Calculus I

Natural numbers: the minimum principle, mathematical induction. Rational numbers, existence of irrational numbers, the completeness axiom and related properties. Existence of the square root, the integer part of a decimal number. Density of rational and irrational numbers on real numbers, approximation of real numbers by using rationals, classical inequities.

Sequences of real numbers. Convergent sequences, monotone sequences, the nested interval property. Recursive sequences. Subsequences: definition and examples. Bolzano-Weierstrass theorem. Limit points of a sequence, upper and lower bound. Basic sequences. Some basic concepts of the theory of sets.

Functions. Basic definitions. Bounded functions. Monotone functions. Inverse functions. Basic algebraic functions (trigonometric, exponential) and their most important properties.

Limits of functions: Limit points, fixed points. The meaning of the limit of a function. Uniqueness. Transfer principle. Algebraic properties, limit of composition. Lateral limits.

Continuity of functions: Transfer principle. Continuity of basic functions. Continuity and local behaviour. Intermediate value theorem. Existence of maximum and minimum value for continuous functions defined on closed intervals. Continuity of a function on a fixed point. Non-continuity of a monotone function. Continuous and 1-1 (one to one) functions. Inverse function of a continuous and 1-1 one. Inverse trigonometric functions. Logarithmic function.

Derivative. Introduction with examples from Geometry and Physics. Definition of a derivative. Derivative rules. Derivatives of basic functions. Mean value theorem. Darboux theorem. Monotone functions and the first derivative test. Criteria of local extrema. L’Hospital’s rule. Convex and concave functions. Turning points. Graphs of functions and their derivatives. Complete analysis of a function.

• Apostol, T. M. (1967). Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra (2^nd Edition). John Wiley & Sons, Inc.
• Briggs, W., Cochran, L. & Gillett, B. (2015). Calculus (2^nd Edition). Pearson Education, Inc.
• Finney, R. L., Weir, M. D. & Giordano, F. R. (2001). Thomas’ Calculus (10^th Edition). Addison Wesley Longman.