How to Find the Gradient of a Curve by Drawing a Tangent?

To find the gradient of a curve at a specific point by drawing a tangent, you follow these steps:

1. Identify the Point: Locate the point on the curve where you want to determine the gradient.

2. Draw the Tangent: Carefully draw a straight line that touches the curve only at the point of interest. This line is the tangent to the curve at that point. Imagine a ruler balanced at the point; that's the idea. The tangent should approximate the slope of the curve very closely near that point.

3. Choose Two Points on the Tangent Line: Select two distinct points on the tangent line that are easy to read off the graph. These points should be relatively far apart to minimize errors in reading their coordinates.

4. Determine the Coordinates: Note the (x, y) coordinates of the two points you selected. Let's call them (x₁, y₁) and (x₂, y₂).

5. Calculate the Rise and Run: Calculate the difference in the y-coordinates (rise) and the difference in the x-coordinates (run):

   • Rise = y₂ - y₁
   • Run = x₂ - x₁

6. Calculate the Gradient: Divide the rise by the run to find the gradient:

   • Gradient = Rise / Run = (y₂ - y₁) / (x₂ - x₁)

Example:

Suppose you draw a tangent to a curve at x = 0.5. You then select two points on the tangent line: (1, 2) and (3, 6).

1. Rise = 6 - 2 = 4
2. Run = 3 - 1 = 2
3. Gradient = 4 / 2 = 2

Therefore, the gradient of the curve at x = 0.5 is approximately 2.

Important Considerations:

• Accuracy: The accuracy of this method depends heavily on the accuracy of the tangent you draw and how accurately you read the coordinates of the points on the tangent line. Draw the tangent carefully and choose points that are easy to read.
• Scale: Pay close attention to the scale of the axes on the graph.
• Approximation: This method provides an approximation of the gradient. Calculus provides more precise methods for finding the exact gradient of a curve.