Mathematical induction. Completeness of the real numbers. Functions. Limits. Continuity, theorems of continuous functions. Uniform continuity. Differentiation, derivative of inverse functions, derivatives of trigonometric functions, differential. Applications of derivatives, extreme values of functions, concavity, curve sketching, Cauchy mean value theorem, L’ Hopital rule, graphical method of solving autonomous differential equations, Newton’s approximation method. Integral, indefinite, definite, techniques of integration. Volume of solids of revolution. Improper integrals. Transcendental functions. Taylor’s formula. Differential equations of first order (separable, homogeneous, linear, Bernoulli, Ricatti, exact, Euler integrating factor, equations of special form, orthogonal trajectories).

The purpose of the course is to give a complete and working knowledge of differential and integral calculus, covering and expanding material presented in the last years of the school. After the successful fulfilment of the course, the student:

• will have a solid knowledge of the analysis of functions of a single variable as this is presented with the necessary mathematical rigor through the proofs of most of the theorems and propositions,
• will have the ability to treat the limit of a function or to study its continuity and differentiability through the classical ε-δ definition,
• will have the ability of the direct applications of the abstract knowledge to a number of problems from everyday life, from geometry (areas, volumes) or from physics realizing the vivid and practical aspect of calculus,
• will have the knowledge of the definition of the definite integral as a limiting summation,
• will have the ability to use a variety of techniques to compute complicated indefinite integrals or generalized integrals,
• will have the ability to use Taylor expansion to approximate the value of a function,
• will have the knowledge of the notion of the differential equation of first order and its solution within the context of differential and integral calculus,
• will have the skills to recognize and solve various classes of useful and characteristic differential equations of first order and to act on his/her own for solving differential equations that will face during his/her future studies and career.

Νot required.

1. Finney R.L, Weir M.D, Giordano F.R., Thomas’ Calculus, Vol I, Crete University Press.
2. Kravvaristis D. Analysis Courses, Tsotras Publications, 2017.
3. Instructor’s notes.

1. Calculus, Vol I, S. Negrepontis, S. Giotopoulos, E. Giannakoulias, Symmetria Edt.
2. Calculus, M. Spivak, Publish or Perish, Inc.
3. Answer Book for Calculus, M. Spivak, Publish or Perish, Inc.
4. A first course in Calculus, S. Lang, Springer.

Systematic development and explanation of the theory (and through examples), methods of solutions of exercises, solutions of exercises in the teaching hours and in the problem session hours, final written exam.

The course evaluation derives from:

• 3 compulsory exercises during the semester: counting 30% in the final grade.
• final exam: counting 70% in the final grade.

Final Grade = (0.3*M.V. Exercises) + (0.7*Final Exam)

A student must have: Final Grade >= 5

Face-to-face.