Tangents, redefined for calculus

Topic:
Calculus
In the previous activity, we explored some unsatisfactory definitions of a tangent. So how should we define a tangent for the purposes of calculus? Well, one good option is to say that if is differentiable at , then...

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

Good Definition #1

If is differentiable at , then we say the line tangent to at is .

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

But there's another approach, and this will be a little more foreign to you.

Good Definition #2 (the best linear approximation approach)

The line tangent to at is the line passing through that best approximates near .

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?

Check all that apply

How do you know?

So which definition should you carry around in your head henceforth? For our purposes (a first-year calculus class with an interest in the AP curriculum), you can't go wrong with Good Definition #1. And if you add the requirement that is differentiable at to Good Definition #2, these two definitions are equivalent!