The Tangent Problem
The Slope of the Tangent Line
This is from your textbook but I wanted to make it a bit more interactive.
Suppose we have a function and let the point have coordinates (1,1). Now suppose we have another point on the graph of with coordinates (x,f(x)), where .
Now, we can construct the secant line through the points and . We can use these two points to then calculate the slope of this secant line using the formula
Now, the question is, what happens as the point approaches the point ? That is, what happens as the secant line approaches the tangent line?
Use the slider to investigate what happens to the slope of the secant line as it approaches the tangent line.
We say that the slope of the tangent line is the limit of the slopes of the secant lines, that is
In terms of the formula we have for our slope from above, this limit can be written in the form
From your investigation from above, you should come to the conclusion that this limit is 2.
You may now ask yourself, is there an easier way to do this?!
Yes, of course, but that is something we will get into very soon!