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Hello, my name is Mr Clasper and today we're going to be estimating the gradient of a curve.

How would you find the gradient of the line? Well remember, the gradient tells us how steep a line is.

And to do this, we need to find a change in Y, and divide this by the change in X.

So if I pick two points on my line, and use this triangle to help me identify my change in Y and my change in X, I should be able to calculate my gradient.

If I look at my change in Y, I have a change of 30 units.

And my change in X is four.

Therefore to calculate my gradient, I would need to calculate 30 divided by four, and this would give me a gradient of 7.

5.

Here's an image of a curve.

Why can't you find the gradient of a curve? This is because the gradient at each point on the curve is different.

So as we move to different points on the curve, the gradient will change.

What we can do is estimate gradients on curves.

Let's have a look at this question.

We're going to estimate the gradient of the curve at the point where X is equal to two.

So the first thing I'm going to do, is to draw a tangent on my curve where X is equal to two.

Now using this tangent, because a tangent is a straight line, we could find the gradient of my tangent, and this will give me my estimate for the gradient at the point where X is equal to two on my curve.

So if I found my difference in Y, and my difference in X, I can calculate my gradient.

So the estimate for my gradient would be 7.

2 divided by 1.

8, which is four.

So this means that the gradient at the point where X is equal to two on my curve would be approximately four.

Here is a question for you to try.

Pause the video to complete your task, and click resume once you're finished.

And here are your solutions.

So remember for each part you need to draw a tangent and find the gradient of each of the tangents.

This should mean that you'll get an answer of approximately six for the gradient of part A, and for part B you should get a gradient of approximately eight.

Here is a 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 Max says the gradient at the curve at the point where X is equal to one is equal to three, and without working out anything, we can see that it's not correct, as the point where X is equal to one would have a negative gradient on our curve, therefore the answer can not have a positive gradient.

Here are parts B and C.

Pause the video to complete your task, and click resume once you're finished.

And here are your solutions.

So for part B, again your estimates are established from drawing tangents, so you need to draw a tangent at the point where X is equal to four, and a tangent at the point where X is equal to negative three, and this should give you approximate gradients of five and negative nine.

And for part C, the question was do you think the gradient at X is equal to 10 will be greater than at X is equal to four? Explain your reasons.

Well, the gradient at X is equal to 10 would be greater, and this is because the gradient is increasing as the value of X increases.

And that brings us to the end of our lesson.

So today you've learnt how to estimate the gradient of a curve.

Why not give the exit quiz a go, and show off your new skills.

I'll hopefully see you soon.