Academic Year 2023/2024 - Teacher: ANDREA SCAPELLATO

Expected Learning Outcomes

The aim of the course of Mathematical Analysis I - Module B is to give the basic skills on Differential Calculus and Integral Calculus for real functions of one real variable and some types of ordinary differential equations.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

  • Knowledge and understanding: The student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures, as derivative and integrals for real functions of one real variable and some types of ordinary differential equations.
  • Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Engineering.
  • Makin​g judgements: The student will be stimulated to autonomously deepen his/her knowledge and to carry out exercises on the topics covered by the course. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
  • Communication skills: The frequency of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for communicating clearly in the scientific field.
  • Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

Required Prerequisites

Arithmetic, Algebra, Analytical Geometry, Trigonometry and contents of Module A.

Detailed Course Content

For topics marked with an asterisk, proofs are not required.

Differential Calculus.
    • Definition of differentiable function and definition of derivative. Geometric and kinematic meaning of the first derivative. Relationship between continuity and differentiability. First finite increment formula. Derivatives of the elementary functions. One-sided derivatives. Point of non-differentiability. Definition of differential. Rules of differentiation. Theorem on the differentiability of the composite function. Theorem on the differentiability of the inverse function.
    • Fundamental theorems of Differential Calculus and their consequences. Fermat's Theorem, Rolle's Theorem, Lagrange's Theorem and its consequences (second finite increment formula, Characterisation of the functions with identically null derivative on an interval, monotonicity test and local extrema, Test for the determination of the extrema, Characterisation of stricly monotone functions). De L'Hôpital's Theorem*. Theorem on the limit of the derivative. Limit of the derivative and points of non-differentiability. Higher-order derivatives. Taylor formula with Peano's remainder* and Taylor formula with Lagrange's remainder*. Concave functions and convex functions: concave functions and convex functions under the differentiability assumption, inflection points, Characterisation of the concavity and the convexity with the monotonicity of the first derivative, Necessary condition for the inflection points, relationship between the concavity/convexity and the sign of the second derivative, general definition of concave function and convex function, test for the inflection points. Higher-order derivative test for the study of stationary points. Qualitative study of a function.
    • Applications. Approximation of functions and data: climatology, finance, biomechanics, robotics. Approximation by Taylor's polynomials. Polynomial interpolation. Numerical differentiation: problem and examples (hydraulics, optics, electromagnetism, demography). Approximation of derivatives. Minimization of real functions of one real variable with the golden section and quadratic interpolation method.
Integral Calculus.
    • Indefinite integrals. Antiderivatives of a function on an interval and indefinite integral. Rules of indefinite integration: linearity property, integration by parts, integrations by substitution. Integration of rational functions. Some integrals that can be transformed into integrals of rational functions. Trigonometric integrals. Integrals of irrational functions.
    • Definite integrals. Riemann integral: lower sum and upper sum, definition of Riemann integrable function, condition of Riemann integrability. Classes of Riemann integrable functions. Properties of the Riemann integral. Integral extended to an oriented interval. Definition of integral average and its geometric meaning. Mean value Theorem. Fundamental Theorem of Integral Calculus. Integration by parts and integration by substitution for definite integrals.
    • Improper integrals. Improper integrals on unbounded intervals. Improper integrals on bounded intervals. Convergence criteria: algebra of improper integrals, integrability of nonnegative functions, comparison test, absolute convergence test, asymptotic comparison test.
    • Applications. Numerical integration: midpoint formula, trapezoidal formula, Simpson formula.
Ordinary differential equations.
    • Basic definitions. 
    • Methods for solving some types of ordinary differential equations. First-order differential equations with separable variables, first-order differential equations with non-constant coefficients, differential equations of order n with constant coefficients, Bernoulli first-order differential equations, homogeneous first-order differential equations.
    • Applications. Euler methods and Crank-Nicolson method.