Mathematics I

1. Numbers
a. Natural, integer, rational, and real numbers
b. Properties of real numbers. Completeness axiom of ℝ
c. Decimal representation of real numbers
d. Bounded and unbounded sets of real numbers. Supremum and infimum of a set. Maximum and minimum. The symbol "infinity". Bounded and unbounded intervals
e. n-th root of a non-negative real number. Absolute value of a real number
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2. Introduction to Real Functions of a Real Variable
a. Domain, image, graph, sign, and zeros of a function
b. Graphs of elementary functions: first- and second-degree polynomials, absolute value of x, square root of x, sign function of x, integer part of x. Power functions with natural, integer, rational, and real exponents. Trigonometric functions
c. Injective functions. Inverse function. Monotonic functions. Relationship between monotonic and injective functions. Concave and convex functions
d. Upper and lower bounded functions. Supremum, infimum, absolute maximum and minimum of a function over a set
e. Elementary operations with functions. Composition of functions. Translations, symmetries. Even and odd functions. Graphs of elementary functions composed with translations and symmetries
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3. Sequences
a. Neighborhood of a real number. Right and left neighborhoods. Neighborhood of ±infinity. Isolated point, accumulation point
b. Bounded and unbounded numerical sequences. Monotonic sequences. Convergent and divergent sequences. Regular and indeterminate sequences. Infinitesimal sequences
c. Theorems on limits of sequences: uniqueness of limit for convergent sequences (*), convergent sequences are bounded (*), monotonic sequences are regular (*), permanence of sign(*), comparison theorem (*), product of a bounded sequence and an infinitesimal sequence
d. Operations with limits of sequences and indeterminate forms. Euler's number (e). Limit tests: ratio test, n-th root test
e. Exponential and logarithmic functions. Comparison between infinite sequences: factorials, powers, exponentials, logarithms. Asymptotic Landau notation
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4. Limits of Functions
a. Right/left-hand limits, from above/below. Horizontal and vertical asymptotes. Relationship between limits of functions and limits of sequences. Limits of composed functions
b. Asymptotic Landau notation for functions. Notable limits and first-order expansions with little-o notation. Oblique asymptotes
c. Continuity of elementary functions. Classification of discontinuity points. Bolzano's theorem (intermediate value theorem). Darboux's theorem (intermediate value property). Weierstrass's theorem.
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5. Differential Calculus
a. Functions differentiable at a point. Geometric meaning of the derivative. Tangent line to the graph of a function at a differentiable point (*). Relationship between differentiability and continuity (*). Derivatives of elementary functions
b. Non-differentiable functions at a point: examples of vertical tangents, corners, cusps
c. Differentiation rules (sum, difference, product, quotient). Derivative of a composite function. Differentiability of the inverse function and its derivative
d. Sufficient condition for differentiability at a point: continuity + equality of (finite) left and right limits of the derivative
e. Relative maximum and minimum points
f. Fermat's Theorem (*). Rolle's Theorem (*). Lagrange's Theorem (Mean Value Theorem) (*)
g. Consequences of Lagrange's Theorem: functions with zero derivative on an interval are constant(*)functions with identical derivatives on an interval differ by an additive constant (*)relationship between monotonicity and the sign of the derivative on an interval (*)
h. Cauchy's Theorem
i. Higher-order derivatives. Relationship between concavity/convexity and the sign of the second derivative. Graph of a function
j. De L'Hôpital's Rule
k. Taylor's formula with Peano and Lagrange remainders
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6. Integral Calculus
a. Indefinite integral. Antiderivative of a function. Some elementary antiderivatives
b. Properties of the indefinite integral. Definite integrals and areas. Properties of the definite integral. Integration methods: by parts and by substitution. Integration of rational functions
c. Mean Value Theorem for integrals (*). Fundamental Theorem of Integral Calculus (*). Fundamental Formula of Integral Calculus (*).
d. Improper integrals over bounded but not closed intervals and closed but unbounded intervals. Criteria for improper integrability of non-negative functions: comparison test, asymptotic comparison test. Absolute convergence of improper integrals
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7. Numerical Series
a. Geometric series. Harmonic series. Generalized harmonic series
b. Series with positive terms, convergence tests: comparison test, asymptotic comparison test, n-th root test, ratio test
c. Series with terms of any sign. Absolute convergence. Leibniz's test for alternating series. Absolute convergence criterion
d. Numerical series and improper integrals
e. Power series. Radius of convergence and determination criteria. Derived series of a power series. Power series with non-zero radius and differentiability of any order of the sum function. Term-by-term differentiation and integration of a power series with non-zero radius
f. Introduction to analytic functions or functions expandable into Taylor series. Sufficient criterion for expandability into a Taylor series. Examples of analytic functions. Comparison between formula and Taylor series
g. Introduction to Fourier series

1. Numbers
a. Natural, integer, rational, and real numbers
b. Properties of real numbers. Completeness axiom of ℝ
c. Decimal representation of real numbers
d. Bounded and unbounded sets of real numbers. Supremum and infimum of a set. Maximum and minimum. The symbol "infinity". Bounded and unbounded intervals
e. n-th root of a non-negative real number. Absolute value of a real number

2. Introduction to Real Functions of a Real Variable
a. Domain, image, graph, sign, and zeros of a function
b. Graphs of elementary functions: first- and second-degree polynomials, absolute value of x, square root of x, sign function of x, integer part of x. Power functions with natural, integer, rational, and real exponents. Trigonometric functions
c. Injective functions. Inverse function. Monotonic functions. Relationship between monotonic and injective functions. Concave and convex functions
d. Upper and lower bounded functions. Supremum, infimum, absolute maximum and minimum of a function over a set
e. Elementary operations with functions. Composition of functions. Translations, symmetries. Even and odd functions. Graphs of elementary functions composed with translations and symmetries

3. Sequences
a. Neighborhood of a real number. Right and left neighborhoods. Neighborhood of ±infinity. Isolated point, accumulation point
b. Bounded and unbounded numerical sequences. Monotonic sequences. Convergent and divergent sequences. Regular and indeterminate sequences. Infinitesimal sequences
c. Theorems on limits of sequences: uniqueness of limit for convergent sequences (*), convergent sequences are bounded (*), monotonic sequences are regular (*), permanence of sign(*), comparison theorem (*), product of a bounded sequence and an infinitesimal sequence
d. Operations with limits of sequences and indeterminate forms. Euler's number (e). Limit tests: ratio test, n-th root test
e. Exponential and logarithmic functions. Comparison between infinite sequences: factorials, powers, exponentials, logarithms. Asymptotic Landau notation

4. Limits of Functions
a. Right/left-hand limits, from above/below. Horizontal and vertical asymptotes. Relationship between limits of functions and limits of sequences. Limits of composed functions
b. Asymptotic Landau notation for functions. Notable limits and first-order expansions with little-o notation. Oblique asymptotes
c. Continuity of elementary functions. Classification of discontinuity points. Bolzano's theorem (intermediate value theorem). Darboux's theorem (intermediate value property). Weierstrass's theorem.

5. Differential Calculus
a. Functions differentiable at a point. Geometric meaning of the derivative. Tangent line to the graph of a function at a differentiable point (*). Relationship between differentiability and continuity (*). Derivatives of elementary functions
b. Non-differentiable functions at a point: examples of vertical tangents, corners, cusps
c. Differentiation rules (sum, difference, product, quotient). Derivative of a composite function. Differentiability of the inverse function and its derivative
d. Sufficient condition for differentiability at a point: continuity + equality of (finite) left and right limits of the derivative
e. Relative maximum and minimum points
f. Fermat's Theorem (*). Rolle's Theorem (*). Lagrange's Theorem (Mean Value Theorem) (*)
g. Consequences of Lagrange's Theorem: functions with zero derivative on an interval are constant(*)functions with identical derivatives on an interval differ by an additive constant (*)relationship between monotonicity and the sign of the derivative on an interval (*)
h. Cauchy's Theorem
i. Higher-order derivatives. Relationship between concavity/convexity and the sign of the second derivative. Graph of a function
j. De L'Hôpital's Rule
k. Taylor's formula with Peano and Lagrange remainders

6. Integral Calculus
a. Indefinite integral. Antiderivative of a function. Some elementary antiderivatives
b. Properties of the indefinite integral. Definite integrals and areas. Properties of the definite integral. Integration methods: by parts and by substitution. Integration of rational functions
c. Mean Value Theorem for integrals (*). Fundamental Theorem of Integral Calculus (*). Fundamental Formula of Integral Calculus (*).
d. Improper integrals over bounded but not closed intervals and closed but unbounded intervals. Criteria for improper integrability of non-negative functions: comparison test, asymptotic comparison test. Absolute convergence of improper integrals

7. Numerical Series
a. Geometric series. Harmonic series. Generalized harmonic series
b. Series with positive terms, convergence tests: comparison test, asymptotic comparison test, n-th root test, ratio test
c. Series with terms of any sign. Absolute convergence. Leibniz's test for alternating series. Absolute convergence criterion
d. Numerical series and improper integrals
e. Power series. Radius of convergence and determination criteria. Derived series of a power series. Power series with non-zero radius and differentiability of any order of the sum function. Term-by-term differentiation and integration of a power series with non-zero radius
f. Introduction to analytic functions or functions expandable into Taylor series. Sufficient criterion for expandability into a Taylor series. Examples of analytic functions. Comparison between formula and Taylor series
g. Introduction to Fourier series