Zoom-In Slope: the Rate at One Instant From Shrinking Stretches, From Both Sides

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Opens on Chapter 6’s example: the dropped ball, d = 16t² feet, at exactly 2 seconds.

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

Stretches separate with commas

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Enter your numbers, then press Work it out.

How it works

Pick a point, step a little to the right, and divide the change in output by the size of the step: an average rate over a short stretch. Do it again with a step ten times smaller, then smaller again. From the right the averages come down toward a number; from the left they climb toward the same number. That number is the instantaneous rate, the slope of the curve at that point, and the line through the point with that slope is the tangent. Zoomed in far enough, a smooth curve and its tangent are hard to tell apart.

From the book, Chapter 6. The ball’s averages from 2 seconds: 65.6, 64.16, 64.016 going forward, 62.4, 63.84 coming up from below: the speed at 2 seconds is 64 ft/s, about 43.6 mph. On y = x² at 3: 6.1, 6.01, 6.001, settling on 6. Thea’s crest at 40 ft: 2.2%, 2.38%, 2.40%, a 2.4% grade. 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.