M:/DeptDocuments/Program learning outcomes learning outcomes.pdf · Solve linear equations algebraically and graphically 20. Recognize how many solutions a linear equation should

Department of Mathematics

Fort Lewis College

September 1, 2014

1 Mathematics Department Mission Statement

In a world where numerical proficiency is becoming ever more vital, the Fort LewisCollege Mathematics Department sees itself as having a threefold mission with theoverarching goal to foster critical thinking and problem solving skills in all coursesand for all students.

1. To provide students in STE departments with the mathematical tools neededfor success in their chosen majors, thereby helping to create strong majors inall STE disciplines;

2. To train future teachers of mathematics, thereby promoting mathematics edu-cation at the elementary and secondary school level in the Four Corners region;

3. To prepare mathematics majors for future graduate study or technical careers,thereby advancing the study and practice of mathematics.

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2 Mathematics Program Learning Outcomes

(with corresponding FLC learning outcome area)

1. Students will demonstrate competency in Algebra. (Learning as Inquiry)

2. Students will demonstrate competency in Geometry. (Learning as Inquiry)

3. Students will demonstrate competency in Calculus. (Learning as Inquiry)

4. Students will be able to write a correct mathematical proof using proper ter-minology and logical structure. (Critical Thinking as Problem Solving, Com-munication)

5. Students will produce and deliver effective written and oral presentations ofmathematical material and ideas. (Communication)

6. Students will apply algebraic, geometric, and computational methods in ab-stract settings. (Learning as Inquiry, Critical Thinking as Problem Solving)

7. Students will apply algebraic, geometric, and computational methods in appliedsettings. (Learning as Inquiry, Critical Thinking as Problem Solving)

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3 Course Specific Learning Objectives

Math 113, Algebra for Calculus

Description of Course

This course is intended to help you develop a strong understanding, both proce-durally and conceptually, of Algebra. The topics covered are standard for a CollegeAlgebra course; the emphasis is on becoming proficient in algebraic procedures andunderstanding their form.

Course Learning Objectives

1. Evaluate algebraic expressions

2. Interpret algebraic expressions

3. Construct algebraic expressions

4. Understand and identify difference between expressions and equations

5. Interpret equations

6. Identify equivalent and inequivalent expressions

7. Manipulate expressions into equivalent expressions, understand and use:

(a) Commutative and Associative Laws

(b) Distributive Law

(c) Expansion and Factoring

(d) Manipulating rational expressions

8. Identify equivalent and inequivalent equations

9. Manipulate equations into equivalent equations

(a) Know the difference between which operations one performs on expres-sions vs. equations

(b) Be able to correctly perform these operations

10. Solve inequalities

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11. Solve equations and inequalities which include the absolute value function

12. Understanding of functions:

(a) definition

(b) terminology

(c) independent vs. dependent variable

(d) evaluating functions both at fixed values and at variable values

(e) multiple representations: algebraic, numerical, graphical, verbal

13. Understand functions as expression: interpretation, evaluation

14. Understand functions in equations: solving, intepreting

15. Direct proportionality, identify which functions represent direct proportional-ity, application

16. Linear Functions:

(a) identify which functions are linear

(b) identify slope and y-intercept

(c) interpret and identify graphically, numerically, and verbally

17. Manipulate into equivalent forms: y = mx + b, Ax + By = C, and y =m(x− x0) + y0

18. Recognize value of and apply each form above

19. Solve linear equations algebraically and graphically

20. Recognize how many solutions a linear equation should have

21. Solve linear equations with parameters

22. Parallel and perpendicular lines

23. Vertical lines

24. solve systems of linear equations by substitution and by elimination

25. applications of systems of linear equations

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26. Exponents

(a) understanding of exponents

(b) ability to correctly use the rules of exponents

(c) understanding of reason for rules of exponents

27. Logarithms

(a) understand and apply definition

(b) apply logarithm rules

(c) use logarithm function to solve exponential equations

28. Power functions:

(a) identify power functions

(b) manipulate form of power functions

(c) identify and interpret coefficient and exponent

(d) know basic form of graph of power functions

(e) understand effect of positive, negative, fractional exponents

(f) solve power equations

29. More details concerning functions:

(a) understanding and ability to find domain and range of functions

(b) compositions and decomposition of functions

(c) affine transformations

(d) inverse functions

30. Quadratic Functions:

(a) Recognize forms of quadratic functions: y = Ax2 + Bx + C, y = A(x −r1)(x− r2), y = A(x− h)2 + k

(b) Ability to manipulate from one form to the others

(c) Understanding of relative value of each form and ability to apply eachform

(d) Solve quadratic equations

(e) Understanding of need for and basic ability to use complex numbers

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Math 121, Precalculus

Functions, Domain, and Range

• Students should be able to solve basic equations for variables and evaluate ex-pressions. Types of equations are linear, quadratic equations in vertex form,and equations involving fractions/powers that are equivalent to linear equa-tions.

• Students should be able to use number sense to find the domain and range ofthese functions.

• Students should understand the graphical representations of solving, evaluat-ing, domain, and range.

Composite and Inverse Functions

• Students should be able to compose functions given algebraically and identifypossible inside and outside functions in composite functions.

• Students should understand how to find formulas for inverses of simple func-tions and how inverse functions may be used to solve equations.

• Students should be able to interpret compositions and inverses verbally usingunits of the input and output, and evaluate an inverse function form a graphof the non-inverted function.

Quadratic Functions and Factoring

• Students should be able to expand factored expressions and factor expandedexpressions.

• Students should understand the relative merits of both factored and standardform of a quadratic function.

• Skills: Students should be able to complete the square by solving the equation

ax2 + bx+ c = a(x− h)2 + k

for the parameters h and k (we assume a is given in the problem).

• Students should be able to solve quadratic equations using factoring, complet-ing the square, or the quadratic formula when appropriate.

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Exponents

• Students shall attain mastery exponent rules, and should probably also getsome order of operations practice in this section

• Students should be able to solve equations after making correct simplificationsto expressions with exponents.

Exponential Functions

• Students should know and understand how to use the terminology initial value,growth factor, and growth rate in the context of exponential functions.

• Students should be able to find a formula for an exponential function interpo-lating two points. The parallel between this procedure and the one for linearfunctions should be emphasized.

• Students should be able to use exponent rules to write any exponential functionin the standard for abt.

• Students should be familiar with the following forms of exponential functions:abt, a(1 + r)t, and aekt. Students should be able to translate from one formto another (finding the continuous growth rate is not covered until logarithmshave been introduced).

• Students should know the general shape of the graph of an exponential func-tion and understand the graphical significance of the parameters a, b, and k.Emphasis is placed on the fact that the graphical significance of each parameterfollows from algebraic/numerical facts.

• Students should know practical settings in which exponential functions arise.

Logarithms

• Students should be able to compute simple logarithms (common and natural)without a calculator, rewrite exponential equations in equivalent logarithmicform, rewrite logarithmic equations in exponential form, and simplify expres-sions using properties of logarithms.

• Students should be able to use logarithms to solve equations involving exponen-tial functions and use exponentiation to solve equations involving logarithms.

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• Students should gain proficiency using exponent rules and properties of loga-rithms to simplify expressions and equations in order to make evaluating andsolving easier.

• Students should be able to find doubling-time or half-life for exponentiallygrowing or decaying quantities, respectively.

• Students should be able graph y = ln(x) and y = log(x) without a calculatorusing their knowledge of exponential functions and the fact that logarithms areinverses of exponential functions.

• Students should be able to find the domain and range of, and graph, generalfunctions involving logarithms.

Graphical Transformations

• Given a formula for a function, with a verbal description of a sequence ofgraphical transformations (including vertical/horizontal shifts, stretches, com-pressions, and reflections across the coordinate axes), students should be ableto write a formula for the transformed function.

• Given a generic transformed formula and a graph of the function being trans-formed, students should be able to draw the graph of the transformed functionas well as describe it verbally in the correct order.

• Students should know the definition of an even or odd function and show afunction is even or odd algebraically, as well as know the graphical interpreta-tion of even- or odd-ness of functions.

Trigonometric and Periodic Functions

• Students should know the definition of a periodic function as well as the period,amplitude, and midline of a periodic function.

• Given the graph or verbal description of a periodic function, students shouldbe able to identify the period, amplitude, and midline. Students should alsobe able to identify functions that are not periodic.

• Students should know the definitions of the sine , cosine, and tangent of anangle in a right triangle.

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• Students should be able to find the sine, cosine, and tangent of an angle giventhe dimensions of a right triangle.

• Students should be able to find dimensions of right triangles in terms of trigono-metric functions.

• Students should be able to use the Pythagorean Theorem to find values oftrigonometric functions and/or dimensions of a right triangle given values ofdifferent trigonometric functions and/or other dimensions.

• Students should know exact values of outputs of sine, cosine, and tangent ofangles whose reference angle is 0◦, 30◦, 45◦, 60◦, and 90◦.

• Students should understand the cosine and sine as the coordinates of a pointon the unit circle.

• Students should be able to find the coordinates of a point on a general circle.

• Students should be able to convert angle measurements from angles to degrees.

• Students should be able to find arc length given an angle measure, and viceversa.

• Students should be able to use values of sine and cosine in the first quadrantportion of the unit circle to fill in a table of values for y = cos(θ) and y = sin(θ)for special values of x, which may be outside the first quadrant portion of theunit circle.

• Students should be able to construct graphs of y = cos(θ) and y = sin(θ) inthe θy-plane.

• Students should be able to graph sinusoidal functions without a calculator.

• Students should be able to answer practical questions based on your knowledgeof the period, amplitude, midline, and horizontal shift of a sinusoidal function.

• Given information about one or more trigonometric functions, students shouldbe able to derive information about other trigonometric expressions by usingalternative expressions. For example, given a value of the sine function andthat the terminal point of the angle is in a given quadrant find values of all theother trigonometric functions.

• Students should know and be able to apply the Pythagorean Identity.

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• Students should be able to evaluate the inverse sine, cosine, and tangent func-tions at known outputs of the sine, cosine, and tangent, respectively.

• Students should be able to solve simple equations involving trigonometric func-tions as follows: Students should be able to find one solution using an inversetrigonometric function and then find additional solutions in any given intervalusing either a graph of a trigonometric function or a picture of the unit circle.

• Students should be able to derive trigonometric identities using known identi-ties.

• Students should be able to solve equations involving trigonometric functionson a given interval by applying trigonometric identities, using inverse trigono-metric functions, using a graph or the unit circle, and possibly using otheralgebraic manipulations.

• Students should be able to use sum and difference identities to find sines andcosines of sums and differences of special angles.

• Students should be able to write a linear combinations of sines and cosines,with the same periods, as a single sinusoidal function

• Students should be able to apply appropriate sun and difference identities tosolve equations involving sums or differences of sines and cosines and of angles.

Power, Polynomial, and Rational Functions

• Students should know and be able to use the following terminology: Power

Function, Proportionality, Inversely Proportional, Directly Proportional, Con-

stant of Proportionality

• Students should be able to find a formula for a power function given points onits graph.

• Students should be able to solve equations involving power functions.

• Students should know the general shape of the graph of a power function andhow its coefficient and exponent influence its features.

• Students should be able to sketch graphs of simple graphical transformationsof power functions.

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• Students should be able to solve word problems involving power functions.

• Students should know and be able to use the following terminology: Polyno-

mial, Degree, Coefficient, Leading Coefficient, Constant Coefficient, Standard

Form, Factored Form, Multiple Zero, Long-Run Behavior

• Students should be able to find a formula for a polynomial given sufficientinformation such as zeros and its y-intercept.

• Students should be able to use factoring to solve simple polynomial equations.

• Students should use standard and factored form of a polynomial appropriately.

• Students should be able to perform arithmetic operations with simple andcomplex fractions involving elementary functions.

• Students should be able to identify the horizontal asymptote of a rationalfunction if it exists.

• Students should be able to write sums, differences, products, and quotients ofrational functions as rational functions.

• Students should find horizontal asymptotes by multiplying and dividing byappropriate powers of the independent variable to arrive at an equivalent ex-pression whose long sun behavior is transparent.

• Students should be able to find a formula for a rational function from its graph,assuming the graph shows all relevant features.

• Students should be able to find the zeros, vertical asymptotes, and long runbehavior of a rational function.

• Students should be able to use the factored and standard forms of the nu-merator and denominator of a rational function to give a rough sketch of itsgraph.

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Math 221, Calculus I

Math 221 is a first semester course in single variable Calculus, addressing limits,the theory of differentiation, antiderivatives of elementary functions, and the funda-mental theorem of calculus. Students who successfully complete this course will beexpected to:

• Interpret limits graphically and describe limiting behavior verbally.

• Compute limits for functions with horizontal and vertical asymptotes, remov-able singularities, and regular points.

• Infer properties of f ′(x) given a graphical or verbal description of a function,f(x).

• Compute the derivative of power, exponential, trigonometric functions, theirinverses, as well as functions that are products, quotients, or compositions ofelementary functions.

• Apply theory and techniques of differentiation in the contexts of implicit dif-ferentiation, linear approximation, optimization, related rates and L’Hopital’srule.

• Approximate areas using Riemann sums, compute definite integrals of piecewiselinear (or circular) functions geometrically, and deduce properties of the definiteintegral from its geometric definition.

• Interpret accumulation functions F (x) =∫

x

0f(t) dt geometrically, and use

the fundamental theorem of calculus to connect accumulation functions as de-scribed above with antiderivates.

• Compute antiderivatives of power, exponential, trigonometric functions, andinverse trig functions, as well as antiderivatives that can be computed usingthe substitution method.

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Math 222, Calculus II

Overarching and general objections:

• Recognize when to use algebra and perform algebra correctly.

Mathematics is a cumulative subject. You must be correctly use the skills ac-quired in previous math classes. While Calculus is the main topic, you cannotcorrectly solve Calculus problems or understand Calculus concepts if you havelazy algebra habits.

• Understand and correctly use notation.

Mathematics is a very precise science. You must think clearly and preciselyabout Calculus and using correct notation will help you do this. Also, whilethe “final answer” is certainly important it is more important to be able tocommunicate your ideas. You must use correct notation to be able to do this.

• Computational and Procedural proficiency in all content objectives listed be-low.

This basically means you can correctly solve basic problems.

• Understand concepts underlying each of the content objectives listed below.

This means you understand why the problem solving techniques make sense.This is evaluated through non-standard and conceptual problems.

Particular content learning objectives

• Integration techniques including substitution, integration by parts, trigono-metric substitutions, and partial fractions. Ability to determine which methodworks and correctly carry out the procedure.

• Improper integral evaluation or determination of convergence. Correctly applyprocedure. Understand and correctly present mathematical argument concern-ing convergence.

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• Applications of definite integrals including: finding areas and volumes, work,pressure, probability. Problem solving. Understanding of general theme of“slicing”. Ability to solve both standard and non-standard problems.

• Sequences and series. Understanding of infinite series. Ability to correctlydevelop and present a rigorous demonstration of convergence or divergence.

• Applications of series: Taylor and Fourier series. Ability to calculate coeffi-cients and apply series to applications.

• Differential equations. Basic understanding of differential equations, solvingseparable differential equations. Application to contextual problem.

• Basic procedural mastery of ancillary topics: parametric equations, complexnumbers, polar coordinates.

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Math 223, Calculus III

Math 223 is a one semester course in Multivariable Calculus, addressing vectors andmultivariable functions, partial derivatives and gradients, double and triple integralsin various coordinate systems, vector fields, line integrals, flux, divergence and Stokes’Theorem. Students who successfully complete this course will be expected to:

• Use vectors as a tool for working with Rn, where n ≥ 3. In particular, stu-

dent are excepted to compute or apply vector addition, scaling, dot and crossproducts within the context of broader questions.

• Identify properties of a function f(x, y), (zeros, limiting behavior, periodicity,etc) that allow them to visualize the graph z = f(x, y). This also includessketching and/or interpreting traces and level sets.

• Compute partial derivatives and directional derivatives of elementary func-tions, and interpret the graphical meaning of partial derivatives and directionalderivatives at a point.

• Find and use the gradient of a function as applicable to tangent planes, normallines, and maximal directional derivatives.

• Approximate volumes bounded by a rectangle in the xy-plane and a surface z =f(x, y) using Riemann sums, and determine whether

∫∫Rf(x, y)dA is positive,

negative, or zero without computing the integral.

• Set up and evaluate double integrals over regions in the xy-plane boundedby curves y = f(x), switch the order of integration for double integrals, andconvert between rectangular and polar coordinates for double integrals.

• Set up, evaluate, and change coordinates for triple integrals in rectangular,cylindrical, and spherical coordinates.

• Identify how properties of a function ~F (x, y) affect the graph of a vector field;parameterize line segments, graphs of functions, and elliptical arcs within R

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and R3 and use parameterizations to compute line integrals.

• Apply the Fundamental Theorem of Line Integrals and Green’s theorem tocompute line integrals, and identify the appropriate times to use these theo-rems.

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• Compute the flux of a constant vector field through a flat surface withoutintegration, and use integration to compute flux of non-constant vector fieldsthrough surfaces z = f(x, y), cylinders, or spheres, and when applicable usethe divergence theorem to compute flux.

• Find the curl a vector field, and use Stokes’ Theorem to find the circulation ofa vector field around a simple closed curve in R

3.

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Math 301, Foundations for Advanced Mathematics

Student Learning Objectives: Students who successfully complete this coursewill be expected to:

• Demonstrate proficiency in using proper mathematical notation and terminol-ogy.

• Develop communication skills necessary for reading and writing mathematicalproofs.

• Understand the basic theory of sets, functions, relations, modular arithmeticand common sets of numbers.

• Understand formal mathematical logic and be able to read and write proofsusing standard methods such as direct, contradiction, and induction.

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Math 313, Combinatorics

Objectives:

• Students will be able to solve problems using various basic counting techniques,including

– the pigeonhole principle

– enumeration of sets

– permutations

– combinations

– partitions

– Binomial Theory

• Students will be able to recognize and analyze the underlying combinatorialand algebraic patterns present in secondary algebra (polynomials)

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Math 311, Matrices and Linear Algebra

Linear Equations, Matrices, and Linear Transformations

• Students will write systems of linear equations in matrix form.

• Students will use elementary row operations to solve systems of linear equa-tions.

• Students will determine whether systems of linear equations have unique solu-tions, infinitely many solutions, or no solutions.

• Students will understand existence and uniqueness of solutions to systems oflinear equations in terms of surjectivity and injectivity of linear transforma-tions.

• Students will understand linear independence and determine whether sets ofvectors are linearly independent.

• Students will know, understand, and apply the Invertible Matrix Theorem, andbe able to prove parts of it.

• Students will be able to use matrices and systems of linear equations in selectedapplications.

Bases, Dimension, and Change of Basis

• Students will know the definition of a basis and determine whether a given setof vectors forms a basis for a vector space.

• Students will be able to find bases of null and column spaces of matrices.

• Students will be able to find the matrix of a linear transformation relative toa given basis.

• Students will use representations of linear transormations relative to advanta-geous bases to solve applied problems.

Eigenvalues, Eigenvectors, and Diagonalization

• Students will know the definition of eigenvalues and eigenvectors, and be ableto find them using the characteristic polynomial.

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• Students will be able to determine if a given matrix is diagonalizable.

• Students will use diagonalization to solve applied problems.

Inner Product Spaces

• Students will know the definition of the inner product on Rn.

• Students will use the Gram-Schmidt process to find orthogonal bases.

• Students will solve least squares problems.

Additional Topics

Additional topics may include:

• Matrix factorizations.

• The Singular Value Decomposition.

• Classification of quadratic forms.

• Constrained optimization.

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Math 316, Number Theory

Learning objective are the following within the context of general topics listed at theend.

• Students should learn to read and understand simple proofs.

• Students should begin to learn to critique other’s (and their own) mathematicalreasoning

• Students should be able to reproduce simple proofs and modify them slightly.

• Students should be able to generate examples which demonstrate a concept -without being told explicitly what the example should be. These should leadto experimentation and conjecture.

• Students should be able to perform basic computations relevant to the specifictopic.

• Students should understand connections between topics.

Contextual specifics:

• Basic properties of integers including the Fundamental Theorem of Arithmetic.

• Divisibility, euclidean algorithm

• Modular arithmetic

• Linear congruences

• Systems of linear congruences and the Chinese Remainder Theorem

• Arithmetic functions and divisor sums

• quadratic congruences and quadratic reciprocity

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Math 325, History of Mathematics

Objectives: to provide mathematicians and teachers of mathematics with:

• Familiarity with the historical development of important mathematical con-cepts over time

• The ability to approach mathematical concepts from many different perspec-tives, including

– problem solving, as a basis for the initial development of many concepts;

– mathematics as a human endeavor, and the role of individuals with theirinsights and idiosyncrasies;

– the impact of social, economic, and cultural forces on mathematical studyand creativity;

– the broad shifts in the mathematical community on what problems aremost important and what a mathematical proof looks like;

– the gradual development of standard notations and terminology;

– the close interrelation of the development of various branches of mathe-matics;

• An appreciation of the dynamic nature of mathematics, including recent de-velopments in modern mathematics and its continuing evolution, as influencedby current technology and culture.

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Math 327, Differential Equations

1. Basic Concept

(a) Students will learn what a differential equation is.

(b) Students will learn what a solution of a differential equation is.

(c) Students will learn what the solution of an initial value problem is.

(d) Students will verify whether or not given functions are solutions of a givendifferential equation or initial value problem.

2. Solving First Order Differential Equations

(a) Students will differentiate between first order and higher order, betweenlinear and nonlinear, and between homogeneous and nonhomogeneous dif-ferential equations.

(b) Students will apply the Existence and Uniqueness Theorem for first orderlinear initial value problems and find largest domains on which it applies.

(c) Students will solve first order linear differential equations analytically us-ing integrating factors.

(d) Students will solve first order linear and nonlinear differential equationsanalytically using separation of variables.

(e) Students will use qualitative methods for constructing solutions, including:

• Sketching and interpreting direction fields and solution curves;

• Sketching and interpreting phase lines.

(f) Students will use numerical methods for constructing solutions, namelyEuler’s Method.

(g) Students will apply the techniques they have learned to model real-worldproblems such as population dynamics, mixing problems, radioactive de-cay, cooling problems, and others.

(h) Possible additional topics: Bernoulli equations; logistic growth model.

3. Solving First Order Linear Systems

(a) Students will learn the matrix theory necessary for solving systems ofdifferential equations.

(b) Students will apply the Existence and Uniqueness Theorem for first orderlinear systems and find largest domains on which it applies.

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(c) Students will rewrite higher order scalar equations as first order systems.

(d) Students will verify whether or not given vectors are solutions of a givensystem or initial value problem.

(e) Students will use the Wronskian to verify whether or not a solution set isa fundamental set of solutions. They will subsequently construct generalsolutions and solve initial value problems. In the process, they will explorethe idea of linearly independent solutions and its necessity in constructinggeneral solutions.

(f) Students will use the eigenvalue method to solve constant coefficient ho-mogeneous systems:

i. Students will find two distinct real eigenvalues and their correspond-ing eigenvectors to build a fundamental set of solutions.

ii. Students will find two distinct complex eigenvalues and their corre-sponding eigenvectors but then build a fundamental set of solutionsconsisting of real-valued solutions (using Euler’s formula).

iii. Students will find one real repeated eigenvalue and its correspondingeigenvector but then construct a second linearly independent solution,in order to build a fundamental set of solutions.

(g) Students will use qualitative methods for analyzing solutions, including:

• Constructing the phase plane as a vector field;

• Constructing solution trajectories on the phase plane as parametriccurves;

• Finding and interpreting horizontal and vertical nullclines;

• Finding and interpreting equilibrium solutions.

(h) Students will solve nonhomogeneous linear systems using the method ofundetermined coefficients.

(i) Possible additional topics: Matrix exponential, numerical methods.

4. Solving Second Order Linear Equations

(a) Drawing a parallel with first order systems, students will construct thegeneral solution of constant coefficient second order homogeneous equa-tions in the case of:

i. a characteristic equation with two distinct real roots.

ii. a characteristic equation with two distinct complex roots.

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iii. a characteristic equation with one real repeated root.

(b) Students will apply their solution techniques to model unforced mechani-cal vibrations. Students will interpret their solutions graphically (requir-ing the phase shift of sinusoidal functions), answer follow-up questions,and determine long-run behavior of solutions. Students will categorizeharmonic oscillators as undamped, underdamped, critically damped, oroverdamped.

(c) Students will use the method of undetermined coefficients to solve non-homogeneous second order linear equations.

(d) Students will apply their solution techniques to model forced mechani-cal vibrations. Students will interpret their solutions graphically, answerfollow-up questions, determine long-run behavior of solutions, and analyzeresonance.

(e) Possible additional topics: Variation of parameters; higher order equa-tions, coupled oscillators.

5. Laplace Transforms

(a) Students will compute Laplace transforms using the improper integraldefinition.

(b) Students will use the Heaviside step function in proving and applying thefirst and second shift theorems.

(c) Students will take Laplace transforms of derivatives. Students will trans-port initial value problems from the time domain to the transform domain,solve them, and then transport the solution back to the time domain.

(d) Students will use tables to find Laplace transforms and inverse Laplacetransforms, including the method of partial fractions.

(e) Students will compute Laplace transforms of periodic functions.

(f) Students will solve second order equations with the approach of systemtransfer functions.

(g) Students will convolve functions and compute the Laplace transform ofthe convolution of two functions.

(h) Students will apply convolutions to the study of the delta function andimpulse response.

(i) Possible additional topics: Solving systems with Laplace transforms.

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6. Possible additional topics: Nonlinear scalar equations; nonlinear systems; morenumerical methods; series solutions.

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Math 342, Modern College Geometry

Objectives: This course will provide future mathematicians and prospective sec-ondary mathematics teachers with:

• Further developed problem solving abilities

• The ability to build on Math 301 and

– clearly communicate mathematics

– make conjectures

– justify and prove results

in a geometric context.

• An understanding of the modern developments in geometry, including ideasfrom

– graph theory

– topology

– non-Euclidean geometries

• An ability to to connect geometry with other areas of mathematics, such as

– Abstract Algebra (group theory and symmetries)

– Number Theory (seeing divisibility patterns geometrically)

– Combinatorics (Graph theory and counting patterns in topology)

– Complex Numbers (in hyperbolic geometry)

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Math 424, Advanced Calculus

It is a basic premise in 400 level math courses that understanding a topic meansthat one can construct proofs about the objects being studied. A student must knowdefinitions, be able to construct examples (with proofs that they are examples), readand understand proofs of theroems, and apply theory in situations that are new tothe student.

Real Numbers

• Students will understand the role of the Completeness Axiom in establishingfundamanetal properties of the real numbers.

• Students will know and use terminology regarding basic topology of the realline.

• Students will prove facts about real numbers using suprema, infima, and ored-ered field properties.

Limits and Continuity

• Students will know and understand the ǫ-δ definition of a limit of a function.

• Students will know and understand the ǫ-δ definition of continuity of a function.

• Students will know the definition of uniform continuity of a function and beable to compare and contrast it with the definition of pointwise continuity.

• Students will know the statements and proofs of the fundamental theoremsregarding limits and continuous functions.

• Students will know how the fundamental theroems regarding limits and con-tinuous functions are applied in a standard Calculus course.

Differentiable Functions

• Students will know and understand the limit definition of the derivative.

• Students will know the statements and proofs of the fundamental theoremsregarding differentiable functions.

• Students will know how the fundamental theroems regarding differentiablefunctions are applied in a standard Calculus course.

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The Riemann Integral

• Students will know and understand the definition of the definite Riemann in-tegral.

• Students will know the statements and proofs of the fundamental theoremsabout the definite Riemann integral.

• Students will know how the fundamental theorems about the definite Riemannintegral are applied in a standard Calculus course.

Additional Topics

Additional topics may include:

• Sequences of numbers and functions, pointwise and uniform convergence.

• Power series representations of functions.

• The precise definition of elementary transcendental functions.

• The Lebesgue criterion for the existence of the definite Reimann integral.

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Math 430, Complex Variables

1. Students will learn the arithmetic of complex numbers in terms of their fieldproperties. Students will learn the geometry of complex numbers in terms ofmodulus and argument. Students’ main task at this level is to learn threerepresentations of complex numbers:

• an indivisible point z,

• rectangular form a+ bi = Re z + i Im z,

• polar form reiθ = |z|ei arg z,

and to know which type of arguments require (or lend themselves most readilyto) which form. Students will compute powers and roots of complex numbers,developing both their algebraic and geometric intuition.

2. Students will grapple with the “branch cut issue” that first arises with thedefinition of a single-valued argument and reappears with the definition of theprincipal value of the logarithm.

3. Functions of a complex variable. Students will interpret functions as map-ping the complex plane to itself. They will determine images under variousmaps both graphically and using set-builder notation. They will learn limitsand continuity in the complex plane.

4. Complex differentiation. Students will carefully explore the notion of an-alyticity. They will compute derivatives from the definition, they will showfunctions to be not differentiable or not analytic using the definition, and theywill interpret the derivative geometrically. Students will prove the necessityand (“almost”) sufficiency of the Cauchy-Riemann equations. Students willapply the Cauchy-Riemann equations to reach various conclusions, such asfunctions being constant on a domain. Specific analytic functions, whose prop-erties students will study in detail, will be polynomials and rational functions,exponential, trigonometric, and hyperbolic functions, and the logarithmic func-tion. Students will determine domains of analyticity for logarithmic and powerfunctions, including “branch picking.”

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5. Complex integration. Students will parametrize contours and compute con-tour integrals. They will interpret the contour integral using parametrizedcurves and show its independence of (admissible) parametrization. Studentswill prove and apply a result on bounding contour integrals of bounded func-tions. Students will prove and apply every direction of the result that a contin-uous function in a domain has an antiderivative throughout the domain if andonly if all its loop integrals in the domain vanish if and only if its integrals arepath-independent throughout the domain. Students will construct counterex-amples if a hypothesis is not met. Students will interpret contour integrals interms of line integrals and use Green’s Theorem to prove Cauchy’s IntegralTheorem, connecting the notion of analyticity to antiderivatives. They willuse this theorem to simplify the computation of integrals. Students will proveCauchy’s Integral Formula, and its generalized version, use it to prove thatanalytic functions have analytic derivatives, and apply it in the computationof contour integrals. Finally, students will prove Liouville’s Theorem in orderto prove the Fundamental Theorem of Algebra.

6. Possible additional topics include the stereographic projection, series and residues,and conformal mapping.

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Math 487, Algebraic Structures

Math 487 is a one semester course in Abstract Algebra, addressing the fundamentalsof group theory, and providing an introduction to the theory of rings and fields.Students who successfully complete this course will be expected to:

• Build a catalog of examples of groups displaying a wide variety of properties:finite versus infinite, cyclic versus more than one generator, abelian versus nonabelian, permutation groups, and their subgroups; for each group within thiscatalog, students should be able to compute, describe structures, prove groupproperties, and apply these examples in other contexts.

• Prove statements where an axiomatic definition appears as either a hypothesisor conclusion of an implication.

• Communicate mathematics clearly using the appropriate terminology, nota-tions and definitions, and use definitions to determine the structure of proofs.

• Describe the concept of isomorphism and properties of isomorphic groups, pro-vide examples of isomorphic groups, and prove that two groups are isomorphic.

• Find and describe the cosets of a subgroup within a group, prove properties ofcosets, and use properties of cosets to understand Lagrange’s Theorem; applyLagrange’s theorem to describe how or which subgroups sit within a group.

• Identify normal subgroups, use properties of normal subgroups to determinefactor groups.

• Describe the concept of homomorphism, prover properties of homomorphismsand the subgroups determined by a homomorphism, including applications ofthe First Isomorphism theorem to classify homomorphism.

• Define and recognize basic properties of rings, integral domains and ideals,and use them in the context of specific examples like polynomials, Gaussianintegers, Z[

√2], or similar.

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