Every day, you look at rates of change in the context of different situations. There could be a word problem on the quiz, or ensuring you stay under the speed limit when driving 🚗. In any case, rates of change are important when discussing the movement of an object, person, or pet. Common rates of change you may be exposed to include speed and velocity. Before proceeding, make sure you know the difference between them.

Velocity

Consider an object that travels in a straight line. The average velocity over a time interval can be defined as the change in position over the change in time. In other words (visuals?):

where d is the displacement and t is the time interval. The triangle symbol ∆ (delta), just means change.

The problem with this equation when attempting to find the velocity of an object at a certain point in time, i.e. when a racecar crosses the finish line, is that there is no change in time. If we look only at one point, there is no time interval, meaning that the denominator is 0, and the velocity is ∞ or DNE. However, we know that the car does have velocity, so how do we find the answer?

A Clever Trick

Rather than scratching our heads wondering why this class is off to such a bad start (it's the first day!), let's generalize the above equation. Although the time interval can't be 0, what if we made it extremely small? If we find the average velocity when the time intervals are very low, won't they predict a pretty close value to the truth and ensure the fraction isn't invalid? Turns out, that's exactly what this chapter is about. 

Quick Problem

Before we get into the thick of it, let's review what we covered earlier by doing a problem. Remember, just manipulate the time interval to very small numbers, gradually approaching zero, so you're able to get a general sense of where the true answer will be. 

Q1: Find the missing average velocities. Then, estimate the Instantaneous Rate of Change at t = 0.7 seconds.

Limits

In the example above, we allowed our time intervals to shrink to near-zero, allowing us to find a point at which we can reasonably estimate the instantaneous velocity to be. So, we can say that the average velocity converges to the instantaneous velocity, or that the instantaneous velocity is the limit of the average velocity. Limits are the point at which the pattern of data seems to converge toward, and this is fundamental to many concepts in Calculus.

In the next article, we'll go over limits in much more depth and show you how to write their notation, calculate them, and help you understand why they're so important. Additionally, be sure to check our our socials down below (we have insta and tiktok) for study tips, time management, and more! Join our Discord to get problems, answer questions, and network with a growing community.