Spring 2015 Study Guide Final Exam 2250-10 (8:05 class) Exam is at 7:15am in WEB 1230 on Wednesday, May 6, 2015 The 2250 final exam consists of at least sixteen problems. The problems are divided by chapters. You are expected to complete one or two per chapter for full credit. Only chapters 1, 2, 3, 4, 5, 6, 7, 9, 10 appear on the exam. Fundamental skills from chapters 1 and 2 are required. This includes the variable separable method in section 1.4, the linear integrating factor method in section 1.5, the stability definition and intuition from the scalar case in section 2.2 and the position-velocity substitution from section 2.3. The following problems will be used as models for the problems that will appear on the final exam. Each problem will have one to five parts, to facilitate division of credit for that problem. In addition, problem types appearing on previous midterm exams, and on final exams for the last 5 years, are fair game. It is expected that you will study the solution keys to final exams for the last 5 years. The old exam keys are available at the course web site. Topics outside the subject matter of these problems will not be tested. However, theoretical questions about the details of the problem may be asked. Generally, proofs of textbook theorems are not part of the final exam. There is no numerical or maple work on the final exam, nor are you asked to know anything other than basic integral tables and derivative formulas. This includes but is not limited to the first 20 integral table entries in the textbook. The basic Laplace table (4 items) is assumed plus the 10 Laplace rules through the convolution theorem, including the unit step and the Dirac impulse. Chapter 1: 1.2-7,8,10; 1.3-15,27; 1.4-15,17,39; 1.5-5,17,23,39; Chapter 2: 2.1-7,17; 2.2-9,17; 2.3-9,23 Quadrature method, Picard theorem on existence-uniqueness, separable equation, applications of first order equations, linear first order, integrating factor method, cascade of two tanks, Verhulst logistic equation, population dynamics, stability: funnel, spout, node, phase diagram, linear drag model, nonlinear drag model, parachute problem. No numerical work will appear on the final exam, which excludes sections 2.4, 2.5, 2.6. Chapter 3: 3.1-16, 3.2-18, 3.2-28, 3.3-18, 3.4-22, 3.4-29, 3.5-21, 3.6-17, 3.6-32, 3.6-39, 3.6-60 Frame sequence to rref. General solution. Reduced echelon system. Free and lead variables. The three possibilities. Matrices. Vectors. Inverses. Rank, nullity. Basis of solutions. Elementary matrices Determinants. Adjugate formula. Cayley-Hamilton theorem. Chapter 4: 4.1-16, 4,1-21, 4.1-31, 4.1-34, 4.2-11, 4.2-13, 4.2-19, 4.3-17, 4.3-23, 4.4-9, 4.4-19, 4.5-9, 4.5-22, 4.6-4, 4.7-7, 4.7-11, 4.7-21 Vector spaces. Subspaces. Basis. Dimension. Orthogonality. Vector space toolkit. Rank. Nullity. Transpose. Theorems 1 and 2 of 4.2. Independence tests: Rank test, Determinant test, Sampling test, Wronskian test. Pivot theorem. Equivalence of bases. Subspace proofs. Finding bases. Topics about row and column spaces in 4.5 are not tested. Section 4.6 is covered lightly, only orthogonality of fixed vectors and independence of orthogonal sets. Chapter 5: 5.1-33 to 5.1-42, 5.3-15, 5.2-21, 5.3-1 to 5.3-20, 5.3-28, 5.3-33 5.4-17, 5.5-4, 5.5-27, 5.5-39, 5.5-49, 5.6-9, 5.6-13, 5.6-17, 5.6-27 Roots. Atoms. General solution from an atom list. Over-damped, critically damped, under-damped. Phase-amplitude solution. Undetermined coefficients. Shortest trial solution. Variation of parameters. Steady state periodic solution. Pure and practical resonance. Beats. Mechanical oscillators. Electric circuits. Pendulum. Tacoma narrows bridge. London Millennium bridge. Wine glass experiment. EPbvp3.7: Electrical circuits, electrical resonance. Chapter 6: 6.1-5, 6.1-13, 6.1-23, 6.1-33 to 6.1-36, 6.2-11, 6.2-17, 6.2-25, 6.2-31 to 6.2-37 Eigenpairs. Eigenpair packages P and D. Complex eigenvalues and eigenvectors. Diagonalization theory AP=PD. Independence of eigenvectors. Similar matrices. Computation of eigenpairs and matrices D, P in diagonalization AP=PD. Slides: Data conversion example. Eigenpair equations. Eigenanalysis history. Fourier's model equivalent to AP=PD. Chapter 7: 7.1-19, 7.1-24, 7.2-15, 7.3-11, 7.3-17, 7.3-27, 7.3-39 Brine tank. Railroad cars. x'=Ax for 2x2, 3x3, 4x4. Linear integrating factor method. Eigenanalysis method for x''=Ax for 2x2, 3x3. The four methods: (1) First-order method for triangular A. (2) Cayley-Hamilton-Ziebur Method to solve u'=Au for any square matrix A. (3) The Eigenanalysis method. (4) Laplace resolvent method for u'=Au and x''=Ax+F(t). Home heating with space heater and furnace. Pollution in 3 lakes. Cascades. Recycled brine tanks. Drug elimination in the human body [mercury, lead, aspirin], which appears in optional Maple Lab 10. Applications: [not on final exam] Earthquakes. Boxcars. Coupled spring-mass system modeling and symmetry. Chapter 9: 9.1-8, 9.1-18, 9.2-2, 9.2-12, 9.2-22, 9.3-28, 9.4-8 Theory: Stability. Autonomous system. Direction field. Phase plane. Equilibria. Unstable. Asymptotically stable. Attractor. Repeller. Spiral. Saddle. Node. Center. Linearization. Jacobian. Classification of almost linear systems. Theorem 2 in 9.2. How to apply Theorem 2 when using the maple 12 phase portrait tool. Applications: [not on the final exam] Predator-prey systems. Competing species. Co-existence. Oscillating populations. Competition. Inhibition. Cooperation. Predation. Hard spring. Soft spring. Damped nonlinear vibrations. Nonlinear pendulum. Undamped pendulum. Damped pendulum. Maple phase portrait tool, Maple DynamicSystems package [enrichment]. Chapter 10: 10.1-11 to 10.1-32, 10.2-5, 10.2-11, 10.2-17 to 10.2-24, 10.3-9, 10.3-19, 10.3-33, 10.3-37, 10.4-17, 10.4-18. Rules: Shift, parts, s-diff, Lerch. Table: 5-line brief Table. Solve y''=10. Solve y'-y=5-2t. Solve a 2x2 system. Solve a second order system x''=10, y''=y'+x. Forward table methods. Partial fractions. Backward table methods. Integral theorem. Periodic function theorem. Convolution theorem. Resolvent methods for u'=Cu and x''=Ax. Transfer function. Unit step, square wave, sawtooth, staircase, ramp. Delta function. Heaviside function. Piecewise defined functions. Solving u'=Cu by the Laplace resolvent method [(sI -C)L(u)=u(0)]. Solving x''=Ax+F(t) by the Laplace resolvent method [(s^2 I - A)L(x)=u'(0)+u(0)s+L(F)]. Maple DynamicSystems package [enrichment]. Maple inttrans package. The second shifting theorem: unit step function solutions and Dirac impulse inputs. EPbvp7.6: Dirac Delta problems. Transfer function. Applications: Hammer hits, Paul Dirac's impulse model. Applications: [not on the final exam] Home heating. Earthquakes. Boxcars. Coupled spring-mass system modeling and symmetry. Final exams for 2250 with solution keys for 2008, 2009, 2010 (none in 2011), 2012, 2013, 2014 appear on the web page http://www.math.utah.edu/~gustafso/index2250.html These exams may be printed and used as a study guide. Other exams from 2006-2007 (exams 1,2,3) are also useful as a study guide, using the above list of problems to filter the likely problem types. Finally, the midterms from this semester are particularly relevant and all problem types that have appeared already are likely to appear on the final exam. ===end===