Five things to push, drag and slice, each one picking up where a chapter's hands-on activity leaves off. Do the paper version first; come here to see it move.
Chapter 1: How fast is it going right now?
A path in space and its velocity
The arrow is r′(t): which way the dot is heading and, by its length, how fast, at one instant.
Speed right now: 2.41. Slowest is 1.60, at the top and bottom of each bump; fastest is 2.41, mid-climb.
A closed loop in space, like a roller-coaster track that goes round in a circle while rising and dipping three times. A dot travels along it and an arrow sticks out of the dot, pointing the way the dot is heading. The arrow's length is the dot's speed at that instant: shortest at the tops and bottoms of the bumps, longest halfway up or down each one.
Chapter 3: Zoom in until it gives up
A tangent plane on a surface
The tile is z=f(a,b)+fx(x−a)+fy(y−b). Zoom in on any smooth surface and it becomes this flat tile, which is why the Earth looks flat.
Height 0.14. Slope going left to right 0.85, going front to back 0.64. One step away the plane misses the surface by up to 0.373; a tenth of a step away, by only 0.0019.
A rolling landscape with one hill and one hollow, and a flat square tile resting on it at the marked point. The tile tilts to match the ground exactly at that point. Drag the point across the landscape, or use the two sliders, and the tile re-tilts: level on the hilltop and at the bottom of the hollow, steep on the slopes between. Close to the point the tile and the ground are almost indistinguishable, which is why the ground under your feet looks flat.
Chapter 4: The slope machine
Derivative as slope
Lay a ruler against the curve at every point and plot its slope underneath. The lower curve is f′(x), the second curve nobody drew.
Drag the point, or focus it and use the arrow keys.
At x = 1.00 the curve is uphill: slope 0.80.
Top: a smooth hill that climbs, rounds over a peak, drops into a dip and climbs again, with a short straight line just touching it at the moving point. Bottom: a second curve that records the slope of that touching line at every place the point has been. Where the top curve climbs, the bottom curve is above zero; at the peak and the dip it crosses zero; where the top curve falls, it is below zero.
Chapter 12: Rectangles that get skinnier
Riemann sum
Trap the area between an under-count and an over-count, then shrink the gap: ∑i=1nf(xi)Δx→∫04f(x)dx.
6 rectangles give 11.851. The true area is 12.760, so the estimate is off by 0.909. Gap between the over-count and under-count: 1.758.
A wavy curve starts at height 2, climbs to about 3.7 near x = 1.4, then sags back to about 3 at x = 4. The region under it is filled with equal-width rectangles. Under-count rectangles sit entirely under the curve, over-count rectangles poke above it, and midpoint rectangles take the curve's height at their centre. As the count goes up the rectangles get skinnier and the gap between the under-count and over-count shrinks toward zero, trapping the true area between them.
Chapter 14: Stacking: volume, rings, and layers
A solid of revolution, sliced
Spin a line around an axis to get a cone, then stack thin discs of area πr2. Thinner discs push the total toward 31πR2H.
The slice is a circle of radius 1.20 with area 4.52. 6 discs hold 5.72, which is 0.421 of the cylinder around the cone. The exact cone is 0.333 of it.
A cone lying on its side, point to the left, drawn as a faint outline. Inside it, a stack of flat discs fills the cone, each disc as wide as the cone gets over its own thickness, so the stack is a little too big, like steps. A highlighted circle marks where a slicing plane cuts the cone; discs past the slice are hidden. More discs make thinner steps, and the stack's share of the surrounding cylinder falls toward one third.