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Graph the curve $ y = e^x $ in the viewing rectangles $ [-1, 1] $ by $ [0, 2] $, $ [-0.5, 0.5] $ by $ [0.5, 1.5] $, and $ [-0.1, 0.1] $ by $ [0.9, 1.1] $. What do you notice about the curve as you zoom in toward the point $ (0, 1) $?

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02:03

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Missouri State University

Oregon State University

Baylor University

University of Nottingham

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Graph the curve $ y = e^x …

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Okay here we have the craft of Y equals E. To the X. And we want to make some observations about the appearance of the graph as we zoom into the 0.0 comma one, the 0.0 common one, X zero Y. It was one, is this point right on the graph. So right now we would observe that white was E. T. X. Is an increasing function. Uh It's positive. Um And you know, informally we could say it's kind of bending or curving up. If we start to zoom in a little bit, We would still say looking at this .001, we would still say of course it's positive. It's still kind of curbing up. And as we continue to zoom in, I pay attention to this curve. Yeah. All right. Here's 2.0 comma one. Obviously the entire function is still positive and curving up. But notice how, you know now that we're really zooming in. Notice how it seems to flatten out a little bit if we zoom in a little bit more. All right. Now, if you look at the point as Euro comma one, you can see that it's really starting to flatten out. So when we were zoomed out, you could tell that this function was curving upwards. But as we zoom in closer and closer to the 0.0 comma one, the curve appears to flatten out. It's not really flat. You know, it's curving up but it appears to flatten out, resuming one more time. You can see that it almost gives you the mistaken impression that it's a straight line, which of course it's not. Um but as you zoom in, the current really does start to flatten out okay.

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