Derivatives: Graphical Representations

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  • 0:06 Instantaneous Rate of Change
  • 1:50 Slopes as a Tangent
  • 2:42 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
Take a graphical look at the definitive element of calculus: the derivative. The slope of a function is the derivative, as you will see in this lesson.

Instantaneous Rate of Change

Average rate of change is zero
rate of change

Let's take another look at Super C, our human cannonball, and let's look specifically at his height as a function of time. Here I'm saying h=f(t). Now, for some specifics, what do we know? We know that his average rate of change is zero, because his end point and his start point are both zero. We also know, by Rolle's theorem, that the instantaneous rate of change somewhere along his path will also be zero. We know that somewhere along this path there's a tangent to this curve that's equal to zero, but how do we find that?

Well, let's start zooming in on some points. Let's look specifically at t=1. Now if we zoom in and want to estimate what Super C's velocity is at t=1, what we might do is see how high he is at t=1, which is f(1), and compare that to how high he is at some small time later, 1 + delta t. To find his average velocity between these two points, we would simply take his height at time 1 + delta t, subtract off his height at t=1 and divide that by delta t. This would give us a pretty good idea of his velocity, which is the slope, the tangent in particular, at t=1.

Slopes as a Tangent

As delta t goes to zero, the region grows smaller until the slope equals the tangent.
slope three

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