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Introduction

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

  • Complex Numbers
  • Multivariable functions
  • Polynomial approximation
  • Numerical analysis
  • Analytic foundation
    Limits, continuous functions, completeness, minimize \( \epsilon - \delta \) arguments
  • Differential equations
  • Linear algebra
    matrices, determinants, ...
  • Probability
  • Line and Surface integrals