To find: the curves
Answer to Problem 56E
Therefore, the horizontal points are
Explanation of Solution
Given:
The curves are
Calculation:
Consider the curve
The objective is to find the points on the given curve where the tangent line is horizontal or vertical.
Every polar point
Rewrite the polar coordinate
Differentiate with respect to “
Rewrite the polar coordinate
Differentiate with respect to “
The slope of the tangent line is,
To find the points where the curve at horizontal tangent, set
Either,
As the period of sin and cos function is
Hence, the value of
Substitute these values of
At
The value of the curve is,
Hence, the value of the curve is
At
The value of the curve is,
Hence, the value of the curve is
At
The value of the curve is,
Hence, the value of the curve is
At
The value of the curve is,
Hence, the value of the curve is
Hence, the horizontal points are
To find the points where the curve has vertical tangent, set
As the period of sin and cos function is
Hence, the values of
At
The value of the curve is,
Here, the value of the curve is 0.
At
The value of the curve is,
Hence, the value of the curve is
At
The value of the curve is,
Thus, the value curve is
Hence, the vertical points are
The graph of the function is shown below:
Conclusion:
Therefore, the horizontal points are
Chapter H.1 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