- Docente: Giovanni Dore
- Credits: 6
- SSD: MAT/05
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mechanical Engineering (cod. 0927)
Learning outcomes
The student knows the basic concepts and the main properties of real functions of a real variable (limits of sequences and functions, continuity, differential calculus, integral calculus) and is able to solve simple exercises on these topics.
Course contents
Functions
Domain and range of a function, injective, surjective and bijective
functions; composition of functions, inverse function.
Elementary functions of a real variable: power, exponential,
logarithm, trigonometric and inverse trigonometric functions,
hyperbolic and inverse hyperbolic functions.
Real sequences
Sequences in R; limit of a sequence; squeeze theorem; algebra of
limits. Limits of monotone sequences. Boundedness and bounds of
subsets of R.
The number e; some remarkable limits of sequences.
Limits and continuity for real functions of a real
variable
Limits for real functions of a real variable; extensions of the
results pertinent to sequences; limit of a composite function. One
sided limits, limits of
monotone functions. Some remarkable limits of functions.
Continuity, algebra of continuity.
Intermediate value theorem, Weierstrass theorem.
Differential calculus for functions of one variable
Definition and basic properties of the derivative; derivation
rules; derivative of elementary functions.
Rolle and mean value theorem, derivative and monotonicity.
L'Hôpital's rule.
Higher order derivatives; Taylor's formula.
Local extrema; convexity, inflexion points.
Integral calculus for functions of one variable
Definition and basic properties of the integral of a continuous
function; mean value theorem for integrals, fundamental theorems of
calculus; primitive. Integration by parts, integration by
substitution; integration of rational functions.
Readings/Bibliography
G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009).
M. Bramanti: Esercitazioni di Analisi Matematica 1, Esculapio (2011).
Teaching methods
Lessons and exercises delivered in a mixed mode, with the possibility for the student to follow in the classroom or remotely.
Assessment methods
The examination consists of a preliminary written test and an oral one.
The written test consists of 11 exercises related to the arguments of the course. 10 exercises are multiple choice, 3 point for each corrects anwer, -1 point for each wrong answer. the test is passed if the total score is at least 10. Last exercise is open-answer, the maximum score is 6, it is added to the score obtained in the multiple choice exercises.
The written test is valid to take the oral test in the same appeal or in the immediately following one, if in the same period of exam (Janary.February, June-July, September).
In order to sustain the written test the student must register at least four days before the test through AlmaEsami.
The oral test follows the written test; it mainly concerns the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions and theorems) and how to connect with each other.
Teaching tools
To carry out the mixed mode, tablet and webcam will be used.
Links to further information
http://www.dm.unibo.it/~dore/Analisi_T-A/index.html
Office hours
See the website of Giovanni Dore