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sowiso logo 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.
Available languages: 
ennl
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
T
3.
Negation, conjunction, and disjunction
PRACTICE
P
4.
Negation, conjunction, and disjunction
5
THEORY
T
5.
Implication
PRACTICE
P
6.
Implication
7
THEORY
T
7.
Compound propositions
PRACTICE
P
8.
Compound propositions
5
THEORY
T
9.
Propositions as variables
PRACTICE
P
10.
Propositions as variables
11
THEORY
T
11.
Truth tables
PRACTICE
P
12.
Truth tables
7
Calculating with propositions
THEORY
T
1.
Equivalence
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PRACTICE
P
2.
Equivalence
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THEORY
T
3.
Rules of calculation for propositions
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PRACTICE
P
4.
Rules of calculation for propositions
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THEORY
T
5.
Order of operations
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PRACTICE
P
6.
Order of operations
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THEORY
T
7.
Verum and Falsum
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PRACTICE
P
8.
Verum and Falsum
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Predicate logic
THEORY
T
1.
Logical quantifiers
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PRACTICE
P
2.
Logical quantifiers
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THEORY
T
3.
Reasoning by induction
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PRACTICE
P
4.
Reasoning by induction
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Logical argumentation
THEORY
T
1.
Mathematical proofs
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PRACTICE
P
2.
Mathematical proofs
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THEORY
T
3.
Modus ponens
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PRACTICE
P
4.
Modus ponens
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THEORY
T
5.
Substitution in logic
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PRACTICE
P
6.
Substitution in logic
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THEORY
T
7.
Proof by contradiction
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PRACTICE
P
8.
Proof by contradiction
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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
T
3.
Set builders
PRACTICE
P
4.
Set builders
6
THEORY
T
5.
Subsets
PRACTICE
P
6.
Subsets
6
THEORY
T
7.
Intervals
PRACTICE
P
8.
Intervals
6
Operations on sets
THEORY
T
1.
Union and intersection of sets
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PRACTICE
P
2.
Union and intersection of sets
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THEORY
T
3.
Rules of calculation for sets
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PRACTICE
P
4.
Rules of calculation for sets
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THEORY
T
5.
Difference and complement of sets
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PRACTICE
P
6.
Difference and complement of sets
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THEORY
T
7.
Rules of calculation with set difference
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PRACTICE
P
8.
Rules of calculation with set difference
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THEORY
T
9.
Cartesian product
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PRACTICE
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10.
Cartesian product
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Relations
THEORY
T
1.
The notion of relation
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PRACTICE
P
2.
The notion of relation
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THEORY
T
3.
Equivalence relation
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PRACTICE
P
4.
Equivalence relation
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THEORY
T
5.
Graphs
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PRACTICE
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6.
Graphs
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THEORY
T
7.
Functions
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PRACTICE
P
8.
Functions
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Real numbers
THEORY
T
1.
Properties of the real numbers
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PRACTICE
P
2.
Properties of the real numbers
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THEORY
T
3.
Ordering of real numbers
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PRACTICE
P
4.
Ordering of real numbers
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THEORY
T
5.
The notion of real number
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PRACTICE
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6.
The notion of real number
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THEORY
T
7.
Ordering and decimal development
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PRACTICE
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8.
Ordering and decimal development
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THEORY
T
9.
Decimal developments of real numbers
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PRACTICE
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10.
Decimal developments of real numbers
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THEORY
T
11.
Calculating with real numbers
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PRACTICE
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12.
Calculating with real numbers
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End of Sets
THEORY
T
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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PRACTICE
P
2.
The notion of function
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THEORY
T
3.
Domain
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PRACTICE
P
4.
Domain
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THEORY
T
5.
Functions and graphs
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PRACTICE
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6.
Functions and graphs
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THEORY
T
7.
The range of a function
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PRACTICE
P
8.
The range of a function
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Operations for functions
THEORY
T
1.
Arithmetic operations on functions
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PRACTICE
P
2.
Arithmetic operations on functions
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THEORY
T
3.
Composition of functions
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PRACTICE
P
4.
Composition of functions
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Range
THEORY
T
1.
Recognizing graphs
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PRACTICE
P
2.
Recognizing graphs
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THEORY
T
3.
Transformations of a graph
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PRACTICE
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4.
Transformations of a graph
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THEORY
T
5.
Symmetry of functions
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PRACTICE
P
6.
Symmetry of functions
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Injectivity
THEORY
T
1.
Injectivity
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PRACTICE
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2.
Injectivity
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THEORY
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3.
Monotonic functions
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PRACTICE
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4.
Monotonic functions
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THEORY
T
5.
The inverse of a function
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PRACTICE
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6.
The inverse of a function
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THEORY
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7.
Properties of inverse functions
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PRACTICE
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8.
Properties of inverse functions
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Applications
THEORY
T
1.
Power functions
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PRACTICE
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2.
Power functions
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THEORY
T
3.
Power functions and equations
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PRACTICE
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4.
Power functions and equations
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THEORY
T
5.
Equations and functions
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PRACTICE
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6.
Equations and functions
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THEORY
T
7.
Substitution for equations
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PRACTICE
P
8.
Substitution for equations
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End of Functions
THEORY
T
1.
Conclusion of Functions
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Polynomials and Rational Functions
Introduction to Pylonomials and Rational Functions
THEORY
T
1.
On the content of Polynomials and Rational Functions
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Quadratic polynomials
THEORY
T
1.
Linear functions
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PRACTICE
P
2.
Linear functions
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THEORY
T
3.
Quadratic functions
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PRACTICE
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4.
Quadratic functions
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THEORY
T
5.
Quadratic equations
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PRACTICE
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6.
Quadratic equations
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THEORY
T
7.
Quadratic inequalities
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PRACTICE
P
8.
Quadratic inequalities
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Polynomials
THEORY
T
1.
The notion of polynomial
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PRACTICE
P
2.
The notion of polynomial
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THEORY
T
3.
Calculating with polynomials
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PRACTICE
P
4.
Calculating with polynomials
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THEORY
T
5.
Division with remainder for polynomials
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PRACTICE
P
6.
Division with remainder for polynomials
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THEORY
T
7.
Other representations of polynomials
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PRACTICE
P
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
P
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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PRACTICE
P
4.
Rules of calculation for gcd of polynomials
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THEORY
T
5.
The Euclidean algorithm for polynomials
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PRACTICE
P
6.
The Euclidean algorithm for polynomials
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THEORY
T
7.
The notion of lcm for polynomials
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PRACTICE
P
8.
The notion of lcm for polynomials
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THEORY
T
9.
The extended Euclidean algorithm for polynomials
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PRACTICE
P
10.
The extended Euclidean algorithm for polynomials
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Factorisation of polynomials
THEORY
T
1.
Factors and zeros
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PRACTICE
P
2.
Factors and zeros
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THEORY
T
3.
Factorisation of polynomials
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PRACTICE
P
4.
Factorisation of polynomials
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THEORY
T
5.
The Fundamental Theorem of Algebra (real version)
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PRACTICE
P
6.
The Fundamental Theorem of Algebra (real version)
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THEORY
T
7.
Factorisation techniques
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PRACTICE
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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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PRACTICE
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2.
The notion of rational function
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THEORY
T
3.
Calculating with rational functions
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PRACTICE
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4.
Calculating with rational functions
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THEORY
T
5.
Standard form for rational functions
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PRACTICE
P
6.
Standard form for rational functions
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THEORY
T
7.
Partial fraction decomposition of rational functions
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PRACTICE
P
8.
Partial fraction decompositions of rational functions
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End of Polynomials and Rational Functions
THEORY
T
1.
Polynomial interpolation
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PRACTICE
P
2.
Polynomial interpolation
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THEORY
T
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
T
1.
Unit circle and angles
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PRACTICE
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2.
Unit circle and angles
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THEORY
T
3.
Sine and cosine
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PRACTICE
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4.
Sine and cosine
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THEORY
T
5.
Sinusoids
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PRACTICE
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6.
Sinusoids
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THEORY
T
7.
Right-angled triangles and trigonometric functions
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PRACTICE
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8.
Right-angled triangles and trigonometric functions
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THEORY
T
9.
Symmetry of trigonometric functions
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PRACTICE
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10.
Symmetry of trigonometric functions
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Calculating with sine and cosine
THEORY
T
1.
Special values of trigonometric functions
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PRACTICE
P
2.
Special values of trigonometric functions
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THEORY
T
3.
Addition formulas for trigonometric functions
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PRACTICE
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4.
Addition formulas for trigonometric functions
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THEORY
T
5.
Triangles and trigonometric functions
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PRACTICE
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6.
Triangles and trigonometric functions
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More trigonometric functions
THEORY
T
1.
Tangent
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PRACTICE
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2.
Tangent
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THEORY
T
3.
Reciprocal trigonometric functions
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PRACTICE
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4.
Reciprocal trigonometric functions
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THEORY
T
5.
Inverses of trigonometric functions
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PRACTICE
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6.
Inverses of trigonometric functions
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End of Trigonometric Functions
THEORY
T
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
T
1.
The notion of exponential function
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PRACTICE
P
2.
The notion of exponential function
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THEORY
T
3.
Rules of calculation for exponential functions
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PRACTICE
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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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PRACTICE
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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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PRACTICE
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2.
The notion of logarithm
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THEORY
T
3.
Rules of calculation for logarithms
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PRACTICE
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4.
Rules of calculation for logarithms
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THEORY
T
5.
Equations with logarithms
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PRACTICE
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6.
Equations with logarithms
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Growth
THEORY
T
1.
Exponential growth
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PRACTICE
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2.
Exponential growth
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End of Exponential and Logarithmic Functions
PRACTICE
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1.
Applications of Exponential and Logarithmic Functions
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THEORY
T
2.
Conclusion of Exponential and Logarithmic Functions
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Limits
Introduction to Limits
THEORY
T
1.
On the content of Limits
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Definition of limit
THEORY
T
1.
The notion of limit
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PRACTICE
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2.
The notion of limit
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THEORY
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3.
Limits and infinity
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PRACTICE
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4.
Limits and infinity
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Calculating with limits
THEORY
T
1.
Limits of rational functions
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PRACTICE
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2.
Limits of rational functions
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THEORY
T
3.
Rules of arithmetic calculation with limits
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PRACTICE
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4.
Rules of arithmetic calculation with limits
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THEORY
T
5.
Composition rule for limits
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PRACTICE
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6.
Composition rule for limits
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THEORY
T
7.
Sandwich rule for limits
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PRACTICE
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8.
Sandwich rule for limits
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Asymptotes
THEORY
T
1.
Vertical asymptotes
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2.
Vertical asymptotes
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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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Limits of some non-rational functions
THEORY
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1.
Limits of exponential functions
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2.
Limits of exponential functions
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THEORY
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3.
Limits of trigonometric functions
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4.
Limits of trigonometric functions
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THEORY
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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
THEORY
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1.
Conclusion of Limits
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Sequences and Series
Introduction to Sequences and Series
THEORY
T
1.
On the content of Sequences and Series
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Definition of sequence and series
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
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3.
Arithmetic sequences and series
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4.
Arithmetic series
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THEORY
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5.
Geometric series
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6.
Geometric series
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Limits of sequences
THEORY
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1.
Convergence of sequences
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2.
Convergence of sequences
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THEORY
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3.
Divergence of sequences
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4.
Divergence of sequences
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THEORY
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5.
Rules for limits of sequences
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Rules for limits of sequences
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THEORY
T
7.
Monotonic sequences
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8.
Monotonic sequences
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Convergence of series
THEORY
T
1.
Absolute convergence and ratio test for series
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2.
Absolute convergence and ratio test for series
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THEORY
T
3.
Alternating and comparison test for series
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4.
Alternating and comparison test for series
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THEORY
T
5.
Condensation test for series
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6.
Condensation test for series
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Power series
THEORY
T
1.
The notion of power series
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2.
The notion of power series
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THEORY
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3.
The natural exponential function
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4.
The natural exponential function
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THEORY
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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
THEORY
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1.
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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THEORY
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3.
Limits and suprema
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4.
Limits and suprema
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THEORY
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5.
Cauchy sequences
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6.
Cauchy sequences
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End of Sequences and Series
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Applications of Sequences and Series
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THEORY
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2.
Conclusion of Sequences and Series
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Continuity
Introduction to Continuity
THEORY
T
1.
On the content of Continuity
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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.
Standard continuous functions
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4.
Standard continuous functions
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THEORY
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5.
Continuous extension
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6.
Continuous extension
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Three theorems on continuous functions
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.
The Intermediate Value Theorem
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4.
The Intermediate Value Theorem
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THEORY
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5.
Uniform continuity
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6.
Uniform continuity
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Limits and continuity
THEORY
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1.
Limits and continuous functions
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2.
Limits and 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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Continuity in geometry
THEORY
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1.
Curves in the plane
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Curves in the plane
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THEORY
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3.
Length of a curve
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4.
Length of a curve
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THEORY
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5.
Regions
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Regions
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THEORY
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7.
Area of a region
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8.
Area of a region
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End of Continuity
THEORY
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1.
Conclusion of Continuity
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Differentiation
Introduction to Differentiation
THEORY
T
1.
On the content of Differentiation
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Definition of differentiation
THEORY
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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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The notion of differentiation
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THEORY
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5.
The derivative and the tangent
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6.
The derivative and the tangent
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Calculating derivatives and tangent lines
THEORY
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1.
Derivative of a power function
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2.
Derivative of a power function
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THEORY
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3.
Sum rule for differentiation
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THEORY
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4.
Video Explanation (Sum rule)
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Sum rule for differentiation
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Product rule for differentiation
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7.
Video Explanation (Product rule)
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8.
Product rule for differentiation
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THEORY
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Chain rule for differentiation
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10.
Video Explanation (Chain rule)
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11.
Chain rule for differentiation
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THEORY
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12.
Quotient rule for differentiation
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THEORY
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13.
Video Explanation (Quotient rule)
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14.
Quotient rule for differentiation
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THEORY
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15.
Calculating tangent lines
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16.
Calculating tangent lines
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Derivatives of special functions
THEORY
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1.
Derivatives of trigonometric functions
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Derivatives of trigonometric functions
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THEORY
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3.
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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6.
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
THEORY
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1.
Summary of differentiation
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THEORY
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2.
The De L'Hôpital rule
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3.
The De L'Hôpital rule
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THEORY
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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
THEORY
T
1.
Local minima and maxima
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PRACTICE
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2.
Local minima and maxima
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THEORY
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3.
The Mean Value Theorem
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PRACTICE
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4.
The Mean Value Theorem
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THEORY
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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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THEORY
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3.
Applications of higher derivatives
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4.
Applications of higher derivatives
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Implicit derivatives
THEORY
T
1.
Implicit differentiation
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PRACTICE
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2.
Implicit derivatives
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THEORY
T
3.
Derivatives of bivariate functions
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PRACTICE
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4.
Derivatives of bivariate functions
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THEORY
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5.
Implicit function theorem
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PRACTICE
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6.
Implicit function theorem
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THEORY
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7.
Tangent line to a curve
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8.
Tangent line to a curve
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Approximation with polynomials
THEORY
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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
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5.
Taylor series of some known functions
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6.
Taylor series of some known functions
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End of Analysis of Functions
PRACTICE
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1.
Applications of Analysis of Functions
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THEORY
T
2.
Conclusion of Analysis of Functions
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Integration
Introduction to Integration
THEORY
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1.
On the content of Integration
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Antiderivatives
THEORY
T
1.
The notion of an antiderivative
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PRACTICE
P
2.
The notion of an antiderivative
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THEORY
T
3.
Antiderivatives of some known functions
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PRACTICE
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4.
Antiderivatives of some known functions
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Definite integrals
THEORY
T
1.
Riemann sums
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2.
Riemann sums
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THEORY
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3.
The definite integral of a function
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PRACTICE
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4.
The definite integral of a function
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THEORY
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5.
Rules of calculation for integrals
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PRACTICE
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6.
Rules of calculation for integrals
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Properties of definite integrals
THEORY
T
1.
Estimates of definite integrals
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2.
Estimates of definite integrals
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THEORY
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3.
The Mean Value Theorem for integrals
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PRACTICE
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4.
The Mean Value Theorem for integrals
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THEORY
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5.
The Fundamental Theorem of Calculus
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6.
The Fundamental Theorem of Calculus
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THEORY
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7.
Improper integrals
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PRACTICE
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8.
Improper integrals
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Calculating with definite integrals
THEORY
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1.
Area between graphs
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2.
Area between graphs
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THEORY
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3.
Length of a curve revisited
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4.
Length of a curve revisited
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THEORY
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5.
Volume in space
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6.
Volume in space
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THEORY
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7.
Series and integrals
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PRACTICE
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8.
Series and integrals
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Finding antiderivatives
THEORY
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1.
Substitution method
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PRACTICE
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2.
Substitution method
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THEORY
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3.
Trigonometric integrals
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4.
Trigonometric integrals
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THEORY
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5.
Inverse substitution
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6.
Inverse substitution
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THEORY
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7.
Integration by parts
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8.
Integration by parts
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THEORY
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9.
Integration by parts, advanced usage
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PRACTICE
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10.
Integration by parts, advanced usage
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Antiderivatives of rational functions
THEORY
T
1.
Known antiderivatives of rational functions
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PRACTICE
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2.
Known antiderivatives of rational functions
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THEORY
T
3.
Fraction decomposition for integration
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PRACTICE
P
4.
Fraction decomposition for integration
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THEORY
T
5.
Existence of antiderivatives of rational functions
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PRACTICE
P
6.
Existence of antiderivatives of rational functions
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THEORY
T
7.
Finding antiderivatives of rational functions
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PRACTICE
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8.
Finding antiderivatives of rational functions
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End of Integration
PRACTICE
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1.
Applications of Integration
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THEORY
T
2.
Conclusion of Integration
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