Zoom In: Slope of a Curve

Lesson 9 of 10 · 7 min · Free, no account needed

What you will learn

Zoom in far enough and a smooth curve looks like a straight line: local linearity.
The slope of that tiny line is the slope of the curve at that point. The line itself is the tangent.
For y = x² the slope at x is 2x: 0, 2, 4, 6 at x = 0, 1, 2, 3.

Slope only works for straight lines. So does a curve have a slope?
Lay the ruler along y = x² at x = 1, the way the curve seems to go there. Drag the ruler's end.
Zoom in on the point (1, 1).
Turn the zoom dial all the way. Watch the curve, and the yellow step's rise over run.
Roll the point along the curve. Find where the slope is 4.
The lens shows the zoomed view. The panel shows the slope right there.
The line a curve hides.
Up close a smooth curve looks straight. That's local linearity.
The straight line it looks like is the tangent line. It touches the curve at one point and runs the same way.
Its slope is the slope of the curve at that point. At (1, 1) on y = x², it's 2.
Build the step under the zoom at x = 0, 1, 2 and 3.
Drag the top of the yellow step until it lands on the curve. The table collects each slope.
Game: slope sprint.
Four sticks, 45 seconds. Slide each one along the curve until it touches without crossing.
An AI model's error depends on one weight. Get it to the bottom.
The yellow line is the slope under the ball. Tap the arrow the ball should roll. Repeat until it rests.
No zoom, no grid. Tilt the ruler so it touches y = x² at x = 3.
Then Check. Within half a step counts.
A fountain jet, still hidden. Where will the water land?
Slide the pin along the ground to your guess. You'll find out in the next lesson.

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