Prerequisite Course(s):
A grade of "C" or better in MTH 252 or equivalent
Required Text(s):
Stewart, Calculus: Concepts and Contexts, 3^{rd} Edition, 2005, Thomson
TenEyck and Ferguson, Linear Algebra Packet
Other Requirements:
Graphing calculator . TI-86, TI-89, or TI-92 recommended
Course Description:
Combines topics from linear algebra and infinite series. Includes geometric,
Taylor and
Fourier Series work with applications; and systems applications using
matrices and
determinants.
Performance Based Learner Outcomes:
Upon successful completion of the course, students should be able to:
1. Create mathematical models of abstract and real world situations using
geometric,
power , Taylor and Fourier series, and situations involving linear
systems using matrices
and determinants.
2. Use inductive reasoning to develop mathematical conjectures involving
infinite series
models and linear systems modeled with matrices. Use deductive
reasoning to verify
and apply mathematical arguments involving these models.
3. Use mathematical problem solving techniques involving infinite series and
linear
systems using matrices.
4. Make mathematical connections and solve problems from other disciplines
involving
infinite series and linear systems using matrices.
5. Use oral and written skills to individually and collaboratively communicate
about
infinite series and their behavior, and about linear systems using
matrices and
determinants.
6. Use appropriate technology to enhance mathematical
thinking and understanding, to
solve mathematical problems involving infinite
series models and to solve problems
involving linear systems using matrices and
determinants.
7. Do project that encourage independent, nontrivial exploration of infinite
series and
linear systems application and models.
Course Content:
I. Number Sense
A. Determine subsequent values of a given sequence according to apparent pattern
B. Write symbolic model (general nth term) of series given numeric data
C. Predict convergence of series given numeric data
D. Evaluate determinants of matrices
II. Symbolism
A. Given the general term of sequence or series, write numeric list or sum of
series
B. Evaluate sums of series symbolically
C. Test for convergence of series symbolically
D. Write the Taylor series approximations for given functions
E. Given symbolic form of a square wave , write its Fourier series approximation
F. Use Taylor series approximations of functions in differentiation,
integration, and
evaluation of limits of function
G. Apply the algebra of matrices , including scalar and matrix multiplication ,
matrix
addition, transposition, evaluation of determinants and determination of eigenvalues
and eigenvectors
III. Problem Solving
A. Use graph analysis to predict convergence of series
B. Solve problems modeled by geometric series
C. Use graph analysis to solve linear systems of equations in R ^{2}
D. Use
inductive and deductive reasoning to generalize solutions in R^{2} to linear
systems
in R^{n}
E. Create graphical interpretation of solution of linear systems in R^{3}
F. Solve
a variety of problems in linear algebra including linear programming,
difference
equations and Markov chain applications
IV. Technology
A. Use calculator or CAS programs to sum series
B. Use calculator or CAS to predict convergence of sequences and series
C. Use CAS to evaluate and graph Fourier approximation of a wave
D. Solve equations using appropriate technology
E. Use graphing calculator or CAS to determine the solutions of systems of
linear
equations, and interpret and investigate the characteristics of
solutions, individually
and in groups .
V. Communication
A. Express interpretations of local and global behavior of
a series in written and oral
form
B. Express interpretations of solutions of linear systems in written and oral
form
C. Explain the reasoning used to arrive at a mathematical conclusion
D. Read, write, hear and speak mathematical ideas, individually and in groups
Course Content Outline:
I. Infinite Series
A. Sequence and series patterns
B. Tests for Convergence (including n^{th} term, ratio, comparison, and integral)
C. Geometric Series
D. Binomial Series
E. Taylor Series
F. Fourier Series
G. Applications of infinite series
H. Error analysis
II. Systems of Linear Equations
A. Row Reduction and Echelon Forms
B. Vector Equations
C. Matrix Equations
D. Solution Sets
E. Linear Independence
F. Linear Transformations
III. Matrix Algebra
A. Matrix Operations
B. Inverses and Their Characteristics
C. Subspaces of R^{n}
IV. Determinants
V. Vector Spaces
A. Dimensions
B. Nullspace
C. Bases
D. Coordinate Systems
E. Rank
VI. Eigenvalues and Eigenvectors
A. Characteristic Equation
B. Diagonalization
C. Linear Transformations