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) $?
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.