Slope Machine: a Function's Graph, Its Slope Graph Beneath It, and the Tangent at Any Point

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Opens on Chapter 8’s example: a summer day’s temperature, 55 + 0.9t² − 0.06t³ (t in hours after 6 a.m.), with the tangent at t = 5.

Formula one letter, numbers, + − × ÷ ^ ( ), ² ³, e, π, √, ln

Tangent at

Graph from

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How it works

For each x the derivative f′(x) is the slope of the curve there: the limit of (f(x + h) − f(x)) ÷ h as h shrinks. For x² that tidies to 2x + h, so f′(x) = 2x. Read aloud, f′(x) is "f prime of x" and dy/dx is "dee y by dee x". Its units are the output's units per input unit: gallons per minute, dollars per pound. Read the two graphs together: the slope graph is above zero where the curve climbs, below where it falls, and crosses zero at the curve's peaks and valleys.

From the book, Chapter 8. The slope of x² is 2x: 10 at x = 5. The ramp’s slope at 3 is 1.5. The day warms fastest around t = 5 (4.5°F an hour) and stops warming at t = 10 (0). The tub drains at 16.2 gallons a minute at 10 minutes. One more pound of coffee costs about $6 at 100 pounds (exactly $6.01). The data plan’s slope is 3 everywhere. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.