## 25 Jul 2012

### Transcendental Equations in Edwards & Penney's Calculus

presentation of Edwards & Penney's work, by Corry Shores
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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.]

Transcendental Equations in Edwards & Penney's Calculus

What do transcendental equations got to do with you?

Because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.

Brief Summary

The solution to transcendental equations with the form f(x) = g(x) is the intersections of the graphs of the functions.

Points Relative to Deleuze

Again, because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.

Summary of
Edwards & Penney
Calculus

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

Subsection 5: Transcendental Equations

We previously examined trigonometric, exponential, and logarithmic functions. These are all types of transcendental functions. Equations that include transcendental functions within them may have infinitely many solutions. Yet they might also have just a finite number of solutions. Edwards and Penney note one approach to dealing with transcendental equations. We might render them as

f(x) = g(x)

"where both the functions f and g are readily graphed." (p.40d) Wherever graphs y = f(x) and y = g(x) intersect are the solutions to the equation.

Example

Consider these graphs

There is a single point where graphs y = x and y = cos x. This means that the equation x = cos x has only one solution. The graphs also tell us that the solution lies within the interval (0, 1).

Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp.40-41.