To find: The points when the ellipse crosses the x-axis and to show that the tangent line at these points are parallel.
Answer to Problem 49E
The ellipse crosses the x-axis at
Explanation of Solution
Given:
The equation of the ellipse
Derivative rules:
Product rule:
Chain rule:
Calculation:
Obtain the points if the ellipse crosses the x-axis.
The ellipse crosses the x-axis. That is,
Substitute
Therefore, the ellipse crosses the x-axis at
Obtain the slope of the tangent at the points
Differentiate
Apply the product rule and the chain rule.
Separate
Thus, the derivative of the equation is
That is, the slope of the tangent is
The slope of the tangent at
Thus, the slope of the tangent at
The slope of the tangent at
Thus, the slope of the tangent at
Note that, the slope of the tangent lines at the points
Hence, the tangent lines at
Chapter 3 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