## 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?