Calculus and Analysis I

Description

Complex numbers. Real functions of one variable. Topics that will be treated are: Preliminaries, Real functions, Trigonometric functions Limits, Continuity, Intermediate Value Theorem Transcendental Functions, Inverse Functions Differentiation, Chain Rule, Mean Value Theorem Applications of Differentiation, Extreme Values, l'Hôpital's Rule, Taylor Polynomial Integration, Fundamental Theorem of Calculus, Improper Integrals Real functions of several variables. Topics that will be treated are: partial derivatives, chain rule, gradients extreme values, Lagrange multiplier method double integrals, polar coordinates Mathematical reasoning. Topics that will be treated are: Logical statements: conjunction, disjunction, negation, implication; Predicates and (nested) quantifiers; Proof methods: direct, contrapositive, contradiction, case distinction; Proof by induction. Course Objective: The goal of this course is twofold: to master the calculus of functions of one and more variables, to recognize and apply the structure of mathematical reasoning and logic. More specifically, at the end of the course the student will be able to: calculate with complex numbers; calculate limits, derivatives and integrals; determine the existence and uniqueness of solutions to equations; find local and global extreme values, with and without constraints; approximate functions with a given error using the Taylor polynomial; formulate and apply theorems for continuous and/or differentiable functions; manipulate logical statements and quantifiers; perform simple proofs using a variety of methods including mathematical induction. Assessment Information: There will be a midterm exam at the end of period 1 and a final exam at the end of period 2. Details about the topics treated in each exam and the calculation of the final grade will be published in Canvas. If your grade is not sufficient, it is possible to make the resit about all topics in the spring semester. Teaching Methods: Lectures (2x2 hours per week) and tutorials (1x2 hours per week).