This course covers the fundamentals of mathematical analysis: construction of real numbers, the topology of reals, the convergence of sequences and series, limits, continuity, compactness, differentiability and mean value theorem, and Taylor’s theorem. It shows the utility of abstract concepts and teaches an understanding and construction of proofs.
Class Schedule: Tu, Thu 11:30-12:45 (Acad Block A7)
Office Hours: Mon, Wed 3:30-04:00, Tu, Thu 13:00-15:00 (SSE- MATH 9B30)
TAs Office Hours: Check LMS (https://lms.lums.edu.pk/)
Weekly Breakdown:
[Week 1] Sets, Properties of sets, Relations, and their types, Equivalence relation, and partitions.
[Week 2] Algebra on Z_n, Functions, and their types.
[Week 3] Mathematical induction, Finite, and Infinite sets.
[Week 4] Countable sets, Cantor's theorem, Real numbers system, Completeness of R.
[Week 5] Sequences, Basic properties, Monotone sequences.
[Week 6] Bounded sequences, Subsequences, Bolzano Weirstrass Theorem.
[Week 7] Nested Intervals Theorem, Cauchy sequences, Completeness Axioms.
[Week 8] Series, Partial sums, Cauchy criteria, Convergence Tests, Absolute and conditional convergence.
[Week 9] Cluster points, Limit of a function at a cluster point, Limit Laws, Sequential criteria for limits.
[Week 10] Continuous functions, Sequential criteria, Basic laws of continuity.
[Week 11] Composition of continuous functions, Continuous functions on compact intervals, Extreme values theorem.
[Week 12] Derivative of a function at a point, Laws of differentiation, Caratheodory's theorem.
[Week 13] Critical points, First derivative test, monotonicity and derivatives, monotone inverse theorem.
[Week 14] Mean value theorem and applications, L'Hopital Rule and Taylor's theorem.
Final Exam (06-05-24) [Syllabus: Week 5- Week 14]