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Contents
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A Course in One Variable Calculus.
TDP
Contents
Real numbers.
Addition and Multiplication.
Ordering.
Intervals.
Absolute value.
Functions.
Generalities.
Injections.
Surjections.
Bijections.
Real functions of a real variable.
The graph of a function.
Monotonous functions.
Extremal points of a function.
Even functions and odd functions.
Sequences of real numbers.
What is a sequence of real numbers?
Bounded sequences.
Arithmetic sequences.
Geometric sequences.
Monotonous sequences.
Convergence and Divergence.
The algebra of limits.
Ordering and convergence.
Sequences not so far from being arithmetic/geometric.
is an affine function.
is an homographic function.
Limits of functions.
The general definition.
Limit at one real point.
Finite limit at infinity.
Infinite limit at infinity.
One-sided limit at one point.
Limits of functions and convergent sequences.
The algebra of limits.
Algebraic operations.
Numerator and denominator have both a limit equal to 0 at the given point.
Numerator and denominator have both an infinite limit at the given point.
Difference of two functions whose limit at some point is positive infinite.
Product of two functions with respectively an infinite limit and a limit equal to 0 at some point.
Powers.
A short catalogue.
Polynomial functions and rational functions.
Trigonometric functions.
Difference two square roots.
Logarithms and exponentials.
The sandwich theorem.
Vertical asymptotes.
Other asymptotes.
Generalizations.
Continuous functions.
Continuity at one point.
The algebra of continuous functions.
Removable discontinuity.
One-sided continuity.
Continuity on an interval.
Intermediate Value Theorem.
Invertible functions.
Powers and Roots.
Inverse Trigonometric Functions.
Differentiable functions.
Differentiability at one point.
The linear approximation of a function at one point.
First derivatives and tangents.
Differentiability on an interval.
The algebra of differentiable functions.
One-sided derivative at one point.
Continuity and differentiability.
Two global theorems.
Other applications of the (first) derivative.
Higher degree derivatives.
Convexity and concavity of a function.
Application to finding extremal points.
Minimax problems.
Parametric curves in the plane.
Curvature of a plane curve.
L'Hopital's rule.
Implicit differentiation.
The study of a function.
A short general flowchart.
Polynomial and absolute value.
On
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On
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Square root and rational function.
Polynomial and natural logarithm.
Rational function of an exponential.
Exponential of a rational function.
Arctangent of a rational function.
A trigonometric polynomial.
The supremum of two functions.
Integrals.
Primitives - The undefinite integral.
Methods of integration.
The usage of derivation formulas.
Substitution.
Useful substitutions.
Integration by parts.
Rational functions: the development into partial fractions.
Trigonometric polynomials.
Trigonometric rational functions.
Definite integral.
Properties of the definite integral.
The area under a graph.
Arc length.
Volumes of revolution.
Functions defined by integrals.
Improper integrals.
First type.
Second type.
Convergence theorems.
Series of real numbers.
Definition and examples.
The algebra of convergent series.
Series with non negative terms.
Alternating series.
Absolute convergence and conditional convergence.
Series of functions.
Pointwise convergence.
Uniform convergence.
Power series.
Some calculus with power series.
Taylor series.
Taylor polynomials.
Applications of Taylor polynomials.
Fourier series.
Logarithms and exponentials
The natural logarithm.
Exponentials.
Table of Integrals.
Integrals containing the form
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Integrals containing
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Integrals containing
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Integrals containing
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Integrals containing trigonometric functions.
About this document ...
Noah Dana-Picard 2007-12-28
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