NUMERICAL SETS
Definition of set and operations between sets. Definition of function. Injective, surjective, bijective and invertible functions. Monotonic functions. Composition between functions. Set of natural, integer, rational, real numbers and their properties.

ELEMENTARY FUNCTIONS
Exponentiation with natural and real exponent and its properties. Root extraction and its properties. Exponential function and its properties. Logarithm function and its properties. Trigonometric functions and properties. Inverse trigonometric functions. Function graphs.

LIMITS AND CONTINUITY FOR REAL FUNCTIONS OF A REAL VARIABLE
Limits of functions of a real variable and properties. Operations with limits. Remarkable limits. Continuity. Right limit and left limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and forms of indecision. Infinite and infinitesimal. Vertical, horizontal and oblique asymptotes. Continuous functions. Weierstrass theorem, the theorem of intermediate values, theorem of zeros.

DERIVATIVES FOR REAL FUNCTIONS OF A REAL VARIABLE
Definition of derivative and geometric interpretation of the derivative. Derivative of elementary functions. Differentiation rules. Derivative of composed function. Derivative of the inverse function. Rolle's theorem, Cauchy's theorem, Lagrange's theorem. Non-derivability. Critical points, monotony, concavity and convexity. The de l'Hôpital theorem. Study of the graph of a function. Taylor-MacLaurin formula with Peano remainder and with Lagrange remainder.

INTEGRAL
Definition of integral. Classes of integrable functions. Integral properties. Integral mean theorem. Fundamental theorem of integral calculus. Primitives and calculation of Riemann integrals. Immediate integrals by decomposition, by replacement. Integration of rational functions. Integration by parts. Integration of trigonometric functions. Integration of irrational functions.

DIFFERENTIAL EQUATIONS
The concept of differential equation. Cauchy problem. Cauchy-Lipschitz theorem and Peano theorem for ordinary first order differential equations. Ordinary linear differential equations of the first and second order homogeneous and non-homogeneous, with constant coefficients.

"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.

NUMERICAL SETS
Definition of set and operations between sets. Definition of function. Injective, surjective, bijective and invertible functions. Monotonic functions. Composition between functions. Set of natural, integer, rational, real numbers and their properties.

ELEMENTARY FUNCTIONS
Exponentiation with natural and real exponent and its properties. Root extraction and its properties. Exponential function and its properties. Logarithm function and its properties. Trigonometric functions and properties. Inverse trigonometric functions. Function graphs.

LIMITS AND CONTINUITY FOR REAL FUNCTIONS OF A REAL VARIABLE
Limits of functions of a real variable and properties. Operations with limits. Remarkable limits. Continuity. Right limit and left limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and forms of indecision. Infinite and infinitesimal. Vertical, horizontal and oblique asymptotes. Continuous functions. Weierstrass theorem, the theorem of intermediate values, theorem of zeros.

DERIVATIVES FOR REAL FUNCTIONS OF A REAL VARIABLE
Definition of derivative and geometric interpretation of the derivative. Derivative of elementary functions. Differentiation rules. Derivative of composed function. Derivative of the inverse function. Rolle's theorem, Cauchy's theorem, Lagrange's theorem. Non-derivability. Critical points, monotony, concavity and convexity. The de l'Hôpital theorem. Study of the graph of a function. Taylor-MacLaurin formula with Peano remainder and with Lagrange remainder.

INTEGRAL
Definition of integral. Classes of integrable functions. Integral properties. Integral mean theorem. Fundamental theorem of integral calculus. Primitives and calculation of Riemann integrals. Immediate integrals by decomposition, by replacement. Integration of rational functions. Integration by parts. Integration of trigonometric functions. Integration of irrational functions.

DIFFERENTIAL EQUATIONS
The concept of differential equation. Cauchy problem. Cauchy-Lipschitz theorem and Peano theorem for ordinary first order differential equations. Ordinary linear differential equations of the first and second order homogeneous and non-homogeneous, with constant coefficients.

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

di Paolo Marcellini e Carlo Sbordone

Liguori Editore.

NUMERICAL SETS
Definition of set and operations between sets. Definition of function. Injective, surjective, bijective and invertible functions. Monotonic functions. Composition between functions. Set of natural, integer, rational, real numbers and their properties.

ELEMENTARY FUNCTIONS
Exponentiation with natural and real exponent and its properties. Root extraction and its properties. Exponential function and its properties. Logarithm function and its properties. Trigonometric functions and properties. Inverse trigonometric functions. Function graphs.

LIMITS AND CONTINUITY FOR REAL FUNCTIONS OF A REAL VARIABLE
Limits of functions of a real variable and properties. Operations with limits. Remarkable limits. Continuity. Right limit and left limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and forms of indecision. Infinite and infinitesimal. Vertical, horizontal and oblique asymptotes. Continuous functions. Weierstrass theorem, the theorem of intermediate values, theorem of zeros.

DERIVATIVES FOR REAL FUNCTIONS OF A REAL VARIABLE
Definition of derivative and geometric interpretation of the derivative. Derivative of elementary functions. Differentiation rules. Derivative of composed function. Derivative of the inverse function. Rolle's theorem, Cauchy's theorem, Lagrange's theorem. Non-derivability. Critical points, monotony, concavity and convexity. The de l'Hôpital theorem. Study of the graph of a function. Taylor-MacLaurin formula with Peano remainder and with Lagrange remainder.

INTEGRAL
Definition of integral. Classes of integrable functions. Integral properties. Integral mean theorem. Fundamental theorem of integral calculus. Primitives and calculation of Riemann integrals. Immediate integrals by decomposition, by replacement. Integration of rational functions. Integration by parts. Integration of trigonometric functions. Integration of irrational functions.

DIFFERENTIAL EQUATIONS
The concept of differential equation. Cauchy problem. Cauchy-Lipschitz theorem and Peano theorem for ordinary first order differential equations. Ordinary linear differential equations of the first and second order homogeneous and non-homogeneous, with constant coefficients.

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

di Paolo Marcellini e Carlo Sbordone

Liguori Editore.