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Math 104: Calculus14 chapters | 116 lessons | 11 flashcard sets

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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.

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.

We could write this for any time that the slope is approximately equal to (*h*(*t* + *delta t*) - *h(t)*)/*delta t*. As *delta t* goes to zero, the region we're looking at grows smaller and smaller, and eventually the slope is going to equal the tangent. That's going to happen as *delta t* goes to zero. So we're going to write that the tangent at *h(t)* is equal to the limit as *delta t* approaches zero of (*h*(*t* + *delta t*) - *h(t)*)/*delta t*. This is the tangent to the curve; this is the derivative of *h(t)*.

One of the most fundamental parts of calculus is that the derivative of any point is equal to the tangent of the curve at that point. So when we go back to Super C, the tangent of his height as a function of time, and at any point along that curve, is equal to the derivative at that point.

If we find where the derivative equals zero - where the instantaneous rate of change equals zero - then this graph has a tangent equal to zero and the derivative is zero.

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Math 104: Calculus14 chapters | 116 lessons | 11 flashcard sets

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