1.1 Rates of change in the world: a model is born

The focus of this textbook is understanding rates of change and how you can apply them to model real-world phenomena. Additionally, this textbook focuses on using equations with data, building both your competence and confidence to construct a mathematical model from data and a context.

Perhaps you analyzed rates of change in calculus course when answering the following types of questions:

• If \(y = xe^{-x}\), what is the derivative function \(f'(x)\)?
• What is the equation of the tangent line to \(y=x^{3}-x\) at \(a=1\)?
• Where is the graph of \(\sin(x)\) increasing at an increasing rate?
• What is the largest area that can be enclosed with 100 feet of fencing, with one side being along a wall?
• If you release a ball from the top of a skyscraper 500 meters above the ground, what is its speed when it impacts the ground?

The first three questions do not appear to be connected in a real-world context - but the last two questions do have some context. For the fencing problem, perhaps a person raises chickens and wants to care for their well-being, with a rectangular pen more aesthetically pleasing than a circular pen. In the last example the ball falling off the skyscraper assumes that acceleration of the ball is constant.

The context may reveal underlying assumptions or physical principles, which are the starting point to build a mathematical model. For the chicken coop problem the next step is to use the assumed geometry (rectangle) with the 100 feet of fencing to develop a function for the area as a function of the length of one of the sides of the pen. For the ball problem, the velocity (or the antiderivative of acceleration) can be found, from which the position function can be calculated through antidifferentiation.

Let’s say we have observational data and several different (perhaps conflicting) assumptions about the context at hand, and these assumptions describe models that involve rates of change. Which model is the best one to approximate the data? The short answer: it depends. To understand why, let’s take a look at a problem in context.