Syllabus
Weierstrass extreme value theorem and its applications.
Derivative -- definition, geometric interpretation. Existence and finiteness of derivative, examples. One-sided differentiability. Derivative as a function.
Derivative and continuity, relations and counterexamples. Derivative of continuous function.
Derivative of arithmetic operations, linearity of derivative.
Derivative of inverse and composite function.
Derivative of elementary functions. Calculation of derivative, limit of derivative theorem.
Mean value theorems (two-sided and one-sided versions), their geometric interpretation and applications.
Derivative and local/global monotonicity, isolated points and endpoints.
Derivative a convexity/concavity, isolated points and endpoints.
L'Hospital's rule and its use.
Taylor polynomials -- polynomial approximation of function, algebraic form, Lagrange remainder, applications.
Antiderivative -- definition, uniqueness, properties.
Newton integral and methods of its calculation.
Partitions of interval, lower nad upper Darboux sums, refinement of partition, relations.
Riemann integral (Darboux approach) -- definiton, equivalent condition of existence, examples of existence and nonexistence, improper integral.
Properties of Riemann integral -- linearity, monotonicity, additivity. Extension for lower limit being greater than upper one.
Fundamental theorem of calculus. Existence of antiderivative of function.
Annotation
Basics of differential and integral calculus - derivative, antiderivative, definite integral and their use.