Question

a function where the rate of change varies is called a ________ function.

Explanation:

In calculus, a function with a varying rate of change means its derivative (rate of change) is not constant. A nonlinear function (or more specifically, a non - linear function, and often in the context of calculus, a function whose graph is not a straight line, like a quadratic, cubic, etc., which have changing slopes) has a rate of change that varies. For example, for the function \(y = x^{2}\), the derivative \(y'=2x\) which changes as \(x\) changes, so the rate of change (slope) varies. The term for a function where the rate of change varies is a "non - linear" (or more precisely, in the context of calculus, a function that is not linear; linear functions have a constant rate of change, \(y = mx + b\) where \(m\) is constant). The most appropriate term here is "nonlinear" (or "non - linear"), but also, in the context of calculus, we can think of functions like polynomial functions (other than linear), exponential, logarithmic, etc., which are all non - linear and have varying rates of change. However, the standard term for a function with a varying rate of change (as opposed to linear functions which have constant rate of change) is a "nonlinear" function. Also, in calculus, we study functions with varying rates of change, and such functions are called non - linear functions (or more specifically, functions that are not linear).

Answer:

nonlinear (or non - linear)