LEARNING OUTCOMES

The course aims to provide the basic knowledge preparatory to other courses that require mathematical methods and tools.

AIMS AND LEARNING OUTCOMES

The course provides basic notions about Functions of one variable, integration of continuous functions and first order ODE’s in view of applications in Physics and Nautic Engineering.

PREREQUISITES

Basic notions in elementary mathematics.

TEACHING METHODS

90 hours  of lectures . Distance lectures until the end of sanitary emergency an subsequently frontal lectures.

SYLLABUS/CONTENT

1) Real numbers and the real line. Cartesian coordinates in the plane. Functions and their graphs. Limits and continuity. Theorems about continuous functions. The derivative. Differentiation rules: product, reciprocal, quotient and chain rule. Monotone functions: the inverse function theorem. Derivatives of some elementary functions. Theorems about differentiable functions: Rolle, Lagrange, Cauchy. Higher order derivatives. Extreme values, convexity and inflection. Sketching the graph of a function.

2) Integration. Sums and sigma notation. Areas as limits of sums. The definite integral. Properties of the definite integral. The fundamental Theorem of Calculus. Changes of variable: the method of substitution. Areas of plane regions. Integration by parts. Integrals of rational functions.

3) L’Hopital’s rule.  Taylor’ formula and its applications.

3) Ordinary differential equations. First order equations: separable equations, linear equations. Cauchy problems: existence and uniqueness theorem. Linear ODE’s with constant coefficients.

RECOMMENDED READING/BIBLIOGRAPHY

 R.A. Adams, Calcolo Differenziale 1 Casa

Editrice Ambrosiana.

Notes & Exercises in  aulaweb ( a.a. 10/11 and the following)