[color=#000000]The applet below shows a [/color][color=#980000][b]brown boat[/b][/color][color=#000000] in the water. A [/color][color=#0000ff][b]blue rope of fixed length[/b][/color][color=#000000] (that's always held taught) connects the [/color][color=#980000][b]brown boat[/b][/color][color=#000000] to a [/color][color=#ff0000][b]red point[/b][/color][color=#000000]. This [/color][color=#ff0000][b]red point[/b][/color][color=#000000] stays on the [/color][color=#444444][b]gray dock[/b][/color][color=#000000] and is allowed to move vertically along the dock. [br][br]Go ahead and drag the [/color][color=#ff0000][b]red point[/b][/color][color=#000000] upwards and observe the path traced out by the the [/color][color=#980000][b]brown boat[/b][/color][color=#000000]. [br]Feel free to do this for different values of taught rope length (adjustable by the [/color][color=#0000ff]blue slider[/color][color=#000000].) [br][br]After interacting with this applet for a few minutes, please answer the questions that appear below it. [br][/color][br]

[color=#000000]1) The [/color][color=#980000]tractrix[/color][color=#000000] is also called a [/color][i][color=#980000]curve of pursuit[/color][/i][color=#000000]. [br] Can you explain, from your observations, why this label makes sense? [br][br]2) Suppose [/color][color=#980000]point P (the boat)[/color][color=#000000] has coordinates [/color][color=#980000](x, y)[/color][color=#000000]. [br] Suppose the [/color][color=#ff0000]red point [i]A[/i] [/color][color=#000000]has coordinates [/color][color=#ff0000](0,a). [br][/color][color=#000000] Write an expression for the slope of the [/color][color=#0000ff]blue rope (segment [i]AP[/i])[/color][color=#000000] in the applet above. [br][br]3) Solve the differential equation you wrote for (2) above. [/color]