Secants and Tangents
In this figure, we see how secant lines (lines defined through two points on a curve) can be used to approximate a tangent line. In particular, if and are two points on the graph of a function and if is very close to , then the secant line through and approximates the line tangent to through .
In the figure, select an example and drag point to see how you can change the secant line. When is very close to , the secant line becomes very close to the tangent line through .
Click "Show details," and you'll see a different way to modify the secant line. In this situation, you control , which is the horizontal displacement of from . When gets very close to 0, this makes get very close to and once again the secant line approximates the tangent line. Observe how the labels on the axes change. There are two ways to modify : in the graphing window you can drag the large green point on the -axis, or in the control window you can drag the green slider.