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Calculus (NEW)
First and second-year calculus course for university and college STEM majors. Contains topics ranging from limits, sequences & series to differentiation and integration.
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Course content
Logic
Introduction to Logic
THEORY
T
1.
On the content of Logic
THEORY
T
2.
On the content of Logic
Propositional logic
THEORY
T
1.
Propositions and truth
PRACTICE
P
2.
Propositions and truth
6
THEORY
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3.
Negation, conjunction, and disjunction
PRACTICE
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4.
Negation, conjunction, and disjunction
5
THEORY
T
5.
Implication
PRACTICE
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6.
Implication
7
THEORY
T
7.
Compound propositions
PRACTICE
P
8.
Compound propositions
5
THEORY
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9.
Propositions as variables
PRACTICE
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10.
Propositions as variables
11
THEORY
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11.
Truth tables
PRACTICE
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12.
Truth tables
7
THEORY
T
13.
Truth tables
THEORY
T
14.
Truth tables
Calculating with propositions
THEORY
T
1.
Equivalence
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P
2.
Equivalence
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T
3.
Rules of calculation for propositions
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4.
Rules of calculation for propositions
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T
5.
Order of operations
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6.
Order of operations
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THEORY
T
7.
Verum and Falsum
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8.
Verum and Falsum
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THEORY
T
9.
Truth tables
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Predicate logic
THEORY
T
1.
Logical quantifiers
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PRACTICE
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2.
Logical quantifiers
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3.
Reasoning by induction
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4.
Reasoning by induction
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Logical argumentation
THEORY
T
1.
Mathematical proofs
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2.
Mathematical proofs
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3.
Modus ponens
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4.
Modus ponens
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5.
Substitution in logic
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6.
Substitution in logic
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THEORY
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7.
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8.
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End of Logic
THEORY
T
1.
Conclusion of Logic
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Sets
Introduction
THEORY
T
1.
On the content of Sets
Sets
THEORY
T
1.
The notion of set
PRACTICE
P
2.
The notion of set
5
THEORY
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3.
Set builders
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4.
Set builders
6
THEORY
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5.
Subsets
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6.
Subsets
6
THEORY
T
7.
Intervals
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8.
Intervals
6
Operations on sets
THEORY
T
1.
Union and intersection of sets
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2.
Union and intersection of sets
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3.
Rules of calculation for sets
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4.
Rules of calculation for sets
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5.
Difference and complement of sets
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6.
Difference and complement of sets
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7.
Rules of calculation with set difference
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8.
Rules of calculation with set difference
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9.
Cartesian product
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10.
Cartesian product
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Relations
THEORY
T
1.
The notion of relation
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2.
The notion of relation
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3.
Equivalence relation
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4.
Equivalence relation
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5.
Graphs
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6.
Graphs
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7.
Functions
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8.
Functions
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Real numbers
THEORY
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1.
Properties of the real numbers
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2.
Properties of the real numbers
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3.
Ordering of real numbers
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4.
Ordering of real numbers
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5.
The notion of real number
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6.
The notion of real number
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7.
Ordering and decimal development
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8.
Ordering and decimal development
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9.
Decimal developments of real numbers
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10.
Decimal developments of real numbers
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11.
Calculating with real numbers
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12.
Calculating with real numbers
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End of Sets
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1.
Conclusion of Sets
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Functions
Introduction
THEORY
T
1.
On the content of Functions
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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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3.
Domain
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4.
Domain
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5.
Functions and graphs
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6.
Functions and graphs
6
THEORY
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7.
The range of a function
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8.
The range of a function
11
Operations for functions
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1.
Arithmetic operations on functions
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2.
Arithmetic operations on functions
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3.
Composition of functions
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4.
Composition of functions
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Range
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1.
Recognizing graphs
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2.
Recognizing graphs
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3.
Transformations of a graph
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4.
Transformations of a graph
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5.
Symmetry of functions
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6.
Symmetry of functions
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Injectivity
THEORY
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1.
Injectivity
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2.
Injectivity
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3.
Monotonic functions
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4.
Monotonic functions
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5.
The inverse of a function
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6.
The inverse of a function
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7.
Properties of inverse functions
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8.
Properties of inverse functions
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Applications
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1.
Power functions
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2.
Power functions
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3.
Power functions and equations
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4.
Power functions and equations
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5.
Equations and functions
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6.
Equations and functions
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7.
Substitution for equations
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8.
Substitution for equations
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End of Functions
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1.
Conclusion of Functions
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Polynomials and Rational Functions
Introduction to Polynomials and Rational Functions
THEORY
T
1.
On the content of Polynomials and Rational Functions
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1.
Linear functions
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2.
Linear functions
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3.
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5.
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6.
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7.
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8.
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Polynomials
THEORY
T
1.
The notion of polynomial
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2.
The notion of polynomial
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3.
Calculating with polynomials
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4.
Calculating with polynomials
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5.
Division with remainder for polynomials
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6.
Division with remainder for polynomials
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T
7.
Other representations of polynomials
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8.
Other representations of polynomials
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Greatest common divisor
THEORY
T
1.
The notion of gcd for polynomials
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PRACTICE
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2.
The notion of gcd for polynomials
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THEORY
T
3.
Rules of calculation for gcd of polynomials
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4.
Rules of calculation for gcd of polynomials
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T
5.
The Euclidean algorithm for polynomials
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6.
The Euclidean algorithm for polynomials
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7.
The notion of lcm for polynomials
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8.
The notion of lcm for polynomials
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T
9.
The extended Euclidean algorithm for polynomials
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10.
The extended Euclidean algorithm for polynomials
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Factorisation of polynomials
THEORY
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1.
Factors and zeros
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2.
Factors and zeros
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3.
Factorisation of polynomials
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4.
Factorisation of polynomials
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T
5.
The Fundamental Theorem of Algebra (real version)
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6.
The Fundamental Theorem of Algebra (real version)
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7.
Factorisation techniques
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8.
Factorisation techniques
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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.
Calculating with rational functions
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4.
Calculating with rational functions
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5.
Standard form for rational functions
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6.
Standard form for rational functions
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7.
Partial fraction decomposition of rational functions
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8.
Partial fraction decompositions of rational functions
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End of Polynomials and Rational Functions
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1.
Polynomial interpolation
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2.
Polynomial interpolation
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3.
Conclusion of Polynomials and Rational Functions
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Trigonometric Functions
Introduction to Trigonometric functions
THEORY
T
1.
On the content of Trigonometric functions
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The functions sine and cosine
THEORY
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1.
Unit circle and angles
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2.
Unit circle and angles
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3.
Sine and cosine
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4.
Sine and cosine
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5.
Sinusoids
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6.
Sinusoids
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7.
Right-angled triangles and trigonometric functions
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8.
Right-angled triangles and trigonometric functions
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9.
Symmetry of trigonometric functions
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10.
Symmetry of trigonometric functions
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Calculating with sine and cosine
THEORY
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1.
Special values of trigonometric functions
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2.
Special values of trigonometric functions
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Triangles and trigonometric functions
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Triangles and trigonometric functions
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More trigonometric functions
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1.
Tangent
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2.
Tangent
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Reciprocal trigonometric functions
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Reciprocal trigonometric functions
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5.
Inverses of trigonometric functions
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6.
Inverses of trigonometric functions
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End of Trigonometric Functions
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1.
Conclusion of trigonometric functions
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Exponential and Logarithmic Functions
Introduction to Exponential and Logarithmic functions
THEORY
T
1.
On the content of Exponential and Logarithmic Functions
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Definition Exp
THEORY
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1.
The notion of exponential function
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2.
The notion of exponential function
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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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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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3.
Rules of calculation for logarithms
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4.
Rules of calculation for logarithms
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5.
Equations with logarithms
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Equations with logarithms
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Growth
THEORY
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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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2.
Conclusion of Exponential and Logarithmic Functions
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Limits
Introduction to Limits
THEORY
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1.
On the content of Limits
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Definition of limit
THEORY
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1.
The notion of limit
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2.
The notion of limit
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3.
Limits and infinity
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4.
Limits and infinity
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Calculating with limits
THEORY
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1.
Limits of rational functions
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2.
Limits of rational functions
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3.
Rules of arithmetic calculation with limits
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4.
Rules of arithmetic calculation with limits
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5.
Composition rule for limits
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6.
Composition rule for limits
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7.
Sandwich rule for limits
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8.
Sandwich rule for limits
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Asymptotes
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1.
Vertical asymptotes
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2.
Vertical asymptotes
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Horizontal asymptotes
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Horizontal asymptotes
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5.
Oblique asymptotes
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6.
Oblique asymptotes
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Limits of some non-rational functions
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Limits of exponential functions
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2.
Limits of exponential functions
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Limits of trigonometric functions
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4.
Limits of trigonometric functions
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5.
Limits of inverse functions
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6.
Limits of inverse functions
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End of Limits
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1.
Conclusion of Limits
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Sequences and Series
Introduction to Sequences and Series
THEORY
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1.
On the content of Sequences and Series
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Definition of sequence and series
THEORY
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1.
The notions of sequence and series
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2.
The notions of sequence and series
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3.
Arithmetic sequences and series
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4.
Arithmetic series
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Geometric series
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Geometric series
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Limits of sequences
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Convergence of sequences
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Convergence of sequences
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Divergence of sequences
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Divergence of sequences
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Rules for limits of sequences
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Rules for limits of sequences
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Monotonic sequences
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Monotonic sequences
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Convergence of series
THEORY
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Absolute convergence and ratio test for series
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Absolute convergence and ratio test for series
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Alternating and comparison test for series
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Alternating and comparison test for series
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Condensation test for series
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Condensation test for series
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Power series
THEORY
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1.
The notion of power series
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2.
The notion of power series
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3.
The natural exponential function
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The natural exponential function
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5.
Limits involving exponential functions
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Limits involving exponential functions
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Completeness of the real numbers
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Infima and suprema for sets of real numbers
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2.
Infima and suprema for sets of real numbers
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3.
Limits and suprema
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Limits and suprema
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Cauchy sequences
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Cauchy sequences
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End of Sequences and Series
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Applications of Sequences and Series
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2.
Conclusion of Sequences and Series
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Continuity
Introduction to Continuity
THEORY
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On the content of Continuity
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Definition of continuity
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1.
The notion of continuity
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The notion of continuity
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Standard continuous functions
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Standard continuous functions
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Continuous extension
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Continuous extension
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Three theorems on continuous functions
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The Min-max Theorem
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2.
The Min-Max Theorem
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3.
The Intermediate Value Theorem
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The Intermediate Value Theorem
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5.
Uniform continuity
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Uniform continuity
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Limits and continuity
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Limits and continuous functions
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Limits and continuous functions
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3.
Rules for continuity
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Rules for continuity
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Continuity in geometry
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Curves in the plane
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Curves in the plane
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Length of a curve
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Length of a curve
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Regions
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Regions
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Area of a region
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Area of a region
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End of Continuity
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Conclusion of Continuity
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Differentiation
Introduction to Differentiation
THEORY
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1.
On the content of Differentiation
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Definition of differentiation
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1.
The notion of difference quotient
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The notion of difference quotient
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The notion of differentiation
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The notion of differentiation
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The derivative and the tangent
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The derivative and the tangent
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Calculating derivatives and tangent lines
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Derivative of a power function
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Derivative of a power function
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3.
Sum rule for differentiation
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Video Explanation (Sum rule)
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Sum rule for differentiation
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Product rule for differentiation
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Video Explanation (Product rule)
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Product rule for differentiation
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Chain rule for differentiation
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Video Explanation (Chain rule)
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Chain rule for differentiation
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Quotient rule for differentiation
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Video Explanation (Quotient rule)
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Quotient rule for differentiation
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Calculating tangent lines
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Calculating tangent lines
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Derivatives of special functions
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Derivatives of trigonometric functions
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Derivatives of trigonometric functions
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Derivatives of exponential functions
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Video Explanation (Derivatives of exponential functions)
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Derivatives of exponential functions
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Derivatives of inverse functions
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Derivatives of inverse functions
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Derivatives of logarithmic functions
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Derivatives of logarithmic functions
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End of Differentiation
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Summary of differentiation
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2.
The De L'Hôpital rule
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The De L'Hôpital rule
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4.
Conclusion of Differentiation
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Analysis of Functions
Introduction
THEORY
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1.
On the content of Analysis of Functions
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Minima and maxima
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1.
Local minima and maxima
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2.
Local minima and maxima
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3.
The Mean Value Theorem
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4.
The Mean Value Theorem
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Monotonicity
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Monotonicity
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Higher derivatives
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Higher derivatives
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Higher derivatives
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Applications of higher derivatives
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Applications of higher derivatives
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Implicit derivatives
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Implicit differentiation
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Implicit derivatives
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Derivatives of bivariate functions
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Derivatives of bivariate functions
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Implicit function theorem
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Implicit function theorem
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7.
Tangent line to a curve
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Tangent line to a curve
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Approximation with polynomials
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Linear approximation
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Linear approximation
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3.
Taylor series
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Taylor series
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5.
Taylor series of some known functions
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Taylor series of some known functions
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End of Analysis of Functions
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Applications of Analysis of Functions
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2.
Conclusion of Analysis of Functions
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Calculus: Integration
Introduction to Integration
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1.
On the content of Integration
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Antiderivatives
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1.
The notion of an antiderivative
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2.
The notion of an antiderivative
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Antiderivatives of some known functions
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Antiderivatives of some known functions
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Definite integrals
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Riemann sums
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Riemann sums
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3.
The definite integral of a function
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4.
The definite integral of a function
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5.
Rules of calculation for integrals
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Rules of calculation for integrals
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Properties of definite integrals
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Estimates of definite integrals
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Estimates of definite integrals
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3.
The Mean Value Theorem for integrals
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The Mean Value Theorem for integrals
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5.
The Fundamental Theorem of Calculus
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The Fundamental Theorem of Calculus
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Improper integrals
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Improper integrals
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Calculating with definite integrals
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Area between graphs
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Area between graphs
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Length of a curve revisited
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Length of a curve revisited
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Volume in space
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Volume in space
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Series and integrals
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Series and integrals
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Finding antiderivatives
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Substitution method
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Substitution method
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3.
Trigonometric integrals
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Trigonometric integrals
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Inverse substitution
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Inverse substitution
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Integration by parts
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Integration by parts
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Antiderivatives of rational functions
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Known antiderivatives of rational functions
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Known antiderivatives of rational functions
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3.
Fraction decomposition for integration
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Fraction decomposition for integration
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Existence of antiderivatives of rational functions
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Existence of antiderivatives of rational functions
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Finding antiderivatives of rational functions
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Finding antiderivatives of rational functions
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End of Integration
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Applications of Integration
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Conclusion of Integration
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