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Calculus: Functions (Part 1 of x)

Preface
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.

Introduction
This article will introduce functions.

There are many calculus references, the one I like to use is Calculus by Larson, Edwards and Hostetler.

This also assumes you are familiar with Python or can stumble your way through it.

Concepts
Variable: a symbol used to hold a value or coordinate.

Set: a combination of variables. In the form of (a,b,c,...,n) and usually (x,y) and (x,y,z).

Coordinate: a particular value taken on by a set used to indicate the position of a point, line, or plane.

Function: a real-valued function f maps X onto Y such that for every coordinate in X there is one and one only output on Y; where X and Y are sets of real numbers such that X=(a1,b1,c1,...,z1) and Y=(a2,b2,c2,...,z2).

Vector: A set of variables, often direction and magnitude.

Intercept: a coordinate at which one coordinate value in the set is zero (e.g. f(a,b,0,d) or g(0,b,c)).

Intersection: a coordinate at which two sets of data (or functions) intercept.

Graph: a visual representation of coordinate sets, most often in 2 to 4 variables.

Cartesian Coordinate System: a system in which the horizontal line on a graph is the x axis and the vertical is the y axis.

Slope: rate of change of a function m=(y2-y1)/(x2-x1), x1 != x2

Inverse: A function g is the inverse of the function f if f(g(x)) = x for each x in the domain of g AND g(f(x)) = x for each x in the domain of f. All inverse functions are reflective about the line y=x.

Vector Functions

General Form r(t)=f(t)i+g(t)j+h(t)k
Addition u(t) + v(t)=<ui + vi, uj + vj>
Scalar Product cu(t)=<cui,cuj>
Dot Product
Cross Product
Commutative u + v=v + u
Associative (u+v)+w=u+(v+w)
Additive Identity u+0=u
Additive Inverse u+(-u)=0
Distributive (c+d)u=cu+du
Distributive 2 c(u+v)=cu+cv
Scalar 1(u)=u
Scalar 2 0(u)=0

Where f, g and h are functions of t; r is a vector function of t; and i, j and k are vectors.

All the properties of the functions below can be components of a vector function. Therefore all the properties of the above properties apply to the functions below.

Linear Functions

General Form Ax+By+C=0
Vertical Line x=a
Horizontal Line y=b
Point-slope Form y-y1=m(x-x1)
Slope-intercept Form y=mx+b

Where A, B, C, y1, x1a and b are coordinates; m is the slope; x and y are variables.

Log Functions

General Form ln x = ∫ba (1/t) dt, for x>0
1 ln(1) = 0
Multiple ln(ab) = ln(a) + ln(b)
Power ln(an) = n ln a
Division ln(a/b) = ln(a) - ln(b)

Where a, b and n are coordinates; x is a variable.

Exponential Functions

General Form ex = f-1[ln(x)]
Inverse 1 ln(ex) = x
Inverse 2 eln(x) = x
Multiple eaeb=ea+b
Division ea/eb=ea-b

Where a, b and n are coordinates; x is a variable.

Function Transformations

Original Graph (Reference) y=f(x)
Horizontal Shift y=f(x±c)
Vertical Shift y=f(x)±c
Reflection (about x) y=-f(x)
Reflection (about y) y=f(-x)
Reflection (about origin) y=-f(-x)

Axis Intercepts
To determine where a function crosses the x or y axis evaluate the following function:

f(0) (for y-intercepts)
f(x)=0 (for x-intercepts)

If the above formula fails try using Newton's Method (not illustrated here).

Next Up
Continuity and Limits of functions.

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