The rate of change is a fundamental concept that helps us understand how one quantity changes with respect to another. In this context, we're interested in how the length of a metal rod changes as the temperature changes. This can give us insight into the material's properties.
To find the rate of change, we use derivatives. When dealing with functions, the derivative gives the rate at which one variable changes as another variable changes. In our case, the length of the rod, denoted by \(L\), is a function of temperature \(t\). By differentiating this function, we can determine \(\frac{dL}{dt}\), the rate of change of length with respect to temperature.
- Derivatives help measure how fast one quantity changes in relation to another.
- The formula used to find this rate is obtained by differentiating \(L\) with respect to \(t\).
This particular rate of change will tell us how many millimeters the rod's length changes for each degree increase in temperature.