Spring 2012 Study Guide
Final Exam 2250-1 (7:30 class)
Exam is at 7:30am in JTB 140 on Friday, 27 April 2012
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 3, 4, 5, 6, 7, 8, 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.
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 Heaviside function and the Delta function.
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, 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 8: 8.1-4, 8.1-12, 8.1-38, 8.2-4, 8.2-19
THEOREM: u(t)=e^(Ct)u(0) solves u'=Cu. Fundamental
matrix. Matrix exponential e^(At). Nilpotent matrix. The
e^(At) series. THEOREM: Putzer's e^(AT) formula for 2x2.
THEOREM: e^(At)=Z(t)Z(0)^(-1) [Z(t)=fundamental matrix].
ANS CHECKS: maple exponential. Undetermined coefficients.
Variation of parameters. Laplace resolvent. Vector equation
Transfer Function. Cayley-Hamilton Theorem as a basis for
Spectral methods [enrichment]. Jordan form and generalized
eigenvectors [enrichment].
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: step function solutions and
Dirac delta inputs.
EPbvp7.6: Delta function problems. Transfer function.
Applications: Hammer hits, Paul Dirac's impulse function.
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 (Europe
2011) 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 (1,2,3) are also useful as a study guide, using the above list
of problems to filter out likely problem types. Finally, the three
midterms from this semester are particularly relevant and all problem
types that have appeared already are likely to appear on the final exam.
Chapters 8, 9 are new in the syllabus since Fall 2008, so there are
fewer past exam questions to study. Sample problems are listed above and
exam questions will use the same problem type.
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