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

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley, do you remember “slope?’
Riley
Most definitely. “Rise over run.”
Devyn
You know it.
Riley
“Change in over change in .”
Devny
That’s right.
Riley
Brought to you by the letter “.”
Devny
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.

Which of the following approximate the slope of the “zoomed line”? (You can select more than one.)
Let . Zoom in on the curve around so that . Use one of the formulations in the problem above to approximate the slope of the curve. The slope of the curve at is approximately…
Repeat the previous problem for , , and . Choose a formulation that will give you a positive answer for the slope. The (positive) slope of the curve at is approximately…
Zoom in on the curve near again. By looking at the graph, what is your best guess for the actual slope of the curve at zero?
impossible to say zero one infinity