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Calculus

Mathematics for university students. Contains polynomials, trigonometric functions, sequences and series, differentiation and more.

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Functions
Sets
THEORY
T
1.
The notion of set
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2.
The notion of set
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THEORY
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3.
Operations for sets
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4.
Operations for sets
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THEORY
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5.
Intervals
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6.
Intervals
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Functions
THEORY
T
1.
The notion of function
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2.
The notion of function
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THEORY
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3.
Operations for functions
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4.
Operations for functions
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Range
THEORY
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1.
The range of a function
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2.
The range of a function
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THEORY
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3.
Functions and graphs
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4.
Functions and graphs
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THEORY
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5.
Transformations of the axes
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6.
Transformations of the axes
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THEORY
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7.
Symmetry of Functions
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8.
Symmetry of functions
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Injectivity
THEORY
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1.
Injective functions
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2.
Injective functions
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3.
The inverse of a function
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4.
The inverse of a function
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THEORY
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5.
Power functions
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6.
Power functions
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THEORY
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7.
Equations and functions
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8.
Equations and functions
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End of Functions
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1.
Applications of Functions
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THEORY
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2.
List of known functions
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THEORY
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3.
End of Functions
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Polynomials and Rational Functions
Polynomials
THEORY
T
1.
The notion of polynomial
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2.
The notion of polynomial
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THEORY
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3.
Calculating with polynomials
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4.
Calculating with polynomials
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THEORY
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5.
Division with remainder for polynomials
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6.
Division with remainder for polynomials
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Linear Polynomials
THEORY
T
1.
Linear functions
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2.
Linear functions
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1.
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2.
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3.
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4.
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5.
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6.
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Factorization of Polynomials
THEORY
T
1.
The notions gcd and lcm for polynomials
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2.
The notions gcd and lcm for polynomials
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THEORY
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3.
Rules of calculation for gcd and lcm of polynomials
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4.
Rules of calculation for the lcm and gcd of polynomials
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THEORY
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5.
The Euclidean algorithm for polynomials
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6.
The Euclidean algorithm for polynomials
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THEORY
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7.
Factorization of polynomials
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8.
Factorization of polynomials
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THEORY
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9.
The Fundamental Theorem of Algebra (real version)
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10.
The Fundamental Theorem of Algebra (real version)
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THEORY
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11.
Polynomial interpolation
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12.
Polynomial interpolation
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THEORY
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13.
The extended Euclidean algorithm for polynomials
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14.
The extended Euclidean algorithm for polynomials
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Rational functions
THEORY
T
1.
The notion of rational function
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2.
The notion of rational function
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3.
Normal form for rational functions
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4.
Normal form for rational functions
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THEORY
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5.
Partial fraction decomposition for rational functions
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6.
Partial fraction decompositions of rational functions
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End of Polynomials and Rational Functions
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1.
Applications of Polynomials and Rational Functions
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THEORY
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2.
End of Polynomials and Rational Functions
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Trigonometric Functions
Basics
THEORY
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1.
Definitions of sin and cos
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2.
Definition of sin and cos
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THEORY
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3.
Right triangles and trigonometric functions
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4.
Right triangles and trigonometric functions
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THEORY
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5.
Periodicity of trigonometric functions
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6.
Periodicity of trigonometric functions
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Calculation
THEORY
T
1.
Special values of trigonometric functions
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2.
Special values of trigonometric functions
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THEORY
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3.
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4.
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THEORY
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5.
Triangles and trigonometric functions
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6.
Triangles and trigonometric functions
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More trigonometric functions
THEORY
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1.
Tangent and cotangent
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2.
Tangent and cotangent
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THEORY
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3.
Inverse trigonometric functions
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4.
Inverse trigonometric functions
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End of Trigonometric Functions
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1.
Applications of trigonometric functions
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THEORY
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2.
End of Trigonometric Functions
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Exponential and Logarithmic Functions
Definition Exp
THEORY
T
1.
The notion of exponential function
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2.
The notion of exponential function
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THEORY
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3.
Rules of calculation for exponential functions
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4.
Rules of calculation for exponential functions
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THEORY
T
5.
Equations with exponential functions
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6.
Equations with exponential functions
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Definition Log
THEORY
T
1.
The notion of logarithm
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2.
The notion of logarithm
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THEORY
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3.
Rules of calculation for logarithms
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4.
Rules of calculation for logarithms
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THEORY
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5.
Equations with logarithms
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6.
Equations with logarithms
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Growth
THEORY
T
1.
Exponential growth
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2.
Exponential growth
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End of Exponential and Logarithmic Functions
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1.
Applications of Exponential and Logarithmic Functions
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THEORY
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2.
End of Exponential and Logarithmic Functions
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Limits
Definition
THEORY
T
1.
The notion of limit
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2.
The notion of limit
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THEORY
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3.
The notion of limit and infinity
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4.
The notion of limit and infinity
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THEORY
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5.
Limits of rational functions
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6.
Limits of rational functions
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THEORY
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7.
Vertical asymptotes
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8.
Vertical asymptotes
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Rules for calculating limits
THEORY
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1.
Rules for limits
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2.
Rules for limits
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THEORY
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3.
Horizontal asymptotes
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4.
Horizontal asymptotes
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THEORY
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5.
Oblique asymptotes
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6.
Oblique asymptotes
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THEORY
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7.
Squeeze theorem for limits
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8.
Squeeze theorem for limits
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Exponential and trigonometric functions
THEORY
T
1.
Limits of exponential functions
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2.
Limits of exponential functions
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THEORY
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3.
Trigonometric limits
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4.
Trigonometric limits
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End of Limits
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1.
Applications of Limits
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THEORY
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2.
End of Limits
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Sequences and Series
Definition
THEORY
T
1.
The notions of sequence and series
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PRACTICE
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2.
The notions of sequence and series
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THEORY
T
3.
Arithmetic series
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4.
Arithmetic series
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THEORY
T
5.
Geometric series
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6.
Geometric series
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7.
The notions of sequence and series_copy
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Convergence
THEORY
T
1.
Convergence
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2.
Convergence
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THEORY
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3.
Monotonic sequences
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4.
Monotonic sequences
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THEORY
T
5.
Divergence
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6.
Divergence
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Rules
THEORY
T
1.
Rules for limits of sequences
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2.
Rules for limits of sequences
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Power Series
THEORY
T
1.
Power series
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2.
Power series
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THEORY
T
3.
Convergence criteria
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4.
Convergence criteria
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Length
THEORY
T
1.
Length
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2.
Length
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End of Sequences and Series
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1.
Applications of Sequences and Series
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THEORY
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2.
End of Sequences and Series
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Continuity
Definition of continuity
THEORY
T
1.
The notion of continuity
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2.
The notion of continuity
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THEORY
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3.
Global minimum and maximum
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4.
Global minimum and maximum
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THEORY
T
5.
Continuous extension
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6.
Continuous extension
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Min-max and Intermediate Value Theorem
THEORY
T
1.
The Min-Max Theorem
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2.
The Min-Max Theorem
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THEORY
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3.
Intermediate Value Theorem
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4.
Intermediate Value Theorem
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Limits
THEORY
T
1.
Limits of continuous functions
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2.
Limits of continuous functions
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THEORY
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3.
Rules for continuity
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4.
Rules for continuity
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End of Continuity
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1.
Applications of Continuity
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THEORY
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2.
End of continuity
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Differentiation
Definition
THEORY
T
1.
The notion of difference quotient
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2.
The notion of difference quotient
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THEORY
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3.
The notion of differentiation
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4.
The notion of differentiation
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THEORY
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5.
A simple derivative
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6.
A simple derivative
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Simple rules
THEORY
T
1.
The derivative of a sum function
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2.
The derivative of a sum function
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THEORY
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3.
The derivative of a polynomial
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4.
The derivative of a polynomial
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THEORY
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5.
The product rule for differentiation
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6.
The product rule for differentiation
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7.
Tangent lines
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More rules
THEORY
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1.
The chain rule for differentiation
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2.
The chain rule for differentiation
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THEORY
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3.
Derivatives of trigonometric functions
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4.
Derivatives of trigonometric functions
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THEORY
T
5.
The quotient rule for differentiation
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6.
The quotient rule for differentiation
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THEORY
T
7.
Derivatives of inverse functions
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8.
Derivatives of inverse functions
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Exp and Log
THEORY
T
1.
The natural logarithm
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2.
The natural logarithm
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THEORY
T
3.
Derivatives of exponential and logarithmic functions
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4.
Derivatives of exponential and logarithmic functions
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End of Differentiation
PRACTICE
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1.
Applications of differentiation
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THEORY
T
2.
End of Differentiation
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Analysis of Functions
Minima and maxima
THEORY
T
1.
Local minima and maxima
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2.
Local minima and maxima
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THEORY
T
3.
The Mean Value Theorem
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4.
The Mean Value Theorem
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THEORY
T
5.
Monotonicity
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6.
Monotonicity
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Higher derivatives
THEORY
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1.
Higher derivatives
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2.
Higher derivatives
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Implicit derivatives
THEORY
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1.
Implicit derivatives
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2.
Implicit derivatives
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Approximation with Polynomials
THEORY
T
1.
Linear approximation
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2.
Linear approximation
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THEORY
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3.
Taylor series
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4.
Taylor series
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THEORY
T
5.
Taylor series of some known functions
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6.
Taylor series of some known functions
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De L'Hôpital
THEORY
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1.
The De L'Hôpital rule
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2.
The De L'Hôpital rule
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End of Analysis of Functions
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1.
Applications of Analysis of Functions
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THEORY
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2.
End of Analysis of Functions
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Integration
Antiderivation
THEORY
T
1.
The notion of an antiderivative
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2.
The notion of an antiderivative
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THEORY
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3.
Antiderivatives of some known functions
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4.
Anti-derivatives of some known functions
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THEORY
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5.
Integration by parts
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6.
Integration by parts
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Area
THEORY
T
1.
Area
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2.
Area
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Integral
THEORY
T
1.
Riemann sums
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2.
Riemann sums
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THEORY
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3.
The integral of a function
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4.
The integral of a function
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THEORY
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5.
Rules of calculation for integrals
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6.
Rules of calculation for integrals
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Estimates
THEORY
T
1.
Estimates of integrals
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2.
Estimates of integrals
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THEORY
T
3.
Mean value theorem for integrals
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4.
Mean value theorem for Integrals
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The Fundamental Theorem of Calculus
THEORY
T
1.
The Fundamental Theorem of Calculus
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2.
The Fundamental Theorem of Calculus
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End of Integration
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1.
Applications of Integration
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THEORY
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2.
End of Integration
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