Mathematics, Tourism, HTW Chur, Thomas Borer, 2010/11

Objectives

Basics

• understand what a set, an element of a set, a subset, an intersection, a union, a complement, an empty set is.
• know the set of real numbers, rational numbers, integers, natural numbers.
• be able to perform basic set operations.
• be able to perform basic algebraic transformations of powers and fractions.

Functions and equations

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

• understand what a linear function is.
• know the general form of a linear function.
• know and understand the terms "slope" and "intercept".
• 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 the term "simple interest".
• be able to perform simple interest calculation.
• know and understand the terms "fixed costs", "variable costs", "total cost", "total revenue", "total profit" and "break-even value".
• 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 the general form of a quadratic function.
• know the vertex form of a quadratic function.
• know that the graph of a quadratic function is a parabola.
• be able to determine the position of the vertex out of the vertex form of a quadratic function.
• be able to graph a quadratic function out of the vertex form.
• know and understand the relation between a quadratic function and a quadratic equation.
• 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 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 a quadratic function out of the coordinates of three points of the corresponding parabola.
• know some applied examples of quadratic functions.
• know and understand the terms "supply", "demand", and "market equilibrium".
• be able to treat applied tasks in economics by means of quadratic equations or systems of quadratic equations.

• know and understand the term "compound interest".
• understand the difference between simple and compound interest.
• be able to calculate the future capital that is invested at an interest rate which is compounded annually.
• know the general form 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 treat compound interest tasks.
• be able to graph an exponential function out of its formula.
• be able to determine the general form 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 logarithm, a natural logarithm is.
• be able to calculate a common logarithm, 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.
• be able to calculate the future capital that is invested at an interest rate which is compounded more than once per year.
• know and understand the terms "nominal annual interest rate" and "effective annual interest rate".
• be able to treat specific compound interest tasks.
• know and understand the term "annuity".
• understand the difference between ordinary annuities and annuities 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 or the rate of change of a function at a specific value of the variable is.
• know and understand what the derivative of a function is.
• know and understand that the derivative of a function is a function.
• be able to determine rates of change of a constant /linear function.
• be able to determine rates of change of a basic power/exponential function.
• be able to determine the derivative of a constant/linear function.
• be able to determine the derivative of a basic power/exponential function.
• know the coefficient, sum, product rule.
• be able to apply the coefficient, sum, product rule to determine the derivative of a function.
• know the chain rule.
• be able to apply the chain rule 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/decreasing of the function.
• understand the relatioin between the second derivative of a function and the concavity of its graph.
• know and understand what a relative maximum/minimum of a function is.
• be able to determine the relative maxima/minima of a function.
• 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/revenue/profit means.
• be able to find the absolute maximum/minimum of a cost/revenue/profit function.
• know and understand what average cost/revenue/profit means.
• be able to find the absolute minimum of an average cost function.
• know what a point of diminishing returns is.
• be able to find the point of diminishing returns of a 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/basic exponential function.
• know the coefficient, sum rule.
• be able to apply the coefficient/sum rule to determine the indefinite integral of a function.
• be able to determine the cost/average cost/revenue/profit function if the marginal cost/average cost/revenue/profit function is known.
• (know and understand what the definite integral of a function is.) *
• (know the fundamental theorem of calculus.) *
• (be able to determine the definite integral of a constant/basic power/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/producer's surplus is.) *
• (be able to determine the consumer's/producer's surplus if the demand and supply functions are basic power functions.) *

* Objectives in parentheses are not relevant for the exam.

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