Functions

A brief introduction to functions and sets. These are
primitive notions, so the discussion is about notation and
and combining things.

Formal Derivative

The derivative of a function is defined in a formal way, with
some insight into the definitions provided by linear
approximations to the functions being considered.
Rules for differentiating sums, products, and powers are developed.

The Antiderivative

The antiderivative is defined as the inverse to the derivative.
Area under a curve plays an analogous role as the linear approximations
for differentiating. An informal discussion of the fundamental
theorem of calculus shows the relation between the integral and
the derivative.

Introduction to Trigonometry

The trignometric functions \(sin\), \(cos\), and \(tan\) are
defined. Heuristic arguments are given for the derivative of
\(sin\) and \(cos\), and then added to the
list of formal derivatives.

The Chain Rule

The chain rule is motivated by using linear approximations.
Then the derivative of \(1 / f \) is also motivated using linear
approximation. The division rule for differentiating
\( f/g \) is derived, and then the derivative of \( tan \) is computed.

Inverse Functions

Inverse functions are defined, and their derivative is computed.
This is applied to \( \arcsin \), \( \arccos \), and \( \arctan \).

Significance of the Derivative

Finding maxima and minima, Rolle's theorem, the intermediate
value theorem, if the derivative is 0, the function is constant.

Logarithm and Exponential

Define \( \log x as \int_1^x 1/t dt \).

Limits, continuous functions, completeness, minimize \( \epsilon - \delta \) arguments

matrices, determinants, ...