**TDP**

- Contents
- Real numbers.

- 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.

- Limits of functions.
- The general definition.
- 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.

- Algebraic operations.
- A short catalogue.
- 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.

- 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.
- 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.
- 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.
- Definite integral.
- Properties of the definite integral.
- The area under a graph.
- Arc length.
- Volumes of revolution.
- Functions defined by integrals.
- Improper integrals.

- 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

- Table of Integrals.
- Integrals containing the form .
- Integrals containing .
- Integrals containing .
- Integrals containing .
- Integrals containing trigonometric functions.

- About this document ...

Noah Dana-Picard 2007-12-28