
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