Course syllabus - Single Variable Calculus

Objectives

The purpose of the course is to give the student the opportunity to acquire basic knowledge about real-valued functions of a real variable and its applications, and to give the student a basis for further studies in mathematics and its applications in science, technology and economics.

Learning outcomes

At the end of a passed course, the student is expected to be able to
1. analyse real-valued functions of a real variable using the concepts domain, range, graph, composition and inverse
2. explain the concept of convergence of sequences and series, and be able to determine limit values ​​for functions and sequences based on standard limit values ​​and counting rules;  in particular, determining and applying the continuity of a function.
3. determine the derivative of a function based on the definition of the derivative, and at a given point be able to determine the tangent line to a graph. For differentiable functions, special emphasis is placed on a derivation technique based on standard derivatives and derivative rules, the use of which should be explicitly acknowledged
4. prove the mean value theorem, and be able to apply this theorem, and the content of paragraphs 1-3, to problems such as estimations and error estimates of function values, determination of extreme values, optimisation and curve sketching
5. give an account of one of the equivalent definitions of the Riemann integral and be able to prove and apply the fundamental theorem of calculus
6. apply techniques such as partial integration, partial fraction decomposition where the zeros  in the denominator have multiplicity one, and variable substitution, in order to determine primitive functions and integrals
7. determine the convergence of generalised integrals and be able to determine the value of those that are convergent
8. apply the integral concept for calculating areas between curves and volumes of solids with known cross section
9. solve separate and/or linear first order ordinary differential equations (ODE), and second order linear ODE with constant coefficients
10. apply Taylor's formula to approximate functions

Course content

- Functions: domain, range, graph, composition, invertibility, standard function, elementary function
- Sequences and series: sequence, series, geometric series as an example of convergent series
- Limits: definition of limits, limit laws, standard limits, continuity, the intermediate value theorem
- Derivatives: definition, standard derivative, derivative rules, implicit differentiation
- The mean value theorem: function value estimation, error estimation, classification of stationary points, singular points and endpoints, extreme value, optimisation, curve sketching
- Integrals: Riemann sums, the definition of the Riemann integral, the mean value theorem for integrals, primitive function, the fundamental theorem of calculus, generalised integral
- Integration techniques: integration rules, integration by parts, partial fraction decomposition, variable substitution
- Applications of the integral concept: area between curves and volume of a solid with known cross section
- Ordinary differential equations: separable differential equations, homogeneous and non-homogeneous first-order linear differential equations, second-order linear differential equations with constant coefficients
- Taylor's formula: Taylor's formula, Taylor polynomial, Lagrange's residual term, Maclaurin series for standard functions, interval of convergence
- Applications of Taylor's formula: approximation of function

Requirements

Mathematics D or Mathematics 4.

Examination

INL1, Assignments, 1.5 credits, written assignments concerning learning outcomes 1-10, grades Fail (U) or Pass (G).
TEN1, Examination, 2.5 credits, written and/or oral examination concerning learning outcomes 1-10, grades Fail (U), 3, 4 or 5.
TEN2, Examination, 3.5 credits, written and/or oral examination concerning learning outcomes 1-10, grades Fail (U), 3, 4 or 5.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3