Download MATHEMATICS Set theory. The sets and the operations between

Contents of Mathematics and Statistics
Degree in: Biosciences and Biotechnology
2014/2015
Teacher: Nadaniela Egidi
MATHEMATICS
Set theory. The sets and the operations between them. The De
Morgan’s laws. The numerical sets: N, Z, Q, R. The real line,
bounded or unbounded intervals, neighbourhoods. The union,
intersection, difference between sets, the complement of a set.
Functions. The domain, the codomain and the image of a function.
The injective, surjective and bijective functions. The composition
of functions, the inverse function. The graph of a function.
The
monotone
functions.
Increasing,
strictly
increasing,
decreasing or strictly decreasing functions and their properties.
The even functions and the odd functions.
The elementary functions. The n-th power function and the n-th
root function. The exponential function and its properties. The
logarithm function and its properties. The absolute value
function, linear functions and their properties. Graphs of
elementary functions
The trigonometric functions. The angles, their radian measure and
their graphical representation. The periodic functions, the
trigonometric
functions:
sin(x),
cos(x),
tan(x),
cot(x),
arcsin(x),
arccos(x),
arctan(x),
arccot(x).
The
fundamental
properties of trigonometric functions.
Limits of functions. Limits of functions, right-hand limit and
left-hand limit. Limits of elementary functions. Theorem on
uniqueness of the limit. Limits of power functions, root
functions,
exponential
functions,
logarithmic
functions,
trigonometric functions. Operations and limits. Indeterminate
forms and some techniques to solve them. Notable special limits.
Significant limits that converge to the Euler number. Significant
limits involving trigonometric functions. The orders of infinity.
Indeterminate forms where the base tends to infinity and the
exponent tends to zero.
Sequences and limits. Definition and examples of sequences,
subsequence. Limit of a sequence. Convergent, divergent or
indeterminate sequences. Properties of sequences.
The continuous functions. Definition of a continuous function at a
point and on an interval. Properties of continuity, examples of
continuous
functions,
examples
of
functions
that
are
not
continuous. The intermediate zero, the Weierstrass theorem, the
bisection method.
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The derivatives. Newton’s difference quotient, the definition of
derivative and notations used in the differentiation. Left
derivative and right derivative. Derivatives of the elementary
functions. The derivation rules of a product and of a quotient.
The chain rule formula, the derivative of the inverse function.
Examples of functions that are not differentiable. Higher order
derivatives,
Rolle’s
theorem,
Lagrange’s
theorem
and
De
L'Hopital’s theorem. Applications of the De L'Hopital’s theorem to
some significant limits, the treatment of some indeterminate
forms.
Geometrical meaning of the derivative and the equation of the
tangent line to the graph of a function, the derivative of the
inverse function of a given function.
The study of function. Determination of the domain of a function.
Determination of the sign of a function and the intersections of
the graph of the function with the coordinate axes. Limit of a
function, horizontal, vertical and oblique
asymptotes. Maximums
and minimums of a function that are relative or absolute.
Correlations between the first derivative of a function and the
maximums, the minimums, or monotonicity of the function. Concave
functions and convex functions in a range, inflection points of a
function. Link between the second derivative of a function to its
concavity and its inflections. Conditions sufficient to obtain the
maximum or minimum for a function.
The integral calculus. Definite integral: geometric meaning,
definition, additive property, linear
property and comparison
between integrals. The theorem of the mean value. The integral
function. The fundamental theorem of calculus. Primitives and
properties. Indefinite integrals and properties. Fundamental
formula of integral calculus. Well known indefinite integral.
Integration by substitution. Integration by parts. Improper
integrals with unbounded domain or unbounded integrating function.
Differential equations. Linear differential equations of the first
order. Cauchy problem for linear differential equations of the
first order, existence and uniqueness of the solution. First order
differential
equations
with
separable
variables.
Bernoulli
differential equation. Second order linear differential equations
with constant coefficients, the solution of the associated
homogeneous equation, and looking for particular solutions. Cauchy
problem for linear differential equations of second order with
constant coefficients.
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STATISTICS
Descriptive Statistics. Content, purpose, components and
terminology in Statistics. The phases of the statistical survey
Measurements scales (nominal, ordinal, interval, ratio) and data
types. Organization of qualitative or quantitative (discrete or
continuous) raw data in tables of frequency distribution.
Graphical representation of frequency distribution: bar charts,
pie charts, histograms, frequency polygons. Measures of central
tendency of variability and of shape: mean, geometric mean,
quadratic mean, harmonic mean, median, mode, mid-range, qquantile. Measures of variability: range, mean absolute deviation,
median absolute deviation, variance, sample variance, standard
deviation, sample standard deviation, coefficient of variation,
moment of order k. Measures of shape: symmetry, kurtosis.
Probability. Random experiment and event. Probability: frequentist
definition,
subjective
definition,
classical
definition
and
axiomatic definition. Probability theory: sample set, opposite
event, union event, intersection event, axioms. Theorem of the
probability
of
the
union
event,
conditional
probability,
statistically independent events, Bayes’ theorem.
Elementary combinatorics. Permutation of n, k-permutation of n,
k-combination of n, the binomial coefficients.
Discrete probability distributions. Probability distribution of a
discrete random variable: mean, variance, cumulative distribution
function. Discrete uniform distribution, binomial distribution,
multinomial distribution, Poisson distribution, hypergeometric
distribution.
Continuous probability distributions. Probability distribution of
a continuous random variable: probability density function,
cumulative distribution function, mean, variance. Continuous
uniform distribution, negative exponential distribution, normal
distribution, Student’s t-distribution, chi-square distribution,
standard normal distribution. Law of large numbers, central limit
theorem. A practical use of the standard normal distribution.
Tables for standard normal distribution, Student’s t-distribution,
chi-square distribution.
Statistical inference. Population, sample, parameter and
statistics. Sampling distributions: sampling distribution of the
mean; sampling distribution of the sample variance; sampling
distribution of the sample proportion; sampling distribution of
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the difference between means; sampling distribution of the
difference between proportions. Interval estimate of a population
parameter: confidence interval for a mean; confidence interval for
a proportion; confidence interval for a difference between means;
confidence interval for a difference between proportions.
Statistical hypothesis test: two-tailed test, one-tailed test on
the left, one-tailed test on the right, errors of type I and type
II, hypothesis test for a mean, hypothesis test for a proportion,
hypothesis test for a difference between means, hypothesis test
for a difference between proportions.
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