Transcendental functions

Definition

Transcendental functions are functions that are not algebraic and cannot be expressed as a finite combination of the basic arithmetic operations (addition, subtraction, multiplication, division) and root extractions. Common examples include exponential, logarithmic, and trigonometric functions.

5 Must Know Facts For Your Next Test

1. Exponential functions have the form $e^x$ or $a^x$, where $e$ is the base of natural logarithms and $a$ is a positive real number.
2. Logarithmic functions are the inverses of exponential functions and can be written as $\log_b(x)$, where $b$ is the base.
3. Trigonometric functions include sine ($\sin$), cosine ($\cos$), and tangent ($\tan$), which are periodic and important in modeling waveforms.
4. The derivatives of transcendental functions often involve other transcendental functions; for example, the derivative of $e^x$ is $e^x$, and the derivative of $\sin(x)$ is $\cos(x)$.
5. Transcendental equations cannot generally be solved using algebraic methods alone; they often require numerical methods or approximations.

Review Questions

• What distinguishes a transcendental function from an algebraic function?
• Give an example of an exponential function and its corresponding logarithmic function.
• How do you differentiate $\sin(x)$?

Related terms

Exponential Function:

$f(x) = e^x$ or $f(x) = a^x$, where $e \approx 2.71828$ and $a > 0$. These functions model growth processes.

Logarithmic Function:

$f(x) = \log_b(x)$ where $b > 0$. This function is the inverse of an exponential function with base $b$.

Trigonometric Function: $\sin(x)$, $\cos(x)$, and $\tan(x)$ represent periodic oscillations. These functions are fundamental in studying angles and waves.