A boy standing at point B on a dock holds a rope of length ℓattached to a boat at point A[ Figur

step 1 All right, so the first part here, we use geometry for part A to show that F prime of X is equal

A boy standing at point B on a dock holds a rope of length...

A boy standing at point $B$ on a dock holds a rope of length $\ell$ attached to a boat at point $A[$ Figure 9$(\mathrm{A})] .$ As the boy walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus, meaning "to pull"). The segment from a point $P$ on the curve to the $x$-axis along the tangent line has constant length $\ell .$ Let $y=f(x)$ be the equation of the tractrix. (a) Show that $y^{2}+\left(y / y^{\prime}\right)^{2}=\ell^{2}$ and conclude $y^{\prime}=-\frac{y}{\sqrt{\ell^{2}-y^{2}}}.$ Why must we choose the negative square root? (b) Prove that the tractrix is the graph of $$x=\ell \ln \left(\frac{\ell+\sqrt{\ell^{2}-y^{2}}}{y}\right)-\sqrt{\ell^{2}-y^{2}}$$

A boy standing at point $B$ on a dock holds a rope of length $\ell$ attached to a boat at point $A[$ Figure 9$(\mathrm{A})] .$ As the boy walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus, meaning "to pull"). The segment from a point $P$ on the curve to the $x$-axis along the tangent line has constant length $\ell .$ Let $y=f(x)$ be the equation of the tractrix.
(a) Show that $y^{2}+\left(y / y^{\prime}\right)^{2}=\ell^{2}$ and conclude $y^{\prime}=-\frac{y}{\sqrt{\ell^{2}-y^{2}}}.$ Why must we choose the negative square root?
(b) Prove that the tractrix is the graph of
$$x=\ell \ln \left(\frac{\ell+\sqrt{\ell^{2}-y^{2}}}{y}\right)-\sqrt{\ell^{2}-y^{2}}$$

Key Concepts

Separation of Variables and Integration

Once the differential equation is established, the technique of separation of variables is employed to solve it. By isolating the terms involving y and its derivative on one side and the x-variable on the other, one can integrate both sides. The integration leads to an implicit solution involving logarithmic and square root functions, which, when rearranged, yields the explicit equation for the tractrix. This process exemplifies common methods used to solve first-order differential equations in calculus.

Choice of Negative Square Root

In the process of simplifying the differential equation, one arrives at the expression y' = -y/?(?² - y²). The negative square root is chosen to ensure consistency with the direction of motion along the curve. In the scenario described, as the point moves along the curve, the y-coordinate decreases, meaning that the derivative (slope) must be negative. This choice reflects the physical behavior of the system being modeled.

Tractrix

The tractrix is a curve defined by a unique geometric property: as an object is pulled by a fixed-length rope along a straight line, the path of the object is such that the segment of the tangent from the point of contact with the curve to the line remains constant. This unusual property gives the curve its distinctive behavior and appearance, and it serves as a classical example in the study of curves and differential equations.

Differential Equation Formulation

Using the constant-tangent-segment property, one can derive a differential equation that characterizes the tractrix. By considering the distance along the tangent from a point on the curve to a fixed axis and relating it to the derivative of the function (which represents the slope of the tangent), the equation y² + (y/y')² = ?² is obtained. This formulation provides a bridge between the geometric description and the analytical representation of the curve.

Tangent Line Properties

The analysis of the tractrix relies heavily on understanding the properties of tangent lines to curves. In this context, the fact that the segment of the tangent from the curve to the x-axis has a constant length ? is used to set up the relationship between the coordinates of the curve and its derivative. This interplay between the geometry of the tangent line and the differential properties of the curve is central to the problem.

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