Introduction to Mathematics:
The student is able to:
Clearly express formulations
- Work with elementary properties of sets and logic
- Construct elementary proofs using basic techniques
- Work with elementary properties of combinatorics
The student is able to:
Work with vectors and elementary properties of functions, especially with the rules of differentiability
- apply elementary vector operations
- calculate dot product and cross product
- determine equations of lines and planes in space
- apply elementary properties of functions
- calculate derivatives using differentiation rules and the derivatives of elementary functions
Work with limits and the definitions of continuity and differentiability and applications, for functions of one variable
- calculate limits
- state and apply the definition of (left, right) continuity
- work with limits involving infinity
- state and apply the definition of differentiability
- calculate and apply linear approximations and differentials
- calculate the absolute extreme values on a closed bounded interval
- apply l'H^opital's rule to indeterminate forms of limits
Investigate functions in two variables
- plot graphs and contour lines
- investigate continuity and differentiability
- calculate partial derivatives
- calculate the tangent plane and linearisation
Introduction to Mathematics:
This introductory course concerns basic mathematical concepts. It contains short introductions in set theory, logic and combinatorics. Much emphasis is laid on precise formulation of statements, logical reasoning and the construction of mathematical proofs. All concepts treated will be needed in successive courses in mathematics, which makes this course of the utmost importance for the bachelor degree programmes in science and engineering.
In set theory, we will define the concept of a set, introduce some well-known mathematical sets (natural numbers, integers, rational numbers, real numbers, intervals) and extreme values of sets (minimum, maximum, in mum and supremum). Furthermore, we will consider subsets, set operations (union, intersection, complement, difference) and Venn diagrams.
In the logic part, propositions, connectives , truth tables, predicates and quantifiers are introduced as well as the concepts of tautology, logical equivalence and logical implication. A connection between set theory and logic is established using membership tables and quantified statements.
The techniques of direct proof, proof by contradiction, proof by cases, mathematical induction and counterexample are treated in detail.
Elementary counting techniques are treated using permutations and combinations. The principle of inclusion and exclusion (up to three sets) and Newton's binomial formula are considered.
The course Calculus 1A is a course in basic mathematics programme of the UT, called the Mathematics Line. This course treats topics that can be applied immediately in other disciplines, such as construction engineering and business administration. It consists of a mix of well-known material and some new concepts. Calculus 1A is divided into four lectures.
Lecture 1: vectors
We focus primarily on vectors in three-dimensional space, but sometimes we will also consider vectors in the plane. Vectors are well known from physics, for instance to describe a force or the velocity of an object. Geometrical and algebraic properties of vectors are discussed. The dot (or inner) product and the cross product of two vectors are introduced. The dot product can be used to measure angles. As a consequence, orthogonality is described with the dot product. The cross product can be used to find normal vectors.
Lecture 2: limits
In this lecture, we discuss the concept of limit both in an informal and in a more precise way. The precise definition (also known as the ε−δ definition) as such is not part of the course, but it is still useful to have a general idea how limits work. Completely ignoring the ε−δ definition, we focus on the computation of limits. We use standard limits and limit laws, and some `crowbar' theorems like the Sandwich Theorem and (in lecture 3) L'Hôpital's rule. Continuity of a function at a point is defined as a limit and methods for analysing continuity are presented.
Lecture 3: differentiation
The derivative of a function of one variable is defined as an instantaneous rate of change, which is the limit of an approximating average rate of change. Calculating derivatives with limits is cumbersome, so in almost all cases derivatives are computed with the differentiation rules (such as the chain rule) and the derivatives of the standard functions. Since this involves prior knowledge from secondary education, we shift our focus on applications, such as linearization. Other applications are: finding extreme values and L'Hôpital's rule.
Lecture 4: multivariate analysis
In the last lecture, functions of more than one variable are introduced. General skills, like graphing and finding the domain, are briefly reviewed. The notions of limits, continuity and partial derivatives are introduced. We finish the lecture with the tangent plane and linearization.
|Thomas' Calculus, Early Transcendentals, 12th edition. ISBN: 9781784498139|
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