Mathematics, Tourism, FH Graubünden, Thomas Borer, 2023/24

Objectives

Basics

• know the definition and elements of the set of natural numbers, integers, rational numbers, and real numbers.
• know and understand what a set, an element of a set, an empty set, a subset, an intersection, a union, and a set difference are.
• be able to perform basic set operations.

Functions and equations

• understand what a function is.
• know and understand what the domain, codomain, and range of a function are.
• be able to determine the domain, codomain, and range of a function.
• know and understand what a value of a function is.
• be able to determine values of a given function.
• be able to judge whether a given relation is a function.
• know four ways to represent a function.
• be able to think of a relation between two quantities as a function.

• know and understand what a linear function is.
• know the general form of the equation of a linear function.
• know and understand what the slope and intercept of a linear function are.
• be able to draw the graph of a given linear function.
• be able to determine slope and intercept of a linear function.
• know some examples of linear functions in economic and everyday life applications.
• know what a constant function is.
• know what a direct proportionality is.
• know and understand what simple interest is.
• be able to perform simple interest calculations.
• know and understand what fixed costs, variable costs, total costs, total revenue, total profit, and break-even value are.
• be able to apply the concept of linear functions to a new problem.
• know what a linear equation is.
• be able to solve a linear equation.
• be able to determine the solution set of a linear equation.
• know and understand what a equivalence operation is.
• know the most common equivalence operations.
• be able to solve a linear equation containing parameters.
• know and understand the relation between a linear function and a linear equation.
• be able to treat applied tasks in economics by means of linear equations.
• be able to solve a linear system of equations.
• know and understand the relation between the intersection point of the graphs of two linear functions and the solution of a linear system of equations.
• be able to treat applied tasks by means of linear systems of equations.

• know and understand what a quadratic function is.
• know the general form of the equation of a quadratic function.
• know the vertex form of the equation of a quadratic function.
• know that the graph of a quadratic function is a parabola.
• be able to graph a quadratic function out of the vertex form of its equation.
• be able to determine the position of the vertex of a parabola out of the vertex form of the equation of the corresponding quadratic function.
• be able to convert the vertex form of the equation of a quadratic function into the general form.
• know, understand, and be able to apply the method of completing the square.
• be able to convert the general form of the equation of a quadratic function into the vertex form.
• know and understand the relation between a quadratic function and a quadratic equation.
• be able to solve a quadratic equation with the method of completing the square.
• be able to solve a quadratic equation by applying the quadratic formula.
• be able to solve special quadratic equations without applying the quadratic formula.
• be able to solve a quadratic equation containing a parameter.
• be able to determine the vertex form of the equation of a quadratic function out of the coordinates of the vertex and the coordinates of another point of the corresponding parabola.
• be able to determine the general form of the equation of a quadratic function out of the coordinates of three points of the corresponding parabola.
• know some applied examples of quadratic functions.
• know and understand what supply, demand, and market equilibrium are.
• be able to treat applied tasks in economics by means of quadratic equations or systems of quadratic equations.

• know and understand what compound interest is.
• understand the difference between simple and compound interest.
• be able to perform compound interest calculations.
• know and understand what a nominal annual interest rate is.
• know the general form of the equation of an exponential function.
• know and understand the graph of an exponential function.
• understand the difference between exponential growth and decay.
• know some applied examples of exponential growth and decay functions.
• be able to graph an exponential function out of its equation.
• be able to determine the equation of an exponential function out of the coordinates of two points of the graph.
• be able to treat applied tasks by means of an exponential function.
• know and understand that it takes a new operation to determine the exponent of a power.
• know and understand the definition of a logarithm.
• be able to determine simple logarithms without a calculator.
• be able to solve simple exponential equations without a calculator.
• know and understand what a common and a natural logarithm is.
• be able to calculate a common and a natural logarithm with a calculator.
• be able to apply one of the logarithmic properties in order to solve simple exponential equations.
• be able to treat specific compound interest tasks by means of logarithms.
• know and understand what an annuity is.
• understand the difference between an ordinary annuity and an annuity due.
• be able to calculate the present and the future value of an annuity if constant payments are made at the beginning or at the end of each compounding period.
• be able to treat specific annuity tasks.

Differential calculus

• understand the purpose of finding the slope of the tangent to the graph of a function.
• know and understand what the derivative (rate of change) of a function at a specific value of the variable is.
• know and understand what the derivative (derived function) of a function is.
• know and understand that the derivative (derived function) of a function is a function.
• be able to estimate a derivative (rate of change) out of the graph of a function.
• be able to state a derivative (rate of change) of a constant and a linear function.
• be able to determine the derivative (derived function) of a constant and a linear function.
• be able to determine the derivative (derived function) of a basic power and a basic exponential function.
• be able to determine a derivative (rate of change) of a basic power and a basic exponential function.
• know the coefficient, sum, and product rules.
• be able to apply the coefficient, sum, and product rules to determine the derivative of a function.
• know what a higher-order derivative is.
• be able to determine a higher-order derivative of a function.
• understand the relation between the first derivative of a function and the increasing or decreasing of the function.
• understand the relation between the second derivative of a function and the concavity of its graph.
• know and understand what a local maximum and a local minimum of a function is.
• be able to determine the local maxima and minima of a function.
• know and unterstand what the global maximum and the global minimum of a function are.
• know and understand what a point of inflection of a function is.
• be able to determine the points of inflection of a function.
• know and understand what marginal cost, marginal revenue, and marginal profit are.
• be able to determine the global maximum and the global minimum of a cost, revenue, and profit function.
• know and understand what average cost, average revenue, and average profit are.
• be able to determine the global maximum and the global minimum of an average cost, average revenue and average profit function.

Integral calculus

• know and understand what an antiderivative of a function is.
• know and understand that a function has infinitely many antiderivatives.
• know and understand what the infinite integral of a function is.
• know and understand what the integration constant is.
• be able to determine an antiderivative and the indefinite integral of a constant, basic power, and basic exponential function.
• know the coefficient and sum rules.
• be able to apply the coefficient and sum rules to determine the indefinite integral of a function.
• be able to determine the cost, revenue, and profit functions if the marginal cost, marginal revenue, and marginal profit functions are known.
• know and understand what a definite integral of a function is.
• know and be able to apply the fundamental theorem of calculus.
• be able to determine a definite integral of a constant, basic power, and basic exponential function.
• be able to determine the area between the graph of a basic power function and the abscissa.
• know and understand what a consumer's and a producer's surplus is.
• be able to determine a consumer's and a producer's surplus if the demand and supply functions are basic power functions.

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