Introductory Course in Calculus

Aim

It is important that you acquire general mathematical accuracy and a stable foundation for your continued studies. After the course is completed you should be able to:

• read and comprehend mathematical texts.
• perform standard calculations with accuracy.
• handle calculations with algebraic expressions, inequalities and absolute values.
• solve polynomial equations and equations containing square roots.
• analyze how the concepts domain, range, injectivity and composition relate to particular functions.
• define and draw the graphs of the elementary functions: the natural logarithm, exponential-, power-, trigonometric- and inverse trigonometric functions.
• use and prove laws and formulas for the elementary functions.
• work with complex numbers in cartesian and polar form.
• define the complex exponential function and use and prove Euler's and deMoivre's formulas.
• solve problems concerning straight lines and circles in the plane.
• perform logical arguments and proofs by induction.
• work with geometric and arithmetic sums.
• check results and partial results in order to verify their correctness or reasonableness.

Content

Algebraic expessions, inequalities, modulus, complex numbers. Solving equations. Functions and graphs. Definitions and properties of the elementary functions: natural logarithm, exponential function, power function, trigonometric functions and complex exponential function, arcus functions. The Euler formulas. Basic principles of logic. Different types of proof techniques. Coordinate systems in the plane. Polar coordinates. Lines and circles. The complex plane. Complex numbers in polar form. Inverse trigonometric functions.