Stage 4 · Change and Relations / #30
The Derivative: Zoom In Until the Curve Is Straight
Planned
The video is in production. Follow along on GitHub to hear when it is out.
The idea
The tangent is the line the secants approach as h goes to zero. Zoom in on a differentiable curve near the point and the bend fades: the curve hugs that tangent more and more closely.
A common mix-up
A tangent is a line that touches the curve at just one point; and any curve turns straight if you zoom in far enough.
Where it stops working
y equals the absolute value of x has a corner at the origin: zoom as far as you like and it stays a corner. Secants on the right always have slope one, on the left minus one, so there is no tangent.
School year
China 高二