Error Propagation Analyzer

Calculus 1 · Axiom Academy

You measured a tank's radius and height — but your tape isn't perfect. How far off could the volume be? Put the differential to work. You're sizing a cylindrical fuel tank: radius r = 2.0 m , height h = 5.0 m , so volume V = r²h 62.83 m³ . But the radius is good only to 0.05 m and the height to 0.1 m — three moves turn those slips into a number you can trust. Nudge the radius by a few centimeters and watch the volume move. A 2.5% slip in r swings the volume by 5% — the error gets amplified , and that amplifier is the derivative V/ r = 2 rh. Both measurements are uncertain at once. The differential dV = ( V/ r)·dr + ( V/ h)·dh turns two input slips into one volume error — set each dial and read off the total the tank's volume could be off by. Which tape should you upgrade? You can afford a sharper instrument for one measurement. Because V/ r (62.83) dwarfs V/ h (12.57), the radius is the loud one — tightening it buys far more accuracy. Drag the radius precision and watch the total error shrink. One relationship, three moves: feel it (a slip gets amplified by V/ r), solve it (the differential adds every contribution), weigh it (the steepest partial decides where precision pays). Anywhere a result is computed from measured inputs — a bridge's stress, a drug dose by body weight, a rocket's fuel margin — these same three moves tell you how much to trust the answer.

This is the written version of the interactive lesson above. See the full Calculus 1 course.