Introduction to FTC Part 2
Up until now we have thought about the FTC as one function as a whole. We've looked at graphs where we have found the area under the curve from time a to time b. From there, we split the area into equal subintervals. This allowed us to create rectangles under the curve to estimate the entire area, and this area tells us the total distance a ladybug crawls during a time interval. However, the second part of the FTC allows us to choose an arbitrary x value while using a new function, f(t). Why is this important?
What the equation above is telling us is that if we want to find the distance covered between a=0 seconds and some x between [a, b], we can take the derivative of the original equation we spoke about in part 1: the integral of the velocity from a to b seconds. Except this time, we will use x as our ending time instead of b. So what if we wanted to find how far the ladybug traveled between the midpoint of 10 seconds? That is, the distance it covered between 0 seconds and 5 seconds? Can we do this? Yes, but how does this formula work?
In this new equation, f(t) and dt act as "dummy variables" or placeholders until we actually plug in the value of x that we want. The "t" could also be represented by any other letter in the alphabet, and it would mean the exact same thing. So this new equation allows us to explore how taking the derivative of the integral of a function gives us the original function. Further, it also allows us to plug in the same beginning and end points to find the distance the ladybug covers, just as before!
Before we explore with an activity, let's cover some basic ideas: Derivatives, Limits, and Inverses.
To learn more about derivatives and limits, click here!