MATH1010: Calculus and Linear Algebra
                                                  Unit Outline, First Semester, 2009
                                  Textbooks:
                                     • M.D. Weir, J. Hass and F.R. Giordano, Thomas’ Calculus, 11th
                                        edition, Addison Wesley
                                     • R.McFeatR.,M.FisherandA.Niemeyer,First-Year Linear Algebra
                                        Notes: 2008 Edition, School of Mathematics & Statistics, UWA.
                               Calculus:
                                     • Logic and induction (4 lectures)
                                        Mathematical statements, connectives and quantifiers. Mathemati-
                                        cal reasoning: direct and indirect proofs. Principle of Mathematical
                                        Induction.
                                     • Real numbers, inequalities (1 lecture)
                                        Basic properties of R as an ‘ordered field’. Inequalities involving real
                                        numbers. Absolute value and its properties.
                                     • Sups and infs, sequences (4 lectures)
                                        Completeness of the real line. Upper and lower bounds of sets of
                                        real numbers, max and min, sup and inf. Sequences, convergence
                                        and divergence of sequences. Limit laws. Squeeze Theorem. In-
                                        creasing/decreasing sequences. Monotone Sequences Theorem. Re-
                                        cursively defined sequences. Infinite limits.
                                     • Limits of functions, continuity (5 lectures)
                                        Functions and their composition, increasing/decreasing functions.
                                        Limits (one-sided and otherwise) and limit laws.            Squeeze Theo-
                                        rem. Infinite limits. Continuity and discontinuity. Continuity of
                                        sums, products, quotients and composite functions. Inertia Prop-
                                        erty.  Continuous images of convergent sequences. Infinite limits.
                                        Intermediate Value Theorem (proof by the Bisection Method).
                                     • Differentiation ( 5 lectures)
                                        Derivatives (one-sided and otherwise) and rules for differentiation.
                                        The Chain Rule and the use of L’Hospital’s rule. Differentiablity
                                        implies continuity. Geometrical meaning of derivatives via tangent
                                        lines. Critical points and inflection points, and their relevance. Mean
                                        Value Theorem and applications. Implicit differentiation, related
                                        rates and applications.
                                     • Integration (7 lectures)
                                        Riemann sums and approximation of areas between curves. Defini-
                                        tion of the Riemann integral. Integrals and their rules. Antideriva-
                                        tives and the Fundamental Theorem of Calculus. Applications of
                                        Riemannsumsandintegralstoareas, volumes, work done by a force,
                                        arc length. Indefinite integrals.
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                Linear Algebra:
                    • Systems of linear equations and matrices (6 Lectures)
                     Gaussian elimination, consistent systems of linear equations, matri-
                     ces and matrix operations, algebraic properties of matrix operations,
                     matrix inverses.
                    • Vector geometry (5 Lectures)
                     Vectors in space, parallel and coplanar vectors, lines and planes, dot
                     products, cross products, review of vectors in the plane.
                    • Euclidean n-space (11 Lectures)
                     Vectors in Rn, subspaces and spanning sets, nullspaces, linear in-
                     dependence, basis and dimension, row and column space, rank of a
                     matrix, rank-nullity Theorem for matrices.
                    • Determinants and eigenvalues (4 Lectures)
                     Definition of eigenvalues and eigenvectors of an n×n matrix. Defini-
                     tion and properties of determinants, and methods of calculation. Use
                     of determinants in calculating eigenvalues and eigenvectors. Defini-
                     tion of eigenspaces.