## Calculus I

course ID

Lecturer

CFU

12

Length

14 Weeks

Semester DD

First

##### Course details

- Natural, integer, rational numbers. The real numbers: order relation and the continuity axiom.

- Functions: graph, composition and inversion, the graph of the inverse function. The (local)maximum and the minimum of a function. Monotone functions, even and odd functions, periodic functions. Examples of elementary functions.

- Limits of functions and limits of sequences. Special limits.

- Continuity: theorem of the permanence of the sign, of intermediate values, continuity of the inverse function and Weierstrass theorem.

- Derivability. Definition, algebraic properties and derivatives of elementary functions.

The mean value theorem. The theorems of Rolle, Lagrange and the characterization of monotony by the sign of the derivative. Determination of local max/min.

- Higher derivatives. Convex and concave functions. Study of the graph of a function. Taylor's formula. Application of Taylor's formula to the calculation of limits. The rest of Lagrange.

- Riemann integral. The integrability of continuous functions and of monotone functions. Fundamental theorem of integral calculus. Integration techniques: substitution, by parts and decomposition into simple fractions for rational functions. Integrals reducible to the integration of rational functions. Improper integrals: the criterion of asymptotic comparison; absolute convergence.

- Numerical series, basic definition and examples. Criteria: asymptotic comparison and comparison; root; quotient; Integral comparison; Leibniz. Sequences of functions: uniform convergence. Power series and Taylor series.

- Complex numbers. Exponential representation. Roots of a polynomial eqation.

- Differential equations. Introduction to the theorem of existence and uniqueness and properties of solutions. Equations with separable variables of the first order. Linear differential differential equations: general properties and explicit solutions of linear equations with constant coefficients. Application to the case of the harmonic oscillator: damped, forced and resonant. Introduction to systems of linear differential equations and solution of systems with diagonalizable constant coefficients. The case of coupled harmonic oscillators.

Co-teaching: Prof. Berretti Alberto

##### Objectives

TRAINING OBJECTIVES: acquisition of basic concepts on the computation of limits, derivatives, integrals for functions of one variable, and differential equations, and their use for the solution of simple problems; acquisition of some basic logical skills (eg, distinction between the hypotheses and the thesis of a theorem, proofs for the absurdity, etc.).

KNOWLEDGE AND UNDERSTANDING: learning and understanding the basic notions relating to the calculation of limits, example derivatives and integrals for functions of one variable and differential equations; read and calculate basic results related to these topics.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: knowing how to calculate limit, derive, integrals of functions of one and solve differential equations; knowing how to apply the notions learned to problem solving (for example: Taylor developments, graphs of functions, convergence of improper integrals). field of mathematical analysis; knowing how to build examples and counter-examples.

COMMUNICATION SKILLS: expose and argue the solution of problems; also be able to discuss and reproduce demonstrations of basic results relating to mathematical analysis.

LEARNING SKILLS: identify solution strategies in situations similar to those faced in the course.