Alright. So now we're going to learn how to calculate the slope of a curve when it's not a straight line. So the idea here on this graph is to see a curve that's not straight, but in this situation, how do we calculate the slope? So when you see a curve like this, the slope is actually changing all the time. So you're going to have a different slope at this point, where it's rising pretty fast, than at this point where it's kind of going up a little slower, right? You would imagine from our last video that those would have 2 different slopes. So the first method when calculating the slope of a curve is to use what's called the point method, and what we do is we draw a tangent line. Right. We draw a tangent line. So a tangent line touches the curve at only one point. Okay. So the idea is we're going to calculate the slope of the line at that point on the graph. Cool. So once we draw the tangent line, we just calculate the slope of the tangent line and then we know what the slope is at that point. So, I'm going to go ahead and do my best to draw a tangent line. It's not very easy to do this by hand. If you were ever to have to calculate this in this class, I'm sure they would give you the tangent line already, and I'll do my best here. It should look something like that. So the idea is that it's only touching the graphical one point. Even if it doesn't look like it, from my example I did my best, but the idea is that it's only touching the graph right there. So the tangent line is just going go go and it touches the graph and it keeps going. Just one point that it touches the graph. So now that we have a tangent line, we can go ahead and calculate the slope of the tangent line and we will know the slope at that point. So, using our same method from finding the slope, let's find 2 points that intersect the graph, and it'll make it easier to calculate. So I see one there. Here's another one right here. Let's go ahead and calculate that slope. So it looks like from the first point to the second point we are going up. Right. Let me do this in a different color. I'll do it in green. It looks like we're going up and from that point to that point, we went up from 4 to 6. So it looks like our rise was 2, and let's do our run now. So it looks like we started with an x value of 3. We got to an x value of 5. So our run was also 2. So let's calculate this slope. So the slope of the tangent line, I'll put slope of tangent line equals still that rise over the run, and in this case, we've got a rise of 2, a run of 2, and that simplifies. 2 over 2 simplifies to 1. So the slope of the tangent line is 1, and that means that the slope at this point where we drew the tangent line right here, the slope of that point equals 1. So the slope of that curve at that point is 1. Remember it's constantly changing but at that point, the slope is 1. So that is how we calculate the slope of a curve, using the point method. Let's move on to the next video.
1. Reading and Understanding Graphs
Slope of a Curve at a Point