Two young mathematicians discuss the novel idea of the “slope of a curve.”

- Devyn
- Riley, do you remember “slope?’
- Riley
- Most definitely. “Rise over run.”
- Devyn
- You know it.
- Riley
- “Change in over change in .”
- Devyn
- That’s right.
- Riley
- Brought to you by the letter “.”
- Devyn
- Enough! My important question is: could we define “slope” for a curve that’s not a straight line?
- Riley
- Well, maybe if we “zoom in” on a curve, it would look like a line, and then we could call it “the slope at that point.”
- Devyn
- Ah! And this “zoom in” idea sounds like a limit!
- Riley
- This is so awesome. We just made math!

The concept introduced above, of the “slope of a curve at a point,” is in fact one of the central concepts of calculus. It will, of course, be completely explained. Let’s explore Devyn and Riley’s ideas a little more, first.

To find the “slope of a curve at a point,” Devyn and Riley spoke of “zooming in” on
a curve until it looks like a line. When you zoom in on a *smooth* curve, it will
eventually look like a line. This line is called the tangent line.