B.4 Lab 1 Introduction: Approximate & Bound Error for df/dx

We're going to introduce a lab activity that is designed so you can explore a fundamental idea in Calculus before we develop it more formally. Essentially we're going to continue to improve our approximation of df/dx for a given function at a particular point, only now with the addition of quantifying our error bound so we know how accurate our approximation is. We'll also see how we can decrease our error to within a given tolerance. Cool stuff!

Quick Recap

So far in this Section focused on average rates of change, we have considered that rate for different functions and over intervals of various sizes. There actually hasn't been any Calculus involved; you could (and may) have computed such average rates of change in earlier courses.

It's now time to bring Calculus back into the picture, and return to the fundamental challenge of the previous Section, and actually of all of Calculus: how do we determine 𝑑𝑓𝑑𝑥∣𝑥=𝑎, the function f's instantaneous rate of change at 𝑥 =𝑎?

As you'll recall from that Section, we obtained successively better approximations to a particular function's instantaneous rate of change by anchoring the first point of the secant line at the point of interest, and then moving the second, "free" end of the secant line closer and closer to that point, thereby shrinking the interval. As we shrunk that interval, the resulting slopes gave us better and better approximations to the function's instantaneous rate of change at the point of interest.

Now that we have the concept of "average rate of change," we can be more systematic about our approach. Specifically, recall that the average rate of change for a function over an interval equals the slope of the secant line that passes through the initial and final points of that interval.

Hence, computing the slope of the secant line is the same as calculating the average rate of change.

This screen is essentially set-up; the next screen has full functionality.

We're now going to work through our first "lab," in which we'll systematically determine the instantaneous rate of change for a particular function, 𝑓(𝑥) =2𝑥, at 𝑥 =1, by considering the average rate of change of the function for various intervals around that point. On this screen we'll introduce the systematic steps you'll use, and then on the following screen you'll cycle through those steps to reach whatever level of accuracy you would like. The steps we'll use lie at the heart of Calculus, and we will return to them again and again.

You'll notice throughout the lab a light lab-notebook-type grid behind everything. This background is meant to remind you that we're exploring a single context throughout these activities.

This lab is based on work done by the CLEAR Calculus Project at Oklahoma State University. From the Project's website: "Project CLEAR Calculus is a research-based effort to make calculus conceptually accessible to more students while simultaneously increasing the coherence, rigor, and applicability of the content learned in the courses."
We at Matheno appreciate the work CLEAR Calculus has done to help students learn Calculus better, and are happy to build off of their efforts.