Calculus I

Objectives:
The course aims at giving an introduction to the most important notions and techniques in mathematical calculus, especially continuity, differentiation and integration, which are needed later in most studies in mathematics and natural sciences. At the same time, the course shall convey how the subject is logically build up and why one needs strict proofs and give insight into how one uses mathematics to depict (models of) the real world.

Contents:
The subject gives an introduction to the concept of limits, continuity, differentiation and integration of real functions of one

real variable, as well as theory of real and complex numbers, with applications to theoretical and practical problems. Central themes are inverse functions, logarithmic and exponential functions, trigonometric functions, Taylor polynomials and Taylor's formula with remainder. Moreover, topics such as implicit differentiation, fixed point iteration and Newton's method, computations of areas in the plane and volumes of solids of revolution, numerical integration and separable and first order linear differential equations are included.

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• is able to compute with and use complex numbers to find real and complex solutions to simple equations
• is able to prove statements using mathematical inductio
• is able to state and apply the mathematical definitions of limits, continuity and derivative, also in theoretical problems.
• is able to state and apply both the formal definition and other techinques such as the limit theorems, squeeze theorem and l'Hôpital's rule to compute limits
• is able to state and apply the Intermediate Value Theorem, the Extremal Value Theorem and the Mean Value Theorem, also in theoretical problems.
• is able to use rules to find derivatives and antiderivatives
• is able to study functions and sketch their graphs
• is able to apply Taylor's formula
• is able to use integration techniques such as substitution, partial integration, as well as polynomial division, method of partial fractions and completing the square, to find antiderivatives.
• is able to apply the Fundamental theorem of Calculus
• is able to solve simple separable and first order linear differential equations
• is able to model simple problems with the help of differential equations and use implicit differentiation and functions to solve simple applied problems
• is able to use numerical methods to find approximative values for roots of equations and definite integrals.

Skills:

The students

• masters fundamental techniques within calculus and how to use these in both theoretical and applied problems
• is able to argue mathematically and present simple proofs and reasoning
• is able to recognize structure and formulate simple problems mathematically

General competence

The student

• is able to work individually and in groups
• is able to formulate in a precise and scientific way on an elementary level
• is able to decide whether simple mathematical arguments are correct