presentation of Edwards & Penney's work, by Corry Shores
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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.]
Because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.
The solution to transcendental equations with the form f(x) = g(x) is the intersections of the graphs of the functions.
Again, because they are all transcendental equations, see the got-to-do entries for trigonometric, exponential, and logarithmic functions.
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.
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