a.
To show: that if the path followed by the boat is the graph of the function y = f(x), then
a.
Explanation of Solution
Given information: A man initially standing at the point O walks along a pier pulling a
rowboat by a rope length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve.
Calculation:
Focus on the rope segment. The length of this rope is always fixed to L .Now draw two lines from this segment, so that the rope becomes the hypotenuse and the two drawn lines are legs of a right triangle.
Since the rope is always tangent to the curve f(x), the slop of this segment is its derivative, f’(x). The dx is obviously x . The dy is not so obvious but can be easily obtained using Pythagoras Theorem since know the lengths of the hypotenuse and another leg.
Hence:
Since the slope is negative, finally have the following differential function
Hence proved.
b.
To find: the function y=f(x).
b.
Answer to Problem 23P
Explanation of Solution
Given information: Given
Calculation:
Chapter 5 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning