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B021058 - CALCULUS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Type of Assessment
Course program
Academic Year 2017-18
Coorte 2017 - 3-years First Cycle Degree (DM 270/04) in STATISTICS
Course year
First year - Annualità singola
Belonging Department
Statistics, IT and its applications "G. Parenti" (DISIA)
Course Type
Single education field course
Scientific Area
MAT/05 - MATHEMATICAL ANALYSIS
Credits
12
Teaching Hours
96
Teaching Term
13/09/2017 ⇒ 01/06/2018
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
Italian
Course Content
Differential and integral calculus
Suggested readings (Search our library's catalogue)
P. Marcellini, C. Sbordone
Elementi di Calcolo
Liguori Editore
P. Marcellini, C. Sbordone,
Esercitazioni di Matematica, volumi 1 e 2
Liguori Editore
Elementi di Calcolo
Liguori Editore
P. Marcellini, C. Sbordone,
Esercitazioni di Matematica, volumi 1 e 2
Liguori Editore
Learning Objectives
Learning basic notions in differential and integral calculus in one and several variables.
Prerequisites
Standard knoweldge of mathematics provided in high school.
Teaching Methods
Theoretical and exercise lectures. The total amount of theoretical lectures will be about the same as that of exercise lectures.
Type of Assessment
Written and oral exam. The written exam is about two hours long and consists in solving some exercises. In the oral part the written part is analyzed and some questions are posed concerning the theoretical content of the course.
Course program
1. Basic notions on real numbers. Rational and irrational numbers. The induction principle. Infimum and supremum of sets of real numbers.
2. Sequences. Sequences of real numbers. Limits. Monotone sequences.
3. Functions of one real variable. The notion of function. Limits. Operations with limits. Continuity. The fundamental theorems on continuous functions: existence of zeroes and intermediate values; the Weierstrass theorem.
4. Differential calculus for functions of one variable. The derivative. Differentiability and continuity. Operations with derivatives. Derivatives of elementary functions. The theorems of Fermat, Rolle and Lagrange. Monotonicity of a function and the sign of its derivative. Second derivative; convexity and concavity. Taylor polynomials; Taylor expansion of some elementary functions.
5. Integral calculus for functions of one variable. Definition of definite integral. Integrability of continuous functions. The fundamental theorem and the fundamental formula of integral calculus. Some tecniques of integration; integration by parts; integration by change of variable.
6. Infinite series. The notion of series. Finite sum of a series. Character of a series. Series with non-negative terms and their behavior. Some convergence critera for series with non-negative terms.
7. Functions of several variables. Basic notions on the Euclidean n-dimensional space. Limts and continuity fr functions of several variables. Partial and directional derivatives; gradient; differentiability. Local extrema and their identification; in particular with the use of the Hessian matrix. Constrained maxima and minima for functions of several variables.
2. Sequences. Sequences of real numbers. Limits. Monotone sequences.
3. Functions of one real variable. The notion of function. Limits. Operations with limits. Continuity. The fundamental theorems on continuous functions: existence of zeroes and intermediate values; the Weierstrass theorem.
4. Differential calculus for functions of one variable. The derivative. Differentiability and continuity. Operations with derivatives. Derivatives of elementary functions. The theorems of Fermat, Rolle and Lagrange. Monotonicity of a function and the sign of its derivative. Second derivative; convexity and concavity. Taylor polynomials; Taylor expansion of some elementary functions.
5. Integral calculus for functions of one variable. Definition of definite integral. Integrability of continuous functions. The fundamental theorem and the fundamental formula of integral calculus. Some tecniques of integration; integration by parts; integration by change of variable.
6. Infinite series. The notion of series. Finite sum of a series. Character of a series. Series with non-negative terms and their behavior. Some convergence critera for series with non-negative terms.
7. Functions of several variables. Basic notions on the Euclidean n-dimensional space. Limts and continuity fr functions of several variables. Partial and directional derivatives; gradient; differentiability. Local extrema and their identification; in particular with the use of the Hessian matrix. Constrained maxima and minima for functions of several variables.