Estimate the gradient of a curve

﻿Hello. My name is Mr. Clasper, and today we're going to be learning how to estimate and interpret the gradient on a curve. Here is a distance-time graph. We're going to use the graph to estimate the gradient of a curve at the point where T all the time is equal to six seconds. So to do this, we need to draw a tangent on our curve at the point where T is equal to six. And we're and we're going to find the gradient of the tangent. So to find the gradient of the tangent, remember, we need to find a difference in Y, and divide this by the difference in X or this case, we're going to divide the distance by the time. This means that we're going to have a calculation of three divided by 4.5. This gives us a gradient of approximately 0.67. Now that we have this, what does the gradient tell you? Well, if we go back, we know that a gradient is calculated with a change in Y divided by a change in X. However, in our example, the Y axis represents our distance in metres, and the X axis represents our time in seconds. So our gradient is actually a change in distance divided by a change in time. And if we think back, distance divided by time is equal to speed. Therefore, our gradient in this example is equal to our speed. So this means that the object was travelling at 0.67 metres per second after six seconds. Let's have a look at these two graphs. What do you think the gradient represents, and what are the units for the gradients? Well, my first graph, the Y axis is represented with velocity, and the X axis represents time. Therefore, my gradient will be velocity divided by time, which is equal to acceleration. And in our case this would be metres per second squared. So if we had an estimate for a gradient on our curve, our answer would be in metres per second squared. If we look at our second graph, the Y axis is represented with temperature in degrees Celsius, and the X axis is represented by time in seconds. Therefore, we would calculate a change in temperature divided by a change in time. This means that the gradient of this graph will tell us what the temperature decreases by every second. And our units would be degrees Celsius per second. Here's a question for you to try. Pause the video to complete your task and click resume once you're finished. And here is the solution. So when we look at the graph, the depth of the water in the tank at the start is the point where T or time is equal to zero. So that means that the depth of water in the tank was three metres. Here is part B. Pause the video to complete your task and click resume once you're finished. And this is a solution to part B. So estimating the gradient at the point where T is equal to three. If we draw a gradient, we should find a gradient of approximately nine at this point on our graph. And here is part C. Pause the video to complete your task and click resume once you're finished. And here is your solution. So, because we have a gradient of approximately nine, this would mean that at the point where T is equal to three or at three seconds, this means that the depth of water is increasing at nine metres per second. Here's another question for you to try. Pause the video to complete your task, and click resume once you're finished. And here is your solution. So again, to estimate a gradient, you need to draw a tangent. So if you draw a tangent at the point where T is equal to four, you should find a gradient of approximately 0.35. And here is your last question. Pause the video to complete your task and click resume once you're finished. And here is your solution. So if we draw a tangent on our graph at the point where T is equal to 10, we should realise that this is negative. Therefore, my gradient must be negative when I calculate this. You should get a figure of approximately negative 0.625. And this brings us to the end of our lesson. So now you can estimate and interpret the gradient of a curve. Why not try out these skills with our exit quiz. I'll hopefully see you soon.

Estimate and interpret gradient

In this lesson, we will estimate and interpret a gradient for a given curve using triangles.

Lesson overview:Estimate and interpret the gradient of a curve

Further graphs (Gradients/Area of curves):