Course syllabus - Single Variable Calculus

Objectives

The course aims at giving a fundamental knowledge about real-valued functions of one real variable and applications thereof, and to give a basis for further studies in mathematics and applications thereof in natural science, technology and economy.
 

Learning outcomes

At the end of a passed course, the student is expected to be able to
- analyse real-valued functions of one real variable based on the concepts domain, range, graph, composition and inverse.
- determine convergence for sequences and series, and also from standard limits and arithmetic rules being able to find limits for functions and sequences. Especially, the continuity of a function should be able to be determined.
- find the derivative of a function based on the definition, and also at a given point be able to find the tangent of a graph. For differentiable functions, a special importance is put on a differentiation technique based on arithmetic rules and standard derivatives, where the arithmetic rules are supposed to be able to be shown.
- prove the mean value theorem, and also be able to apply it and the contents of the points 1-3 on problems which includes estimates and error estimates of function values, finding of extreme values, optimization, curve sketching, and related rates.
- identify limits of Riemann sums as integrals, and also be able to prove and apply the fundamental theorem of calculus.
- apply techniques as partial integration, division in partial fractions, and substitution, with the purpose of being able to find primitive functions and integrals.
- determine convergence of generalized integrals, and be able to calculate those which are convergent.
- apply the integral concept for finding areas between curves, arc lengths, areas of rotation surfaces, and volumes of bodies with known section areas.
- solve first order separable and/or linear, ordinary differential equations (ODE), and also second order linear ODE with constant coefficients.
- apply Taylor’s theorem for approximating functions, determining types of stationary points, and finding limits.

Course content

- Functions: domain, image, graph, composition, invertibility, standard function, elementary function.
- Number sequences and series: number sequence, series, geometrical series, integral test and comparison tests for positive series, ratio and root tests, absolute convergence and conditional convergence, The alternating series test, radius of convergence for a power series.
- Limits: definition, arithmetic rules, standard limits, continuity. Derivatives: definition, standard derivatives, arithmetic rules, implicit derivative, related rates.
- Mean value theorem: approximation of a function value, error estimate, extreme value, optimization, curve sketching.
- Integrals: Riemann sums, the definition of the Riemann integral, mean value theorem for integrals, primitive function, the fundamental theorem of calculus, generalized integrals. Techniques of integration: arithmetic rules, partial integration, partial fraction decomposition, method of substitution, inverse substitution (i.e. trigonometric).
- Applications of the concept of integral: area between curves, arc length, area of surface of revolution, volume of a body with known section areas.
- Ordinary differential equations: separable differential equations, homogeneous and non- homogeneous linear differential equations of first order, second order linear differential equations with constant coefficients.
- Taylor’s formel: Taylor’s formula, Taylor polynomials, Lagrange remainder, small and big-O notation, Maclaurin series for standard functions, convergence interval.
- Applications of Taylor’s formula: approximation of functions, classifying of stationary points, finding limits, l’Hospital’s rule.

Tuition

Teaching is given in the form of lectures and classes.
 

Specific requirements

Mathematics 4 or Basic Course in Mathematics
 

Examination

Exercise (INL1), 1,5 credits, marks Pass (G)
Examination (TEN1), 2,5 credits, marks 3, 4 or 5
Examination (TEN2), 3,5 credits, marks 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

Pass with distinction, Pass with credit, Pass, Fail