Interactive charts

Charts you work with your hands: drag a point, slide a strip count, and watch the numbers answer. Blue is what you move, green is the answer.

Calculus

f(x) = x³ − 3x

Point x = a 0.60 Show the secantGap h 1.00

Slope of the tangent at x = 0.60: −1.920 falling ↘

Secant slope over [0.60, 1.60] = 0.880, off by 2.800. Shrink h and the gap closes.

The power rule on each term gives f′(x) = 3x² − 3. Drag the point: the slope is zero at x = −1 and x = 1, where the curve turns.

Solver: Power Rule Derivative · In the lessons: From secant to tangent, The tangent line (Grade 12 Calculus)

v(t) = t²/4 + 1

Strips n 4

Estimate: 9.250 m Exact: 9.333 m

Under by 0.083 m (0.89%). Double n and watch the error shrink.

A cart's speed over 4 s. The area under the curve is the distance it covers: (m/s) × s = m.

Solver: Definite Integral of a Power Function

Closing in on x = 2 from both sides

f(x) = x²/2 + 1, x ≠ 2

Distance from x = 2 1.000

From the left: f(1.000) = 1.500

From the right: f(3.000) = 5.500

Both sides close in on 3, so the limit as x → 2 is 3.

f(2) is undefined: the open circle is a hole, and the limit never needed it.

Drag toward x = 2: the blue points close in from the left and from the right, and green marks the height each side arrives at. Tap Jump for two sides that disagree.

In the lessons: Closing in, Left, right, both (Grade 12 Calculus)

Steam and water

Steam Diagrams →

4 chartsThe pressure–enthalpy chart, the Mollier chart, the T–s diagram and the saturation dome, each with a live reading wherever you touch.

Metal Precipitation by pH →

hydroxide · sulfideSwitch on the metals in your water and read the pH window that brings every one under your discharge limit.