Matematica e Statistica, le basi per le scienze della vita, ||| edizione; M. Abate, MacGraw-Hill.
Learning Objectives
- Understanding the concept of function, especially in the fundamental case of a real-valued function of a real variable.
- Concepts of limit of a function and of asymptotic behavior (in terms of limits).
- Knowledge of the definitions of derivative and integral of a a real-valued function of a real variable.
- Knowledge of the basic concepts of statistics.
- Applying knowledge and understanding.
- Ability to describe the qualitative behavior of elementary functions and calculate derivative and integral.
- Knowing how to analyze a statistical population, for example by calculating the mean and standard deviation.
Prerequisites
Basic notion of high school mathematics courses. In particular: formal calculus, polynomials, algebraic equations and inequalities, elements of analytic geometry, and of trigonometry.
Teaching Methods
Lectures and exercise sessions.
Further information
See the web page http://www.dma.unifi.it/~poggiolini/didattica/2019-20-MateStat.php
Type of Assessment
The final exam consists in a written test and a subsequent oral exam. Access to the oral exam is reserved to students who have passed the written test.
Course program
- Intuitive set theory.
-Descriptive statistics
- Functions. Domain and range. Image and inverse image. Identity function and constant function. Injective, surjective and bijective. Inverse function. Restriction of a function. Composition of functions. monotone functions. Even and odd functions. central indexes and dispersion indices. Linear functions, and quadratic functions.
- Rational functions. Exponential and logarithmic functions. Periodic functions.
- Radiant definition. Sine and cosine of an angle: properties. Tangent and cotangent. Trigonometric formulas. Inverse trigonometric functions. Elementary trigonometric equations and inequalities.
- The absolute value of a number. Equations and inequalities with absolute value. Triangular inequality.
- Internal and border of a set. Accumulation points and isolated points. Closed and open sets. Circular neighborhood of a point. Left- and right-neighborhood. Formal definition of Limit. Notions of right- and left-limit.
- Limit uniqueness. Sandwich Theorem. Theorem about the existence of the limit for monotone functions. Calculating limits and indeterminate forms. Infinite and infinitesimal and principle of substitution.
- Theorem sign permanence. Intermediate Zero Theorem. Weierstrass Theorem. Intermediate Values Theorem.
- The problem of the tangent. The quotient. The derivative. Differentiability implies continuity. Left and right derivatives. Derivatives of elementary functions. Derivative of the sum, product, quotient of differentiable functions. Derivative of composite and inverse function.
- Tangent line at a given point. Horizontal, vertical and oblique asymptotes. Derivatives of inverse trigonometric functions. Non-differentiable points.
- Fermat Theorem. Theorems of Rolle, Lagrange and Cauchy. Maximum and minimum relative and absolute, concavity and convexity. Inflection points.
- L'Hôpital's rule.
- Study of functions.
- Antiderivatives of a function and their characterization. Indefinite Integral. Functions integrated on an interval: continuous functions can be integrated. Immediate and almost immediate integrals. Integration of rational functions. Integration by parts. Integration of square root function and of trigonometric functions.
- Partition of an interval. Integrable function on intervals and definition of definite integral. Main Properties. Integration by substitution. Integrability of monotone functions. Mean value Theorem for integrals. Fundamental theorem of calculus.
- Differential equations of 1° and 2° order.