Introductory Calculus (5cr)

Objectives

After this course the students can use the basic concepts in set theory, union, intersection, complement and difference, in presenting subsets of real numbers. The students can sketch graphs of basic functions and their compositions, calculate their limits and derivatives, and draw conclusions on function behaviour and extremal values based on them. The students can present complex numbers in both coordinate form and polar form and apply both forms in elementary calculations, find roots of complex numbers and factorise polynomials. The students can calculate integrals of basic functions. They can deliver both oral and written presentations of their solutions.

Content

Core contentSet union, intersection, difference and complement. Introduction of the logic and proofs needed in mathematical analysis.Definition of a function. Monotonicity of a function. Inverse function and combined function. Properties of basic functions. Hyperbolic functions and their inverses.Complex numbers and their basic properties (sum, difference, product, quotient, conjugate and modulus), presenting and calculating complex numbers both in coordinate form and polar form, complex roots.Limit and continuity of a fuction. One-sided and inproper limits, l'Hopital's rule.Derivative as limit of difference quotient. Derivating basic functions, products and quotients, chain rule. Studying the values and extrema of a function based on derivatives..Basics of integral calculus.Partial derivatives and gradient of a multivariable function.Complementary knowledgePreimage, injection, surjection, bijection.Roots of real valued polynomials, factorisation.Intermediate value theorem. Continuity of inverse function. Derivative of inverse function. Applications, for example linear approximations.Specialist knowledgeMean value theorem.

Prerequisites

This course assumes the students to have the mathematical knowledge on the level of Finnish high school mathematics (advanced syllabus).

Further information

This course belongs to the SEFI 1 level of engineering mathematics.