Concept explainers
To find: The number of lines that are normal to the parabola
Answer to Problem 73E
Three normal lines to the parabola that passes through the point
One normal line to the parabola that passes through the point
Explanation of Solution
Derivative rule:
Power rule:
Calculation:
The derivative of the curve
Apply the power rule (1) and simplify the expression,
Thus, the derivative of the curve is
Therefore, the slope of the tangent line to the curve is 2x.
Obtain the slope of the normal line to the curve by using the slope of the tangent line.
Since every point of the parabola is of the form
Here, the tangent line is perpendicular to the normal line. That is, if
This implies that, the slope of the normal line to the curve is
Note that, the slope of the line passing through the points
Here, the normal line passing through the points
The slope of the normal line is computed as follows.
Since the slope of normal line is
The above equation has two solution if
Since the y-axis is normal to the parabola and passes through the point
Therefore, there are three normal lines to the parabola that passes through the point
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