Calculus

Good fit for first and second-year math courses for university STEM majors

Contains polynomials, trigonometric functions, sequences and series, differentiation and more.

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Course content

Chapter 1: Logic

• Propositional logic
1. Propositions and truth
2. Negation, conjunction, and disjunction
3. Implication
4. Compound propositions
5. Propositions as variables
6. Truth tables
• Calculating with propositions
1. Equivalence
2. Rules of calculation for propositions
3. Order of operations
4. Verum and Falsum
• Predicate logic
1. Logical quantifiers
2. Reasoning by induction
• Logical argumentation
1. Methamtical proofs
2. Modus ponens
3. Substitution in logic

Chapter 2: Sets

• Sets
1. The notion of set
2. Set builders
3. Subsets
4. Intervals
• Operations of sets
1. Union and intersection of sets
2. Rules of calculation of sets
3. Difference and complement of sets
4. Rules of calculation with set difference
5. Cartesian product
• Relations
1. The notion of relation
2. Equivalence relation
3. Graphs
4. Functions
• Real numbers
1. Properties of the real numbers
2. Ordering of real numbers
3. The notion of real number
4. Ordering and decimal development
5. Decimal developments of real numbers
6. Calculating with real numbers

Chapter 3: Functions

• Functions
1. The notion of function
2. Domain
3. Functions and graphs
4. The range of a function
• Operations for functions
1. Arithmetic operations on functions
2. Composition of functions
• Range
1. Recognizing graphs
2. Transformations of a graph
3. Symmetry of functions
• Injectivity
1. Injectivity
2. Monotonic functions
3. The inverse of a function
4. Properties of inverse functions
• Applications
1. Power functions
2. Power functions & equations
3. Equations and functions
4. Substitution of equations

Chapter 4: Polynomials & rational functions

1. Linear functions
• Polynomials
1. The notion of polynomial
2. Calculating with polynomials
3. Division with remainder for polynomials
4. Other representations of polynomials
• Greatest common divisor
1. The notion of gcd for polynomials
2. Rules of calculation for gcd of polynomials
3. The Euclidean algorithm of polynomials
4. The notion of lcm for polynomials
5. The extended Euclidean algorithm for polynomials
• Factorisation of polynomials
1. Factors and zeros
2. Factorisation of polynomials
3. The Fundamental Theorem of Algrebra
4. Factorisation techniques
• Rational functions
1. The notion of rational function
2. Calculating with rational functions
3. Standard form for rational functions
4. Partial fraction decomposition for rational functions
• Other
1. Polynomial interpolation

Chapter 5: Trigonometric functions

• The functions sine and cosine
1. Unit circle and angles
2. Sine and cosine
3. Sinusoids
4. Right-angled triangles and trigonometric funcitions
5. Symmetry of trigonometric functions
• Calculating with sine and cosine
1. Special values of trigonometric functions
2. Addition formulas for trigonometric functions
3. Triangles and trigonometric functions
• More trigonometric functions
1. Tangent
2. Reciprocal trigonometric functions
3. Inverses of trigonometric functions

Chapter 6: Exponential & logarithmic functions

• Definition exp
1. The notion of exponential function
2. Rules of calculation for exponential functions
3. Equations with exponential functions
• Definition log
1. The notion of logarithm
2. Rules of calculation for logarithms
3. Equations with logarithms
• Growth
1. Exponential growth

Chapter 7: Limits

• Definition of limit
1. The notion of limit
2. Limits and infinity
• Calculating with limits
1. Limits of rational functions
2. Rules of arithmetic calculation with limits
3. Composition rule for limits
4. Sandwich rule for limits
• Asymptotes
1. Vertical asymptotes
2. Horizontal asymptotes
3. Oblique asymptotes
• Limits of some non-rational functions
1. Limits of exponential functions
2. Limits of trigonometric functions
3. Limits of inverse functions

Chapter 8: Sequences & series

• Definition of sequence and series
1. The notions of sequence and series
2. Arithmetic sequence and series
3. Geometric series
• Limits of sequences
1. Convergence of sequences
2. Divergence of sequences
3. Rules for limits of sequences
4. Monotonic sequences
• Convergence of series
1. Absolute convergence and ratio test for series
2. Alternating and comparison test for series
3. Consideration test for series
• Power series
1. The notion of power series
2. The natural exponential function
3. Limits involving exponential functions
• Completeness of the real numbers
1. Infima and superma for sets of real numbers
2. Limits and superma
3. Cauchy sequences

Chapter 9: Continuity

• Definition of continuity
1. The notion of continuity
2. Standard continuous functions
3. Continuous extension
• Three theorems on continuous functions
1. The Min-Max Theorem
2. The Intermediate Value Theorem
3. Uniform continuity
• Limits & continuity
1. Limits and continuous functions
2. Rules for continuity
• Continuity in geometry
1. Curves in the plane
2. Length of a curve
3. Regions
4. Area of a region

Chapter 10: Differentiation

• Definition of differentiation
1. The notion of difference quotient
2. The notion of differentiation
3. The derivative and the tangent
• Calculating derivatives and tangent lines
1. Derivative of a power function
2. Sum rule for differentiation
3. Product rule for differentiation
4. Chain rule for differentiation
5. Quotient rule for differentiation
6. Calculating tangent lines
• Derivatives of special functions
1. Derivatives of trigonometric functions
2. Derivatives of exponential functions
3. Derivatives of inverse functions
4. Derivatives of logarithmic functions
• Other
1. The De L’Hôpital rule

Chapter 11: Analysis of functions

• Minima and maxima
1. Local minima and maxima
2. The Mean Value Theorem
3. Monotonocity
• Higher derivatives
1. Higher derivatives
2. Applications of higher derivatives
• Implicit derivatives
1. Implicit derivatives
2. Derivatives of bivariate functions
3. Implicit function theorem
4. Tangent line to a curve
• Approximation with polynomials
1. Linear approximation
2. Taylor series
3. Taylor series of some known functions

Chapter 12: integration

• Antiderivatives
1. The notion of an antiderivative
2. Antiderivatives of some known functions
• Definite integrals
1. Riemann sums
2. The definite integral of a function
3. Rules of calculation for integrals
• Properties of definite integrals
1. Estimates of definite integrals
2. The Mean Value Theorem for integrals
3. The Fundamental Theorem of Calculus
4. Improper integrals
• Calculating with definite integrals
1. Area between graphs
2. Length of a curve revisited
3. Volume in space
4. Series and integrals
• Finding antiderivatives
1. Substitution method
2. Trigonometric integrals
3. Inverse substitution
4. Integration by parts
5. Integration by parts, advanced usage
• Antiderivatives of rational functions
1. Known antiderivatives of rational functions
2. Fraction decomposition for integration
3. Existence of antiderivatives of rational functions
4. Finding antiderivatives of rational functions