Calculus, Interactively
Single-variable calculus first, then multivariable - built from local linearity up, carried by a robot on a line and a two-link arm reaching for a target, and ending with the gradient, Jacobian and Hessian that the optimization guides assume.
The method is the one idea that makes the whole subject obvious: zoom in far enough on any smooth curve and it becomes a straight line. Everything else - the derivative, the chain rule, the tangent plane, the Newton step - is a consequence of that single observation, and every part lets you drag the thing being described until you can see it. No part assumes more than high-school algebra.
When you finish, the gradient, Jacobian and Hessian are no longer notation you have seen: they are the derivative in one, many, and second-order form, and they connect directly to the nonlinear optimization guide, whose Gradient Descent, Newton and Gauss-Newton parts pick up exactly where Part 14 leaves off.
The parts
The whole subject in one idea: almost every function is locally linear, and a derivative is just the slope you find when you zoom in far enough.
The epsilon-delta definition as a two-slider game, one-sided limits, and the exact places a function can fail to be continuous.
A secant line sliding into a tangent: drag the point and watch the derivative trace itself out, then see why the slope is the best linear approximation.
Product, quotient and chain rules built as pictures - the product rule as the area of a growing rectangle - plus the derivatives worth memorising.
Composed machines with rate gauges: multiply the gear ratios and the chain rule falls out, the same rule backprop runs in reverse.
Differentiate an equation you cannot solve for y by dragging along the curve it defines, and connect rates that are tied together by a constraint.
Add terms one at a time and watch a polynomial wrap a curve, then use the first-order model to run Newton's method and map its basins of attraction.
A Riemann-sum slider that shrinks rectangles into signed area, with left, right, midpoint and trapezoid rules side by side.
Area-so-far traced live beside the curve: differentiation and integration are inverse operations, and the odometer makes the theorem an observation.
Substitution as the chain rule backwards, integration by parts as the product rule backwards, and when to give up on closed forms and use quadrature.
A surface with draggable slice planes: hold one input fixed and single-variable calculus returns as a partial derivative.
A contour map, a directional-derivative dial, and the reason the gradient points uphill and meets every level set at a right angle.
The arm Jacobian, a quadratic-form explorer, and Hessian eigenvalues that sort every critical point into bowl, ridge or saddle.
The local quadratic model on a surface, the second-derivative test in matrix form, and the shape of the landscape a solver has to descend.