index
More information2.10 Saddles, Nodes, Foci and Centers
2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one
More information154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.
54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and
More informationLecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. Poincare-Bendixson theorem Hopf bifurcations Poincare maps
Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion Poincare-Bendixson theorem Hopf bifurcations Poincare maps Limit
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationProblem set 7 Math 207A, Fall 2011 Solutions
Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase
More informationẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.
4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationMath 234 Final Exam (with answers) Spring 2017
Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More informationSolutions to Dynamical Systems 2010 exam. Each question is worth 25 marks.
Solutions to Dynamical Systems exam Each question is worth marks [Unseen] Consider the following st order differential equation: dy dt Xy yy 4 a Find and classify all the fixed points of Hence draw the
More informationProblem Set Number 5, j/2.036j MIT (Fall 2014)
Problem Set Number 5, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Fri., October 24, 2014. October 17, 2014 1 Large µ limit for Liénard system #03 Statement:
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More informationUniversity of Alberta. Math 214 Sample Exam Math 214 Solutions
University of Alberta Math 14 Sample Exam Math 14 Solutions 1. Test the following series for convergence or divergence (a) ( n +n+1 3n +n+1 )n, (b) 3 n (n +1) (c) SOL: n!, arccos( n n +1 ), (a) ( n +n+1
More information11 Chaos in Continuous Dynamical Systems.
11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47 11 Chaos in Continuous Dynamical Systems. Let s consider a system of differential equations given by where x(t) : R R and f : R R. ẋ = f(x), The linearization
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationNonlinear Control Lecture 2:Phase Plane Analysis
Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationAPPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.
APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.
More informationProblem List MATH 5173 Spring, 2014
Problem List MATH 5173 Spring, 2014 The notation p/n means the problem with number n on page p of Perko. 1. 5/3 [Due Wednesday, January 15] 2. 6/5 and describe the relationship of the phase portraits [Due
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 informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationNonlinear dynamics & chaos BECS
Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More information= e t sin 2t. s 2 2s + 5 (s 1) Solution: Using the derivative of LT formula we have
Math 090 Midterm Exam Spring 07 S o l u t i o n s. Results of this problem will be used in other problems. Therefore do all calculations carefully and double check them. Find the inverse Laplace transform
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More information8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
More informationCALCULUS: Math 21C, Fall 2010 Final Exam: Solutions. 1. [25 pts] Do the following series converge or diverge? State clearly which test you use.
CALCULUS: Math 2C, Fall 200 Final Exam: Solutions. [25 pts] Do the following series converge or diverge? State clearly which test you use. (a) (d) n(n + ) ( ) cos n n= n= (e) (b) n= n= [ cos ( ) n n (c)
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationMath Maximum and Minimum Values, I
Math 213 - Maximum and Minimum Values, I Peter A. Perry University of Kentucky October 8, 218 Homework Re-read section 14.7, pp. 959 965; read carefully pp. 965 967 Begin homework on section 14.7, problems
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt
More informationVANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions
VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the
More informationMath Review for Exam Compute the second degree Taylor polynomials about (0, 0) of the following functions: (a) f(x, y) = e 2x 3y.
Math 35 - Review for Exam 1. Compute the second degree Taylor polynomial of f e x+3y about (, ). Solution. A computation shows that f x(, ), f y(, ) 3, f xx(, ) 4, f yy(, ) 9, f xy(, ) 6. The second degree
More informationProblem Set 5 Solutions
Problem Set 5 Solutions Dorian Abbot APM47 0/8/04. (a) We consider π θ π. The pendulum pointin down corresponds to θ=0 and the pendulum pointin up corresponds to θ π. Define ν θ. The system can be rewritten:
More informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
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 information1. For each function, find all of its critical points and then classify each point as a local extremum or saddle point.
Solutions Review for Exam # Math 6. For each function, find all of its critical points and then classify each point as a local extremum or saddle point. a fx, y x + 6xy + y Solution.The gradient of f is
More informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More information1 The pendulum equation
Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating
More informationNonlinear Control Lecture 2:Phase Plane Analysis
Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2009 Farzaneh Abdollahi Nonlinear Control Lecture 2 1/68
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Lyapunov Stability - I Hanz Richter Mechanical Engineering Department Cleveland State University Definition of Stability - Lyapunov Sense Lyapunov
More informationMATH 2400: Calculus III, Fall 2013 FINAL EXAM
MATH 2400: Calculus III, Fall 2013 FINAL EXAM December 16, 2013 YOUR NAME: Circle Your Section 001 E. Angel...................... (9am) 002 E. Angel..................... (10am) 003 A. Nita.......................
More informationPhase Portraits of Nonlinear Differential Equations
ODE4-net.nb Phase Portraits of Nonlinear Differential Equations Nonlinear Differential Equations: x' = f (x, y) (1) Consider the system where f and g are functions of two y' = g(x, y) () variables x and
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the
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 informationComplex Dynamic Systems: Qualitative vs Quantitative analysis
Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic
More informationMATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.
MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationMATH 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
More information0, otherwise. Find each of the following limits, or explain that the limit does not exist.
Midterm Solutions 1, y x 4 1. Let f(x, y) = 1, y 0 0, otherwise. Find each of the following limits, or explain that the limit does not exist. (a) (b) (c) lim f(x, y) (x,y) (0,1) lim f(x, y) (x,y) (2,3)
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: April 06, 2018, at 14 00 18 00 Johanneberg Kristian
More informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
More informationE209A: Analysis and Control of Nonlinear Systems Problem Set 3 Solutions
E09A: Analysis and Control of Nonlinear Systems Problem Set 3 Solutions Michael Vitus Stanford University Winter 007 Problem : Planar phase portraits. Part a Figure : Problem a This phase portrait is correct.
More informationMath 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall.
Math 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall. Name: SID: GSI: Name of the student to your left: Name of the student to your right: Instructions: Write all answers
More informationMATH 44041/64041 Applied Dynamical Systems, 2018
MATH 4441/6441 Applied Dynamical Systems, 218 Answers to Assessed Coursework 1 1 Method based on series expansion of the matrix exponential Let the coefficient matrix of α the linear system be A, then
More informationMATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:
MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all
More information1 4 (1 cos(4θ))dθ = θ 4 sin(4θ)
M48M Final Exam Solutions, December 9, 5 ) A polar curve Let C be the portion of the cloverleaf curve r = sin(θ) that lies in the first quadrant a) Draw a rough sketch of C This looks like one quarter
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian
More informationBIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs
BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.
More informationMATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation
MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter
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 informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationJim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt
Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain
More information1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:
More informationMath 232, Final Test, 20 March 2007
Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.
More informationCIRCLES PART - II Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x 2 + y 2 = a 2 is
CIRCLES PART - II Theorem: The equation of the tangent to the circle S = 0 at P(x 1, y 1 ) is S 1 = 0. Theorem: The equation of the normal to the circle S x + y + gx + fy + c = 0 at P(x 1, y 1 ) is (y
More informationSample Solutions of Assignment 10 for MAT3270B
Sample Solutions of Assignment 1 for MAT327B 1. For the following ODEs, (a) determine all critical points; (b) find the corresponding linear system near each critical point; (c) find the eigenvalues of
More informationClassification of Phase Portraits at Equilibria for u (t) = f( u(t))
Classification of Phase Portraits at Equilibria for u t = f ut Transfer of Local Linearized Phase Portrait Transfer of Local Linearized Stability How to Classify Linear Equilibria Justification of the
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationFINAL EXAM MATH303 Theory of Ordinary Differential Equations. Spring dx dt = x + 3y dy dt = x y.
FINAL EXAM MATH0 Theory of Ordinary Differential Equations There are 5 problems on 2 pages. Spring 2009. 25 points Consider the linear plane autonomous system x + y x y. Find a fundamental matrix of the
More informationMath 53 Spring 2018 Practice Midterm 2
Math 53 Spring 218 Practice Midterm 2 Nikhil Srivastava 8 minutes, closed book, closed notes 1. alculate 1 y 2 (x 2 + y 2 ) 218 dxdy Solution. Since the type 2 region D = { y 1, x 1 y 2 } is a quarter
More informationNonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points
Nonlinear Control Lecture # 2 Stability of Equilibrium Points Basic Concepts ẋ = f(x) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f( x) = 0 Characterize and
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 informationThere is a more global concept that is related to this circle of ideas that we discuss somewhat informally. Namely, a region R R n with a (smooth)
82 Introduction Liapunov Functions Besides the Liapunov spectral theorem, there is another basic method of proving stability that is a generalization of the energy method we have seen in the introductory
More informationMath 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C
Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line
More informationMath 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will
Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The
More informationMath 273, Final Exam Solutions
Math 273, Final Exam Solutions 1. Find the solution of the differential equation y = y +e x that satisfies the condition y(x) 0 as x +. SOLUTION: y = y H + y P where y H = ce x is a solution of the homogeneous
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS08 SEMESTER: Spring 0 MODULE TITLE: Dynamical
More informationMATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Often, the equation of a curve may not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this section we see how to calculate the
More informationApplied Calculus. Review Problems for the Final Exam
Math135 Study Guide 1 Math 131/135/194, Fall 2004 Applied Calculus Daniel Kaplan, Macalester College Review Problems for the Final Exam Problem 1../DE/102b.tex Problem 3../DE/107.tex Consider the pair
More informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More informationLesson 4: Non-fading Memory Nonlinearities
Lesson 4: Non-fading Memory Nonlinearities Nonlinear Signal Processing SS 2017 Christian Knoll Signal Processing and Speech Communication Laboratory Graz University of Technology June 22, 2017 NLSP SS
More informationME3.6 Sheet 2 Answers
ME36 Sheet 2 Answers (i) We have dx/dt Ý6x 2xy Ý 8 fx,y,dy/dt y 2 Ý x 2 gx,y The critical points satisfy g y x and f Ý6x 2xy Ý 8 Substitute y x to get Ý6x 2x 2 Ý 8 x Ý 4x x 4 or Ý If instead we substitute
More informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
More informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
More informationMath 114: Make-up Final Exam. Instructions:
Math 114: Make-up Final Exam Instructions: 1. Please sign your name and indicate the name of your instructor and your teaching assistant: A. Your Name: B. Your Instructor: C. Your Teaching Assistant: 2.
More informationPhysics 202 Laboratory 3. Root-Finding 1. Laboratory 3. Physics 202 Laboratory
Physics 202 Laboratory 3 Root-Finding 1 Laboratory 3 Physics 202 Laboratory The fundamental question answered by this week s lab work will be: Given a function F (x), find some/all of the values {x i }
More information