Walk towards a point from the left and from the right. A removable hole, a jump and an asymptote, then an ε–δ band you can tighten.

A secant line through two points of a curve, pulled together until it becomes the tangent — with the difference quotient updating live.

The power rule, the product rule as a rectangle growing on two sides, and the chain rule as rates multiplying through a composition.

f, f′ and f″ stacked on one x-axis. Rising and falling, bending up and down, maxima, minima and inflection points, all tied together.

Cut squares from the corners of a sheet and fold it into an open box. Which cut holds the most? Find it where the tangent goes flat.

Follow the tangent down to the x-axis, then do it again. Correct digits double each step near a root — until a bad start sends it cycling or flying off.

Add one term at a time and watch the polynomial hug sin x and eˣ. For ln(1 + x), see exactly where adding terms stops helping.

Slice a surface along x and along y to get the two partial slopes, then see the gradient on the contour map — always perpendicular to the level curves.

A ball stepping downhill on a loss surface in 3D and on its contour map. Slide the learning rate from crawling to converging to zigzagging to blowing up.