Captain Quint is standing at point B on a dock and is holding a rope of length attached to a boat at point A

Captain Quint is standing at point B on a dock and is holding a rope of length ℓ attached to a boat at point A [Figure 14(A)]. As the captain walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus, meaning “pulled”). The segment from a point P on the curve to the x-axis along the tangent line has constant length ℓ. Let y = ƒ(x) be the equation of the tractrix.

(a) Show that y² + (y/y')² = ℓ² and conclude y'= − y /√ℓ² − y² . Why must we choose the negative square root?

(b) Prove that the tractrix is the graph of

A B Dock l ein [ C + = 3 ) - /0 - 1 12 y x = lln X A P = (x, f(x)) f(x) X f(x) f'(x) l X