OUTLINE OF CALCULUS
(oNE vARIABLE)

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

2.1 DERIVATIVE: instantaneous rate of change

2.1.1 Definition, Notation dy/dx

2.1.2 Alternative Name (differential coefficient) and Notation

2.1.3 Derivative = slope of curve = slope of tangent to the curve

2.2 DERIVATIVE: examples, first principles

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

2.3 HIGHER ORDER DERIVATIVES AND NOTATION

2.3.1 Comment on Higher Order Derivatives

2.3.2 Examples - constant, x, x²

2.3.3 Theorem: If dy/dx = 0, then y = constant

2.4 CURVATURE AND RADIUS OF CURVATURE: plain curves

2.4.1 Definitions

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

[Working with rectangular coordinates is difficult.]

2.4.3 Curvature at a Point on a Curve is the Inverse of the Radius of the Circle that Most Closely Conforms to the Curve at the Point

2.5 INSTANTANEOUS VELOCITY AND ACCELERATION

2.5.1 Definition

2.5.2 Formulas

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

2.5.3 Examples

2.6 INTEGRAL

2.6.1 Definite Integral and Area

I₁_®2 = DEF ∫₁_®2 f dx = A₁_®2

2.6.2 Fundamental Theorem of Calculus

2.6.3 Examples: first principles

2.6.4 Velocity and Distance from the Acceleration

... and variations

2.6.5 Uniform Acceleration

2.6.6 Simple Harmonic Motion

3 THE MECHANICS OR "CALCULUS" OF DIFFERENTIATION

3.1 SUM, DIFFERENCE; PRODUCT, QUOTIENT

Derive 1/g

3.2 FUNCTION OF A FUNCTION

Derive f(ax + b)

3.3 INVERSE FUNCTIONS

Derive x^1/2 from x²

3.4 STANDARD DERIVATIVES

3.4.1 Powers and Polynomials

3.4.2 Logarithms and Exponentials

3.4.3 Trigonometric Functions

3.4.4 Hyperbolic Functions

3.4.5 Inverse Trigonometric and Hyperbolic Functions.

3.5 EXAMPLES

3.5.1 A Variety

3.5.2 Curvature Revisited

3.5.3 Other Applications

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

4.2 SUM AND DIFFERENCE

4.3 SUBSTITUTION

4.4 INTEGRATION BY PARTS

4.5 Standard integrals

4.6 EXAMPLES

4.6.1 A Variety

4.6.2 Simple Harmonic Motion

4.6.3 Other Applications