Missing function detail when zoomed out

0:00- [Presenter] What we're gonna do in this video is get a little bit of practice identifying when the graph does not tell you the full picture. So for example, the graph of F of X is equal to all of this business is shown below. Now, if you were to just look at the graph and if you weren't looking to inspect the actual definition of the function, you would say, okay, it looks like we have an asymptote right here in X equals zero.

0:24But then when you look at the actual definition of the function here, the asymptote isn't an X equals zero, X equaling zero isn't what would make the denominator equal to zero. It would be X is equal to 0.001. So that's actually where the asymptote is. So that's an example of you really have to look a little bit closer, zoom in a bunch to recognize that or analytically look at the function right over here to realize that the denominator does not equal zero when X equals zero, but at a slightly larger value.

0:56Now, we could look at another example where you could get in trouble when you don't inspect the definition of the function, or if you don't zoom in. Here we see what is true about F of X but cannot be determined from the graph. So you might guess that something like that is going to occur.

1:12When you look at the graph, it actually just looks like, it looks something like negative X to the fourth where it just gets close, it gets closer and closer to the x axis. Probably it looks like maybe at zero, it might touch it and then it keeps going down again.

1:27But when you look at the actual function itself, you see a potentially different picture, not even using calculus. If we set negative X to the fourth plus 0.04 X squared equals zero. If you wanted to see where it equals zero, you could factor in X squared or maybe even a negative X squared, and you would get X squared minus 0.04 is equal to zero.

1:56So this is going to happen at X equals zero or X is equal to plus or minus, X is equal to plus or minus 0.2. If you just solve this, X squared would be 0.04. And so X is plus or minus 0.02. So you see here that there's not even using calculus, that there's three zeros here.

2:18We're intersecting the X axis, we're touching the X axis three times. And then of course, if we wanted to figure out the minimum and maximum points, I could take the derivative, find out when those equals zero. I could think about where we are concave upwards or downwards, or we could even use this as a hint knowing that we have multiple zeros here, not just one.

2:37And we could zoom in on the graph and we would see this right over here. Then look, we have zeros at negative 0.20 and positive 0.2 just as we predicted. So it isn't just the graph does not look, doesn't look like a straight up X to the fourth that might look something like this.

3:00It actually has this slumpiness into it and it touches the X axis at multiple points. So the whole idea here is don't always just look at a graph and assume like you know what's going on. Sometimes either the function definition itself is telling a different story or a combination of that, or will give you a hint that maybe you should zoom in more to miss to at least see so you don't miss some of the finer detail.