Upon completion of this course the student is able to:
- give and work with precise definitions of some fundamental mathematical concepts and write a coherent and understandable mathematical proof related to these concepts. In particular, the student is able to
- use appropriate techniques such as the method of induction and proof by contradiction to prove mathematical statements,
- explain and use the concepts from the real number system, such as infimum/supremum of a set, (un)countable sets and open/closed intervals,
- explain and use the concepts related to the limit of a sequence of real numbers,
- explain and use the concepts related to real-valued functions (of one variable), such as bijective and inverse functions, and limits of functions,
- explain the concepts of point-wise and uniform continuity and use associated important results such as the intermediate-value and extreme-value theorems,
- explain the concept of differentiability of real-valued functions (of one variable) and use relevant associated results such as the mean-value theorem and Taylor’s theorem,
- properly apply the definitions and important theorems to perform explicit calculation. In particular, the student is able to
- work with the simple properties of the complex numbers, the trigonometric functions, the exponential function and their inverses,
- work with the properties of limits of functions and real-number sequences,
- work with the rules of continuity and differentiability of functions, such as the chain rule, the product rule, Taylor series, L’Hôpital rule, and the extreme value theorem, and apply them to modeling problems.
Within the Analysis-line of Applied Mathematics, one learns the fundamental concepts in Real Analysis that are indispensable for the further study. Another integral part of the Analysis-line is the proper and rigorous manner of proving results. In this first course, the following topics will be treated:
- The real number system and functions
Set theory and logic, Real numbers and axioms, Absolute value and inequalities, Mathematical induction, Definition of a function, Countable and uncountable sets, Complex numbers (roots of polynomials and hyperbolic functions), Polynomials (fundamental theorem of algebra), trigonometric functions, exponential and logarithm, inverse functions and images.
- Sequences in ℝ
Limits of sequences, Limit theorems, Bolzano–Weierstrass theorem, Cauchy sequences, Limits supremum and infimum.
- Continuity of functions on ℝ
Two-sided limits, One-Sided limits and limits at infinity, Continuity, Uniform continuity.
- Differentiability of functions on ℝ
The derivative, Differentiability theorems, The mean value theorem, Taylor’s theorem and L’Hôpital rule, Inverse function theorems.
|Bachelor Applied Mathematics||Verplicht materiaal|
|Calculus: A Complete Course, R.A. Adams & C. Essex, 10th edition. ISBN: 9780135732588|
|An Introduction to Analysis, William Wade, 4th Edition, ISBN: 9781292357874|
|Mathematical Proofs; A Transition to Advanced Mathematics, van Gary Chartrand; Albert D. Polimeni; Ping Zhang, 3rd edition. ISBN:9781292040646|
|Zelfstudie geen begeleiding|
|Written exam: Abstract|