Lecture 2 | Gradient of a Line and a Curve

Gradient of a Straight Line

The steepness or the slope of a straight line, m, is its gradient. It can be found by the following formula:

m = change in y / change in x



You can see that the gradient of the above line is 2. It remains constant for any point on the line.

m = (2-0) / (1-0) = 2 / 1 = 2

Gradient of a Curve

Finding the gradient of a curve, however, is not easy; nor is it as straightforward as the above method, because its gradient is changing from point to point.

In order to address the issue, we can draw tangents to a curve and find the gradient of each tangent as the gradient of the curve at a particular point.

This method creates two of its own problems:

1.      It's not easy to draw a tangent to a curve with the aid of eye

2.      When you draw too many tangents on a grid to a curve, it's going to be messy and unpleasant to look at.

The following diagram illustrates the point:



I have drawn three tangents to find the gradients of the curve at the points, A, B and C; they are -4, -2 and 3.46 respectively. The diagram is already messy with just three tangents. You can imagine the scenario in the event of seeing too many of them on the grid.

With a tangent, however, the gradient of a curve can easily be found.

The following applet lets you practise - or play around - interactively by moving the tangent from point to point while calculating the gradient on the go.

Interactive Practice

Move the slider in order to change the position of the tangent. The slope of each tangent is automatically calculated - and updated.

Question:

What is the gradient of the curve when x = 1.5?