Course syllabus for Introductory course in mathematics

- fluently handle algebraic calculations and the elementary functions, both in problem solving and in theory. The student shall be able to draw graphs and solve equations, both by hand and using MATLAB. The student shall also be able to solve systems of linear equations by hand and using MATLAB and master vector algebra in two and three dimensions.

This means that the student should be able to

- explain the meaning of definition, theorem and proof and interpret and use logical symbols such as implication and equivalence in problem solving and in their own explanations and proofs.
- solve second-degree equations with real coefficients and be able to derive the solution formula.
- explain the relationship between the factors of a second-order polynomial and its zero points and use this in equation solving.
- solve inequalities for rational functions using the character table for the factors.
- calculate real part, imaginary part, absolute value and conjugate to a complex number.
- make computations with complex numbers in Cartesian as well as in polar form.
- interpret the mathematical computing operations geometrically in Argand diagram.
- solve binomial equations.
- define the concept of function and the concepts related to it such as domain, codomain and value set and give examples of functions that meet certain requirements.
- define concepts that relates to the properties of functions such as monotonous, growing, declining, injective, surjective, bijective, invertible and give examples of functions with definite properties.
- define the elementary functions: trigonometric functions, arcus functions, logarithm and exponential functions, hyperbolic functions and also to draw the graphs of these functions.
- utilize the unit circle and the definitions of the trigonometric functions to derive relationships between these functions.
- derive the addition formulas for the trigonometric functions.
- handle the basic functions in problem solving, without the help of formulary, by memorizing the relationships (formulas) or by the ability to derive from more basic relationships.
- provide and use informal definitions of different types of limits, be able to use the rules for calculation of limits and be able to use them in problem solving.
- prove certain standard rules for limit calculation and inequalities for elementary functions.
- define the different concepts of continuity and explain by example.
- apply the theorems for continuous functions in problem solving.
- define the concepts of derivatives, left and right derivatives and be able to give examples of functions that are differentiable and those that are not (at a certain point), be able to use the function's differential for approximation.
- prove that derivability implies continuity.
- prove the product rule and the chain rule for the derivative.
- calculate the derivative of simple functions only by means of the derivative's definition and be able to derive the derivative of some elementary functions.
- calculate the derivative of composite elementary functions using knowledge of the derivative of the simple elementary functions and the derivation rules, including logarithmic derivation, without the aid of a Formulary.
- apply implicit derivation in simple situations.
- determine the tangent and normal lines for different functions.
- formulate and prove the theorem about the significance of the derivative's sign and apply it to determine if a function is growing/decreasing in an interval.
- formulate and prove the theorem about the relationship between stationary points and extreme values for the function.
- formulate and prove the mean value theorem including the special case Rolle's theorem.
- establish mathematical relationships between quantities based on a descriptive text, be able to use exponential functions in growth models.
- determine all asymptotes for a function, be able to determine on what intervals the function is growing or decreasing, being convex or concave, be able to determine all local extremes and inflection points and with the aid of all this be able to draw the function's graph. Be able to decide for yourself with what precision a graph needs to be drawn in order to have a particular question that concerns e.g. the range of the function, the number of zeros, the proof of an inequality, answerable.
- use Newton's equation solution method; with and without MATLAB.
- use MATLAB to solve equations numerically and for graphing.
- solve linear equation systems by applying the elimination method both to the system and to the total matrix of the system.
- explain why the elimination method leads to equivalent systems and what this means.
- explain how the different types of sets of solutions arise and how they can be described.
- use MATLAB to solve systems of equations numerically.
- define and use the basic concepts and operations, including scalar product, vector product and scalar triple product, for vector algebra in 2 and 3 dimensions.
- derive the geometric interpretation of scalar product, be able to calculate orthogonal projections and be able to divide a vector into two perpendicular components.
- explain how a plane or line can be described by equations in different ways and be able to determine equations for planes and lines that are geometrically described.
- calculate the distance between points, lines and planes.