Foundational Calculus 1

Basic properties of real numbers and real functions of a real variable; limits, continuity, differentiation and integration. The fundamental theorem of calculus and its applications are central. There is an emphasis on rigour.

1. Knowledge: The student is familiar with central concepts of real analysis, including convergence; properties of the real numbers and of sequences and continuous, differentiable and integrable functions; linearization; the fundamental theorem of calculus. Moreover, the student is familiar with numerical methods for integration and equation solving. The student has more detailed knowledge of the properties of special functions such as polynomials, exponential functions, trigonometric functions and their inverses.

2. Skills: The student is able to apply techniques of integration and derivation in mathematical modeling, to derive simple mathematical results and to analyze functions. The student is able to set up and analyze simple mathematical models that require elementary optimization. The student is able to choose and implement suitable numeral methods for problems involving integration and equation solving, and to estimate the accuracy of the chosen method. Moreover, the student is able to read and write rigorous mathematical proofs related to the content of the course, including proofs based on induction.

Lectures and compulsory exercises. Parts of this course may be taught in English.