10 Transcendental Functions
10.1 Transcendental Functions
Transcendental functions are functions that cannot be expressed as finite polynomials. They include exponential, logarithmic, trigonometric, and inverse trigonometric functions, and are essential in advanced mathematics, physics, engineering, and applied sciences for modeling complex phenomena. Transcendental functions are used to model complex phenomena in science and engineering in the table Table 10.1.
| KeyConcept | Description | ExampleApplication |
|---|---|---|
| Exponential Functions | \(f(x) = e^x\) or \(a^x\), model growth and decay | RC circuit voltage: \(V(t) = V_0(1 - e^{-t/RC})\) |
| Logarithmic Functions | \(f(x) = \ln x\) or \(\log_a x\), used in scaling | Measuring pH, sound intensity |
| Trigonometric Functions | \(f(x) = \sin x, \cos x, \tan x\), model periodic behavior | Wave motion: \(y(x,t) = A \sin(kx - \omega t)\) |
| Inverse Trigonometric Functions | \(f(x) = \arcsin x, \arccos x, \arctan x\), solving angles | Population oscillations: \(P(t) = P_\text{avg} + A \cos(\omega t + \phi)\) |