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B021058 - CALCULUS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2018-19
Coorte 2018 - 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
13/09/2018 ⇒ 31/05/2019
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 for functions of one variable. Local and constrained optimisation for functions of several variables.
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
Knowledge acquired
Limits and derivatives of functions of one real variable
Optimisation methods for functions of one real variable
Integral calculus for functions of one real variable
Infinite series; convergence criteria
Taylor expansions
Differential calculus for functions of several variables
Optimisation methods for functions of several real variable
Competence acquired
Compute limits and derivatives of functions of one real variable
Determine relative and absolute maxima and minima of functions of one real variable
Solve indefinite and definite integrals
Determine the Taylor polynomial of a given order of an assigned function
Compute partial derivatives of any order of functions of several real variables
Determine local, absolute and constrained maxima and minima of functions of several real variables
Limits and derivatives of functions of one real variable
Optimisation methods for functions of one real variable
Integral calculus for functions of one real variable
Infinite series; convergence criteria
Taylor expansions
Differential calculus for functions of several variables
Optimisation methods for functions of several real variable
Competence acquired
Compute limits and derivatives of functions of one real variable
Determine relative and absolute maxima and minima of functions of one real variable
Solve indefinite and definite integrals
Determine the Taylor polynomial of a given order of an assigned function
Compute partial derivatives of any order of functions of several real variables
Determine local, absolute and constrained maxima and minima of functions of several real 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.
Further information
The teacher will use Moodle platform as a help for the course, and all students will be invited to register to the Moodle course page. On this page the teacher will upload, along the course, exercises that will be subsequently solved and explained during the lectures.
Type of Assessment
The exam consist of two written parts and a final oral part.
In the first written part the student is asked to solve exercises on the topics of the course: limits and derivatives, study of the graph of a function of one variable, Taylor expansions, infinite series, integrals, local and constrained extrema of functions of several variables. This part can be done also through the intermediate written parts that are set during the course. The sufficiency in this part gives access to the second written part.
The second written part is made of question about theoretical results presented in the course. The student is asked to expose: definitions, statements of theorems and the corresponding proofs.
The oral part is a discussion about the two written parts.
The final grade is an appropriate mean of the grades of the two written parts.
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.
In the first written part the student is asked to solve exercises on the topics of the course: limits and derivatives, study of the graph of a function of one variable, Taylor expansions, infinite series, integrals, local and constrained extrema of functions of several variables. This part can be done also through the intermediate written parts that are set during the course. The sufficiency in this part gives access to the second written part.
The second written part is made of question about theoretical results presented in the course. The student is asked to expose: definitions, statements of theorems and the corresponding proofs.
The oral part is a discussion about the two written parts.
The final grade is an appropriate mean of the grades of the two written parts.
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 criteria for series with non-negative terms. Convergence criteria for series with terms of arbitrary sign.
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. The Theorem of Lagrange multipliers.
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 criteria for series with non-negative terms. Convergence criteria for series with terms of arbitrary sign.
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. The Theorem of Lagrange multipliers.