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If you keep moving one of the points on the curve closer and closer to the other fixed point, then the slope of the secant lines converges onto the tangent line. The tangent line is the instantaneous slope of the fixed point on the curve.
I think I get it, somewhat. I'll keep hammering away. Thanks!
daperkins, when you do the worked example in section 1, you will have the opportunity to check out the secant approximation mathlet tool located at
which will provide a visual way to see the secant lines and the tangent line, and how they interact.
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The way I like thinking about this is the following: you can only define a secant and a tangent in the neighborhood of a point in a function.
Around a small interval in that neighborhood, a tangent will touch the function once and a secant will touch the function more than once. Therefore, lines that are tangents to a function on a given point may be secants (or not touch the function anymore) on other points.
I hope that helps!