In this interlude, we will investigate a famous and simple way to model temperature using differential equations, known as “Newton’s Law of Cooling”. Although the situation is much more complicated in reality (see Heat Equation), we can already understand the calculus behind Newton’s Law of Cooling, which is a very useful approximation in many situations.
The notes provide an overview of how to understand what the differential equation for Newton’s Law of Cooling is telling us, and how we can use slope fields to visualize the similarities with exponential decay models, and then guess and check the correct form of the solutions.
In addition to the handout, there is a 12 minute YouTube video on the department YouTube channel which reviews the main ideas.
You can also use the Desmos graph below to experiment with the effect of changing the parameters ‘k’ and ‘a’ has on the slope field and behavior of solutions.