Bounds on error terms: the big-O and little-o notations
Limits:
sequences
series
functions
Completeness of R:
Cauchy sequences
Absolutely convergent series
Nested intervals
inf and sup
Derivatives:
f(x+h)=f(x)+f'(x)+o(h)
rules for calculating derivatives:
the composition of two functions --- the chain rule
the sum of two functions
the product of two functions
the ratio of two functions
The exponential function:
you look for a function satisfying f'(x)=f(x)
e^x = 1+x+x^2/2+x^3/6+...
e^z = 1+z+z^2/2+z^3/6+... for z complex
geometric interpretation of e^z
Trigonometric functions:
sin(x) and cos(x)
Euler's formula
tan(x), sec(x), cotan(x), csc(x)
arctan(x), arcsin(x), arccos(x)
graphs of these functions
derivatives of these functions
Antiderivatives:
easy cases
integration by parts
function of ... times derivative of ...
substitution
trigonometric substitutions
decomposition of rational functions into simpler fractions
Integrals:
definition
calculating them with antiderivatives
improper integrals
Power series
Introduction to differential equations
The Laplace transform
