1. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 7.1 Gradient of a curve

2. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Gradient of a curve You will already be familiar with finding the gradient of a straight line. Identify the coordinates of two points on the line, (x 1, y 1 ) and (x 2, y 2 ). Gradient =

3. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Gradient of a curve The gradient at these three points (and all other points) is different. The gradient of a curve is not as straightforward. This is because the gradient of a curve is not constant, it changes.

4. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Gradient of a curve A B To understand how to find the gradient of a curve at a point (A), consider a straight line passing through A and another point B on the curve. It is clear from the diagram that the gradient of the line passing through AB has not got the same gradient as the curve at A. But what happens as B gets closer to A?

5. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Gradient of a curve A B B1B1 B2B2 As B gets closer to A the gradient of the straight line passing through A and B changes. So the gradient AB is different from the gradient AB 1, which in turn is different from the gradient AB 2. Importantly however, as B gets closer to A, the gradient of the straight line gets closer to that of the curve at A.

6. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Gradient of a curve A B Eventually, when B gets very close to A, we have in effect a tangent to the curve at A. Therefore, the gradient of the curve at A is equal to the gradient of the tangent drawn to the curve at A. The gradient at a point on a curve is equal to the gradient of the tangent to the curve at that point.