# Tangents, redefined for calculus

- Author:
- Charlie Barnes

- Topic:
- Calculus

Wait, what does it mean to say that is differentiable at ?

## Good Definition #1

Explain why it makes sense that is the line tangent to at , incorporating the point-slope form of a line into your explanation.

## Good Definition #2 (the best linear approximation approach)

The applet above contains two windows, each containing the same black graph, red tangent line, and blue square. (Thus, the window on the right is zoomed further in.) Explain in your own words what it means to describe the line tangent to at as "the best linear approximation of near ".

Use the slider for a to move the point of tangency across x=0. (I find it easier to click the dot on the slider and then use the arrow keys to slide under control.)
When I built the applet above, I used GeoGebra's `Tangent() `

command. Based on your observations from the experiment you just performed, which definition was GeoGebra's `Tangent() `

command programmed with?

How do you know?