Slope of a Curve at a Point: Study with Video Lessons, Practice Problems & Examples

To calculate the slope of a curve, use the point method by drawing a tangent line that touches the curve at a single point. The slope of this tangent line represents the slope of the curve at that point. For example, if the rise is 2 and the run is 2, the slope simplifies to 1. This method highlights how slopes can change along a curve, emphasizing the importance of understanding marginal benefits and costs in economics, as well as the concept of equilibrium price in market analysis.

Calculating Slope of a Curve:Point Method

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.

Here’s what students ask on this topic:

What is the slope of a curve at a point?

The slope of a curve at a point is the rate at which the curve is changing at that specific point. To find this slope, you draw a tangent line that touches the curve at only one point. The slope of this tangent line represents the slope of the curve at that point. For example, if the rise is 2 and the run is 2, the slope simplifies to 1. This method is crucial in understanding concepts like marginal benefits and costs in economics.

How do you calculate the slope of a curve using the point method?

To calculate the slope of a curve using the point method, follow these steps: 1) Draw a tangent line that touches the curve at a single point. 2) Identify two points on the tangent line. 3) Calculate the rise (change in y) and the run (change in x) between these two points. 4) Use the formula for slope: $\frac{∆y}{∆x}$. For example, if the rise is 2 and the run is 2, the slope is 1.

Why is the slope of a curve important in economics?

The slope of a curve is important in economics because it helps in understanding marginal changes. For instance, the slope of a demand curve at a point can indicate the marginal benefit, while the slope of a supply curve can indicate the marginal cost. These concepts are essential for analyzing equilibrium prices and making informed economic decisions.

What is a tangent line and how is it used to find the slope of a curve?

A tangent line is a straight line that touches a curve at exactly one point without crossing it. To find the slope of a curve at a specific point, you draw a tangent line at that point. The slope of this tangent line, calculated as the rise over the run between two points on the line, represents the slope of the curve at that point.

Can the slope of a curve change at different points?

Yes, the slope of a curve can change at different points. Unlike a straight line, where the slope is constant, a curve's slope varies depending on the point at which it is measured. This variability is why it is important to use methods like drawing a tangent line to find the slope at a specific point on the curve.