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# Derivatives

# How Tangent Lines Look

The *tangent line to f at a* is the line approached by the secant lines between

*a*and

*a*+

*h*as

*h*approaches 0. This applet lets us watch the secant line approach the tangent line as we drag the point at

*a*+

*h*closer to

*a*:

Tangent lines and secant lines are (usually) different things.

**A secant line must hit two points on a graph.** A secant line between the points on the graph where *x* = *a* and *x* = *b* will hit the graph at *x* = *a* and *x* = *b*.

**A tangent line occurs at one single point on a graph and has the same slope as the graph at that point.**

Visually, the tangent line to *f* at *a* bounces off the graph at *x* = *a*:

A tangent line may also cross through the graph somewhere else. The important thing that makes a tangent line a tangent line is that it grazes the graph at that one special point. This line is tangent to *f* at *x* = *a* because it bounces off there:

However, this line is also a secant line between *x* = *a* and *x* = *b*:

A tangent line usually doesn't "cross over" the graph from one side to the other. However, it may cross over the graph at *x* = *a* in cases like this:

Here, the tangent line isn't so much "bouncing off" as it is "laying along" the graph of *f*.

This phenomenon has to do with something called **concavity.**

When looking to see if a line is tangent to *f* at *a*, we're looking to see if the line "bounces off" or "lays along" the graph of *f* at *a*.

### Sample Problem

This function has a tangent line with infinite slope at *x* = *a*:

### Sample Problem

If we have a function *f* that's already a line, the tangent line to *f* at any point *a* will be *f* again:

.