Functions and Complex NumbersLaajuus (3 ECTS)

Objective

After completing this course the student will understand the concepts associated with functions, graphs of functions, composed functions and inverse functions. The student will be familiar with the common real functions, i.e. polynomial, power, exponential and logarithmic functions, trigonometric and inverse trigonometric functions along with their fundamental properties. The student will be familiar with complex numbers and their representations. The student will be able to solve the second order equation and other equations of a complex unknown.

Content

1) Functions
2) Complex numbers

Qualifications

Comprehensive school mathematics. Particularly it is required that
i) the student has basic knowledge of real numbers and operations and is able to write, manipulate ja interpret formulas,
ii) the student knows how to solve simple equations and inequalities as well as groups of linear equations,
iii) the student is familiar with the fundamental concepts and facts in geometry and trigonometry,
iv) the student is able to make simple computations using a calculator.

Assessment criteria, satisfactory (1)

1) Functions. The student is familiar with the general concepts associated with a real function and its graph. The student is able to isolate the inner and outer function from an expression of a composed function. The student is able to solve simple problems concerning common real functions.
2) Complex numbers. The student is familiar with complex numbers and their representations. The student is able to simplify straightforward expressions consisting of complex numbers. The student is able to solve a second order equation with real coefficients and some other simple equations.

Assessment criteria, good (3)

1) Functions. The student has a good command of general concepts associated with mathematical functions as well as the common real functions and their properties. The student is able to model real-world phenomena, in particular exponential and wave phenomena, using appropriate functions.
2) Complex numbers. The student is able to transform complex numbers from one representation to another. The student is able to solve equations of a complex unknown by using appropriate representations.

Assessment criteria, excellent (5)

1) Functions. The student has got exceptional skills in manipulating functions. The student is able to prove or disprove assertions about functions (e.g. to prove the formula for the logarithm of a product based on power identities).
2) Complex numbers. The student has a profound understanding of the Euler formula. The student is able to apply the Euler's formula to verify certain properties of trigonometric functions. The student understands how real functions generalize to complex functions. The student is familiar with the multivaluedness problematics of this generalization.

Further information

Students should have access to a function calculator.