24 Jul 2012

Powers of Transformation. Exponential Functions in Edwards & Penney's Calculus

Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

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[I your author am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.]

Powers of Transformation
Exponential Functions in Edwards & Penney's Calculus

What do exponential functions got to do with you?

As we are learning how to swim, there is a whole host of powers/abilities/capacities that we are acquiring. For example, we are becoming able to enjoy swimming at the beach, to learn watersports like waterpolo or competitive swimming, to be capable of saving a drowning person, to be fit for sailing, and so on. So in a sense, becoming a swimmer does not merely add a little to our powers, but raises our powers to a whole new level of expression. Then consider if we do take up sailing. As we are becoming a sailor, we then are rising to yet another even higher level; for, we can move great distances on the water, compete in races, have new experiences out at sea, and so forth. If we continue through a sequence of changes in which one increased power raises us up to a far greater level from which we can yet rise yet remarkably further, this is something like growing exponentially. Each increase gives us more powers. But also, each increase gives us more power to increase. So one way we might understand our continuous changing is that we go through a series of states. But maybe we can also judge the changes that we go through and that we put ourselves through on the basis of how they increase our ability to increase our abilities. Who we are changes over time, but there is a sort of 'constant', which is our constantly changing in power.

Brief Summary

In an exponential function, a constant base is raised to a variable power.

Points Relative to Deleuze

Our series of self-transformations is for Deleuze and Orson Welles like a series of forgers or fakes of oneself, yet these are self-creative forgeries or fakeries. Each such self-forgery is like an exponential power increase:

It is Welles who, beginning with The Lady from Shanghai, imposes one single character, the forger. But the forger exists only in series of forges who are his metamorphoses, because the power itself exists only in the form of a series of powers which are its exponents. (Deleuze, Cinema 2, 140b)

Summary of
Edwards & Penney

Chapter 1: Functions, Graphs, and Models
Section 1.4: Transcendental Functions

Subsection 3: Exponential Functions

Before examining exponential functions, we first will review power functions for contrast. In both cases of power functions and exponential functions, we speak of their 'form', which means we give explicit formulation to the categories of its component parts and of their relations. The form that power functions takes is

f (x) = xk (where k is a constant)

So in some specific case of a power function, the base is a variable that can take on one from a range of values, while the exponent is specified as some given numerical value.

Exponential functions, however, take this form:

f(x) = ax

We see that in exponential functions, the base is given as a constant, while the exponent is a variable that may take-on one of a range of values. Edwards and Penney write that in the case of power functions, the variable is raised to a constant power, while in exponential functions, a constant is raised to a variable power. (37c)

Below is a graph (made with geogebra) resembling the diagram in Edwards & Penney, and showing exponential functions y = 2x (blue) and y = 10x (red).

Example 6

Consider exponential functions when they have a base that is greater than one (base a > 1). Its value increases quite rapidly when the exponent x is large. Power functions, however, grow more slowly as x increases.

Edwards and Penney then have us consider smaller values for x2 and 2x. Seeing where their graphs overlap tell us the solutions to the equation x2 = 2x.

They also have us consider when the exponent in the exponential function is negative. The graphs for such functions fall from left to right.

The authors then compare the purposes of trigonometric and exponential functions. We use trigonometric functions to describe "periodic phenomena of ebb and flow;" however, we use exponential functions to describe "natural processes of steady growth or steady decline." (p38bc)

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.37-38.

Deleuze, Gilles. Cinema 2: The Time Image. Transl. Hugh Tomlinson and Robert Galeta. London & New York: 1989.

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