Interactives
Hands-on explorations to build intuition alongside the practice problems.
Calculus 1
Related Rates Explorer
A scroll-through tutorial: a ripple on a pond, a growing sand pile and a funnel draining into a glass, with each rate worked out as you drag.
Related rates Assumptions statedOptimization Explorer
A scroll-through tutorial: a pen by a river, a box folded from a sheet and a can that holds half a liter, with the best size found as you drag.
Optimization Assumptions statedCalculus 2
Taylor Polynomial Explorer
A scroll-through tutorial: build the polynomial one derivative at a time, measure its error, move its center, and find where more terms stop helping.
Taylor and Maclaurin polynomials Assumptions statedPumping Work Explorer
A scroll-through tutorial: cut the water in a tank into slabs, add up the work to lift each one, and watch the sum become an integral for a cylinder, a cone and a sphere.
Work: springs, pumping and cables Assumptions statedCalculus 3
Line Integral Explorer
A scroll-through tutorial: straight and curved paths, flow vs flux, work, a whirlpool and a river, with ∫ F · dr worked out as you drag.
Line integrals and work Assumptions statedGradient Explorer
A scroll-through tutorial on a contour map: turn to face any direction, read the slope, and see why the gradient points uphill, crosses contours at right angles, and vanishes at a saddle.
Gradient and directional derivatives Assumptions stated