OUTLINE OF CALCULUS
(oNE vARIABLE)

ANIL MITRA PHD, COPYRIGHT © 2000, REFORMATTED July 1, 2003

# 1      CALCULUS: THE MATHEMATICS OF NON-UNIFORM VARIATION

Calculus is the mathematics of arbitrary functions

Consideration of functions in-the-small [limits, derivatives…] is a method of calculus rather than an end result; the power of the method is due to the similarity of an arbitrary but sufficiently well behaved function in the small to some linear function

# 2      FUNDAMENTALS

## 2.2           DERIVATIVE: examples, first principles

First principles are laborious. We need a "calculus".

## 2.4           CURVATURE AND RADIUS OF CURVATURE: plain curves

### 2.4.2 Formulas in Terms of Arc-length and Tangent (angle) - s, θ

[Working with rectangular coordinates is difficult.]

## 2.5           INSTANTANEOUS VELOCITY AND ACCELERATION

### 2.5.2 Formulas

v = dv/dy; a =dv/dt = d2y/dt2 = v2dv/dy

## 2.6           INTEGRAL

### 2.6.1 Definite Integral and Area

I1®2 = DEF ∫1®2 f dx = A1®2

### 2.6.4 Velocity and Distance from the Acceleration

... and variations

# 3      THE MECHANICS OR "CALCULUS" OF DIFFERENTIATION

Derive 1/g

Derive f(ax + b)

## 3.3           INVERSE FUNCTIONS

Derive x 1/2 from x2

# 4      MECHANICS OF INTEGRATION

Since integration is "harder" than integration the fundamental theorem of calculus is used to reduce integration to differentiation.

## 4.1           USE THE FUNDAMENTAL THEOREM OF CALCULUS

IF dF/dx = y, then

1®2 y dx = F