Calculus
Course introduction
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Calculus
FILIPPO TERRAGNI
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ESTIMATED LEARNING TIME
Total time needed to get familiar with the theory and understand the concepts, solve the proposed problems, and practice with the self-assessment tests: 160 hours.
PREREQUISITES AND RECOMMENDED PREVIOUS KNOWLEDGE
High-school level Mathematics.
GENERAL DESCRIPTION OF THE COURSE
- Sequences and series of real numbers.
- Differential Calculus of one real variable.
- Taylor polynomial.
- Integral Calculus of one real variable.
OBJECTIVES: KNOWLEDGE AND SKILLS
The global purpose of the course is to provide the student with the ability to solve mathematical problems arising in Engineering, showing a correct and systematic understanding of key concepts, being able to interpret the obtained results. In order to accomplish this main objective, the student should achieve specific knowledge and skills, as detailed below.
Knowledge: be able to apply some methods of proof; be versed in dealing with sequences and series of real numbers; know the properties of elementary functions; understand the concepts of limit, continuity, and differentiability of a real function of one real variable; be familiar with the Taylor polynomial and the related framework; comprehend the concept of integral to use it in various applications.
Skills: ability to work with real functions of one real variable either analytically, graphically, or numerically; capacity to solve problems involving the concepts of derivative and integral; abstract thinking; deduction and communication skills by the correct use of mathematical notation and language.
TEACHING MATERIAL
Lecture notes are provided for the nine topics introduced in the course.
Chapter 1. Sets of numbers.
Chapter 2. Sequences and series of real numbers.
Chapter 3. Real functions: limits and continuity.
Chapter 4. Real functions: derivative.
Chapter 5. The Newton-Raphson method.
Chapter 6. Taylor polynomial.
Chapter 7. Local and global behavior of a real function.
Chapter 8. Integration: fundamental theorems and techniques.
Chapter 9. Improper integrals.
A list of problems, with the corresponding solutions, is provided for each chapter. Several self-evaluation tests are available.
PRACTICAL ASSIGNMENTS AND ASSESSMENT ACTIVITIES
A collection of problems and exercises is provided for each of the nine chapters the course comprises. In addition, a list of solutions to the proposed assignments is facilitated. On the other hand, fifteen evaluation tests (with solutions) covering the essential parts of the course are also available, thus yielding a tool to assess the student’s learning and progress.