What is a Tangent Line?
The red line in the following applet is the tangent line to the graph. 

Move the point of tangency around to see how the tangent line changes.
Try this on all the functions available.

(Note: If you are not seeing the applets below after a few moments then you need the Java 2 PlugIn)



You can use the "Forward" and "Back" buttons to animate the tangent line

Questions to think about:
1.  When do we get a horizontal tangent line?
2.  What do you notice about the graph when the tangent line has a positive slope? (What about a negative slope?)


How do we determine the Tangent Line?
The tangent line is the limit of secant lines that pass through two points of the function's graph. 

To determine the tangent line through a point we take the limit of secant lines passing through the desired
point and another point on the function's graph. As we move the second point closer and closer to the point of  tangency, we get a better and better approximation to the tangent line by the secant line.

The red line in the applet represents the tangent line through the point. To see how secant line (blue line) approaches this line, move the blue point closer and closer to the red point. Notice that the secant line approaches the tangent line as you move the points closer together. 

You may reposition the tangent line and repeat the process.  Try this with all the functions available in the applet.




Question to think about?
How would you express the limit of the slope of the secant lines as a mathematical limit?