Mathematical Analysis
The following basic knowledges are required: equations and inequalities, polynomial division, trigonometry, analytic geometry (line and parabola).
Written and oral exams.
The written part consists of 5-6 exercises on topics contained in the syllabus. Students are allowed to use textbook.
Pass mark is set to 15/30.
The oral exam consists of three theoretical questions picked out from a list published on e-learning. The exam is passed if the final score is at least 18/30.
The course has the purpose to provide the students with both theoretical and practical tools of basic mathematical analysis with special attention to the study of a real function in one real variable, to numerical sequences and series and to the theory of Riemann integrable functions. At the end of this course the students will be able to:
1. identify and properly define the provided theoretical concepts;
2. understand and present with an adequate formalism the given results and the introduced techniques;
3. replicate short proofs and provide examples and counterexamples;
4. use the fundamental calculus techniques for solving exercises that involve the study of a function or the behavior of a series, as well as the computation of limits, derivatives and integrals.
Moreover, at the end of this course the students will have acquired a mathematical formalism and a rigorous scientific methodology that they will be able to apply to the other subjects encountered during their studies.
The contents of the lectures and related exercises sessions are:
Preliminaries on sets and functions (10 hours, learning objectives 1,3)
- Recap on the rational and real numerical sets and their properties
- Notion of infimum/supremum, minimum/maximum of a set
- Notion of function and its properties
- Elementary functions: modulus, power function, exponential function, logarithmic function, inverse trigonometric functions
Limit of a sequence (8 hours, learning objectives 1-4)
- Convergent, divergent, oscillating, bounded sequences and uniqueness of the limit
- Operations with limits and indeterminate forms
- Main theorems about the limit of sequences (e.g., squeeze theorem, permanence of sign, limit of monotone sequences)
- The number of Nepero as limit of a certain sequence
- Infinite/Infinitesimal sequences and use of the asymptotic
- Notion of subsequences and their usage to prove the oscillating character of a sequence
Limit of a function (12 hours, learning objectives 1-4)
- Definition of limit of a function and relation with the limit of a sequence
- Extension of the main theorems on the limit of sequences to the case of a generic function
- Order of infinite/infinitesimal of a function, asymptotic analysis and “little-o” notation
- List of limits for common functions and their formulation as asymptotic relations or by using the “little-o” notation
- Use of the limits to compute the asymptotes of a function
Continuous functions of one real variable (6 hours, learning objectives 1-4)
- Continuity at one point and classification of the discontinuity
- Continuity vs algebraic operations, composition, inversion and monotonicity
- Main theorems concerning continuous functions defined over an interval
Differential calculus (16 hours, learning objectives 1-4)
- Notion of derivative given by its geometrical interpretation and classification of non-differentiability points
- Differentiability and continuity
- Differentiability vs algebraic operations, composition, inversion and monotonicity
- Critical points and search for local extrema
- Main theorems concerning differentiable functions defined over an interval
- De L'Hospital’s rule and its usage to compute limits and derivatives
- Convex/concave functions and their geometric interpretation, inflection points and relation with the sign of the second derivative
- Recap of the steps that characterize the study of a function
- Taylor and MacLaurin polynomials as application of higher-order derivatives and as tool for computing limits of functions
Integration (12 hours, learning objectives 1-4)
- Definition of the Riemann integral by means of inferior and superior sums and its properties
- Integral functions and the fundamental theorem of calculus as a tool for computing integrals through the notion of antiderivative
- Methods of integration by parts and by substitution
Numerical series (6 hours, learning objectives 1-4)
- Series as the limit of the sequence of the partial sums
- Some known series: Mengoli, geometrical, harmonic series
- Necessary condition of convergence
- Series with nonnegative terms and related sufficient convergence criteria
- Series with non-constant sign addends and notion of absolute convergence
- Alternating series and Leibniz criterion
Complex numbers (4 hours, learning objectives 1-3)
- Algebraic and trigonometric form of a complex number
- Operations with complex numbers, modulus, power and conjugate of a complex number
- Roots of a complex number and mention to the fundamental theorem of algebra
Introduction: Numerical sets. Rational and real numbers. Density properties of real numbers. Intervals. Module and its properties. Upper bound, lower bound. Roots and powers. Properties of powers. Powers with real exponent. (4 hours)
Functions: Function, domain, image, graph. Bounded functions, injective, surjective and bijective functions. Composite function. Reverse function. Arcsine, arccosine and arctangent functions. Monotonic functions. Integer part function, Heaviside function, sign function. Monotonicity theorem of a composite function. Even, odd and periodic functions. Exponential function and logarithmic function. (6 hours)
Concept of limit: Euclidean distance. Accumulation points. isolated points. Open and closed sets. Internal, external and boundary points. Closure of a set. Definition of limit (various cases). Uniqueness of the limit (dim.). Right and left limit. Sign theorem for limits (dim.). Comparison theorem (dim.). Functions with finite limit are definitely bounded (dim.). Local maxima and minima. Algebra of limits. Limit of composite functions. (6 hours)
Sequences: Sequences with real values. Convergent, divergent and irregular sequences. Permanence of the sign. Convergent sequences are regular. Comparison theorem. Limits of monotonic sequences. Limits of some particular sequences. Neper's number e. Definition of subsequence. Subsequence limit. Infinities and infinitesimals. Comparison between infinites and infinitesimals. Order of infinity and infinitesimal. Definition of asymptotic. (6 hours)
Additional function limits. "Bridge" theorem and its consequences. forms of indecision. Horizontal, oblique and vertical asymptotes. (2 hours)
Continuous functions of one real variable: Definition of continuous function. Continuity from right and from left. Sign theorem for continuous functions (dim.). Composition of continuous functions. Classification of discontinuity points. Continuity of monotone functions. Theorem of zeros (dim.). Corollary to the zeros theorem (dim.). Intermediate value theorem (dim.). Relationship between monotonicity and invertibility of a continuous function (without dim.). Continuity of the inverse function. Weierstrass theorem (without dim.). (12 hours)
Differential calculus: Secant line and tangent line. Difference quotient and derivative. Geometric meaning of the derivative. Derivatives of elementary functions (calculation through the definition of derivative). Differentiable functions are continuous (dim.). C1 functions. Right and left derivative and their geometric meaning. Corner points and cusps. Algebra of derivatives. Derivative of a product (dim.), of a quotient (dim.) and of a composite function (dim.). Derivative of the inverse function (dim.). Derivative of log(x), arcsin(x), arccos(x),arctan(x) . Derivative of log|x|,
log|x|,ax . Fermat's theorem (dim.). Critical points. Finding the local extremum points of a function. Rolle's theorem (dim.). Lagrange mean value theorem (dim.). Monotonicity of a differentiable function and sign of the derivative (dim.). de l'Hopital's theorem. Corollary to de l'Hopital's theorem (dim.). Higher order derivatives. Convex and concave functions. Convex functions and continuity (without dim.). Geometric meaning of convexity. Convexity and sign of the second derivative. Inflection points. Vertical tangent. Inflection points and second derivative. Study of functions. Taylor polynomial. MacLaurin polynomial of the exponential, logarithm, sine and cosine functions. Peano's theorem (without proof). Applications of Peano's theorem (dim.). Taylor polynomial and computation of limits of functions. Lagrange's error and formula of the remainder. (16 hours)
Integrals: Definition of the Riemann integral. Partition of an interval. Upper and lower sums. Integrability of continuous and bounded functions. Relationship between the definite integral and the area of the subgraph of an integrable function. Properties of the integral. Integral mean value
Class lectures and exercise sessions