What is a tangent line?

by David Radcliffe

In this post I will give my answer to a question from David Wees.

Recall that a secant line to the graph of y = f(x) is a line that intersects the graph in at least one point (p, f(p)). This figure shows a secant line (in blue) to the curve y = x^2 (in red) at the point (1,1).

This particular secant line crosses from above to below, because the secant line lies above the curve when we are immediately left of the intersection point, and it lies below the curve when we are immediately to the right of the intersection point.

We could also draw another secant line that crosses from below to above, as shown here. Note that this new secant line has a greater slope than the previous secant line.

Now we can give a mathematically precise definition of a tangent line. A secant line to the curve y = f(x) at the point (p, f(p)) is said to be a tangent line if the following two conditions are satisfied.

  1. Every secant line with lesser slope crosses from above to below at (p, f(p)).
  2. Every secant line with greater slope crosses from below to above at (p, f(p)).

The following picture shows a tangent line to the graph of y = x^2. Notice that if the slope were increased then the line would cross from below to above, and if the slope were decreased then it would cross from above to below.

It is tempting to suppose that a tangent line cannot cross the curve, but this is not the case.

In this case, the tangent line crosses from above to below. If the slope were decreased, then it would still cross from above to below. However, if the slope were increased, then it would cross from below to above. This is consistent with our definition of the tangent line.

Credit: The idea for this definition of tangent line was inspired by the book Calculus Unlimited by Jerrold Marsden and Alan Weinstein, which develops calculus without the concept of limit.

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