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1. Demonstrate understanding of concepts in linear algebra, relating to vector spaces, linear transformations and symmetric matrices.
2. Demonstrate understanding of concepts in calculus, relating to differential equations, limits and continuity, and Taylor series.
3. Demonstrate understanding of introductory concepts in two-variable calculus.
4. Employ methods related to these concepts in a variety of applications.
5. Apply logical thinking to problem-solving in context.
6. Demonstrate an understanding of the role of proof in mathematics.
7. Use appropriate technology to aid problem-solving.
8. Demonstrate skills in writing mathematics.

1. Poole: Linear Algebra a Modern Introduction 4th ed. (Cengage Learning)
2. Stewart: Calculus 7th ed. (international ed.) (Brooks/Cole)

This course also makes use of online assessment software for mathematics called Maple TA, which we use to provide students with instantaneous formative feedback.

Mathematics IB Summer is taught double-time compared to Semester 1 or 2. The two topics of algebra and calculus detailed below are taught in parallel, with four lectures a week on each. There are two tutorials per week, one on algebra and one on calculus.

Lecture Outline

Algebra

• Further Results on R^n (7 lectures)
  • Revision of subspaces, linear independence, basis, dimension.
  • Row and column space, null space of a matrix. Rank and rank theorem.
  • Scalar product, distance. Length and angle. Orthogonality, Gram-Schmidt process.
• Linear Transformations (4 lectures)
  • Kernel and range, the matrix of a linear transformation.
  • Dimension theorem.
• Symmetric Matrices (4 lectures)
  • General quadratic equation in 2 variables, conics.
  • Revision of eigenvalues, eigenvectors and diagonalization.
  • Orthogonal diagonalization of real symmetric matrix. Applications.

• Differential Equations (5 lectures)
  • First order DE's: separable, linear.
  • Linear second order DE's with constant coefficients. Applications.
  • Modelling. The logistic equation.
• Limits (5 lectures)
  • Definition and uniquess of limit, limit laws.
  • Squeeze Theorem, trigonometric limits, one-sided limits, limits at infinity, unbounded functions.
  • Improper integrals. Linear approximation. L'Hopital's rule.
• Continuity (2 lectures)
  • Continuity, classification of discontinuities, continuity and differentiation.
  • Intermediate Value Theorem. Newton's Method.
• Applications of the Derivative (5 lectures)
  • Extrema of continuous functions, applied max-min problems.
  • Rolle's Theorem, Mean Value Theorem and consequences.
  • Graphing: First and second derivative tests, concavity and inflection points.
• Taylor Series (6 lectures)
  • Taylor and Maclaurin polynomials, Taylor's Theorem, error terms.
  • Power series, geometric series, convergence of power series, interval and radius of convergence.
  • Taylor and Maclaurin series, binomial series, differentiation and integration of power series.
• Calculus of More than One Variable (8 lectures)
  • Surfaces: planes, cylinders, quadric surfaces.
  • Functions of more than one variable, limits and continuity.
  • Partial derivatives, derivatives of higher order. Chain rules, directional derivative, gradient.
  • Tangent planes, local maxima and minima. Second derivative test for functions of 2 variables.

Tutorial 1: Algebra: Subspaces, linear independence. Row, column and null space. Calculus: First order DEs. Second order DEs.

Tutorial 2: Algebra: Rank theorem. Distance, angle and orthogonality. Orthogonality. Gram-Schmidt process, projections, linear transformations. Calculus: Logistic equation. Limits, Squeeze Theorem.

Tutorial 3: Algebra: Linear transformations: kernel, range, standard matrix. Composition of linear transformations, conic sections. Calculus: Limits, improper integrals. L'Hopital's rule. Discontinuities.

Tutorial 4: Algebra: Eigenvalues, eigenvectors, diagonalisation. Standard form for conics and quadrics. Calculus: Newton's Method. Intermediate Value Theorem. Max-min problems. Mean Value Theorem.

Tutorial 5: Functions of 2 variables, limits and continuity. Limits, continuity, partial derivatives of functions of 2 variables. Mean Value Theorems, graph sketching. Taylor and Maclaurin polynomials.

Tutorial 6: Tangent planes to surfaces. Chain rule. Directional derivatives. Maximum rate of change. Classification of critical points. Series, convergence. Taylor and Maclaurin series.

Note: precise tutorial content may vary due to the vagaries of public holidays.

Assignments 1-6 are made up of a written component, worth a total of 5%, and an online (Maple TA) component, worth 10%.

1. All written assignments are to be submitted at the designated time and place with a signed cover sheet attached.
2. Late assignments will not be accepted without a medical certificate.
3. Written assignments will have a one week turn-around time for feedback to students.
4. Online Maple TA assignments provide instantaneous feedback to students.