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 induction

* 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

Written examination: 5 hours

Examination support materials: Non- programmable calculator, according to model listed in faculty regulations and Textbook