Functions and number sets
Introduction: operations among sets. Function; domain, co-domain, image and graph of functions. Injective, surjective, inverse function and composition. Increasing and decreasing, odd and even functions, Number sets N, Z, Q, R.

Elementary functions
Review on lines, parabolas, exponential, logarithmic and trigoniometric functions. Absolute value. Neighborhood of a real number.

Limit and continuity
Finite and infinite limit; sign permanence theorem. Right-side and left-side limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and indeterminate forms. Infinite and infinitesimal. Vertical, horizontal and oblique asymptote. Continuous functions. Weierstrass theorem. Intermediate value theorem. Intermediate zero theorem.

Derivatives
Definition of derivative and its geometric interpretation. Calculation of derivatives. Differentiability and continuity. Point of non differentiability. Higher derivatives. Rolle's and Lagrange's theorem. De L’Hôpital's theorems. Taylor's theorem and McLaurin's expansion. Fermat's theorem. Maximum and minimum points. Convexity and concavity. Inflection point. Study of a function.

Integral
Definition of indefinite integral and its properties. Straightforward anti-derivatives. Integration by parts. Integration by substitution. Definite integral and its properties. The fundamental theorem of calculus. generalized integral. Area.

Differential Equations
Differential Equations: an introduction. Differential Equations of first and second order and Cauchy problems. Separate variables differential equations. Malthus model; bacterial growth; epidemic diffusion; radioactive decay. Logistic growth. Time of the crime.

"Elementi di Calcolo. Versione semplificata per i nuovi corsi di laurea"

di Paolo Marcellini e Carlo Sbordone

Liguori Editore.

Functions and number sets
Introduction: operations among sets. Function; domain, co-domain, image and graph of functions. Injective, surjective, inverse function and composition. Increasing and decreasing, odd and even functions, Number sets N, Z, Q, R.

Elementary functions
Review on lines, parabolas, exponential, logarithmic and trigoniometric functions. Absolute value. Neighborhood of a real number.

Limit and continuity
Finite and infinite limit; sign permanence theorem. Right-side and left-side limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and indeterminate forms. Infinite and infinitesimal. Vertical, horizontal and oblique asymptote. Continuous functions. Weierstrass theorem. Intermediate value theorem. Intermediate zero theorem.

Derivatives
Definition of derivative and its geometric interpretation. Calculation of derivatives. Differentiability and continuity. Point of non differentiability. Higher derivatives. Rolle's and Lagrange's theorem. De L’Hôpital's theorems. Taylor's theorem and McLaurin's expansion. Fermat's theorem. Maximum and minimum points. Convexity and concavity. Inflection point. Study of a function.

Integral
Definition of indefinite integral and its properties. Straightforward anti-derivatives. Integration by parts. Integration by substitution. Definite integral and its properties. The fundamental theorem of calculus. generalized integral. Area.

Differential Equations
Differential Equations: an introduction. Differential Equations of first and second order and Cauchy problems. Separate variables differential equations. Malthus model; bacterial growth; epidemic diffusion; radioactive decay. Logistic growth. Time of the crime.

"Elementi di Calcolo. Versione semplificata per i nuovi corsi di laurea"

di Paolo Marcellini e Carlo Sbordone

Liguori Editore.