Determine whether the following functions are acceptable or...

Determine whether the following functions are acceptable or not as state functions over the indicated intervals.
a. $\frac{1}{x}(0, \infty)$
b. $e^{-2 x} \sinh x(0, \infty)$
c. $e^{-x} \cos x \quad(0, \infty)$
d. $e^{x}(-\infty, \infty)$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

State Function

A state function is a property of a system that depends only on the present state, not on the path taken to reach that state. In mathematics and physical sciences, state functions are associated with quantities whose differentials are exact, meaning that their integrals are path independent when moving between two states.

Domain and Continuity

For a function to be acceptable as a state function over a given interval, it must be well?defined and continuous over that interval. The domain must not include points of discontinuity or singular behavior that can lead to ambiguities in integrating or differentiating the function.

Differentiability and Exact Differentials

A key requirement for state functions is differentiability, which ensures that their differentials are exact. This property guarantees that the value change between any two states is unique regardless of the path taken, a critical aspect when a function represents an equilibrium or state property.

Single-valuedness

A state function must assign a unique value to each state, meaning it must be single?valued over its domain. This ensures consistency in defining the system's state and prevents ambiguities that could arise from functions with multiple values at the same point.

Transcript

Please give Ace some feedback