(a)
To show: If two lines intersect at an angle
(a)
Explanation of Solution
Given:
Two lines are
The slope of the lines
Calculation:
Let
Here, the slopes of the line
The angle between two lines are
If the angle
Take tangent on both sides,
Since the identity
Hence the required result is proved.
(b)
To find: The angle between each pair of curves at each points.
(b)
Answer to Problem 17P
The angle between two curves
the angle between two curves
Explanation of Solution
Given:
The pair of equation are
(i)
(ii)
Calculation:
Obtain the intersection point of
Thus, the intersection is
The slope of tangent line to the curve
Thus, the slope of tangent to the cure
The slope of the line
Substitute
Thus, the slope of the tangent to the curve at
Form part (a),
The value
The another angle is
Therefore, the angle between two curves
(ii)
Obtain the intersection point of
Substitute
Here,
Substitute
Thus, the points of intersection are,
The slope of tangent line to the curve
Therefore, the slope of the tangent is,
Thus, the slope of tangent to the cure
The slope of the line
That is,
Substitute
Substitute
Thus, the slope of the tangent to the curve
At the point
The angle between two lines at
Substitute
The value
At the point
The angle between two lines is ,
Substitute
The value
Therefore, the angle between two curves
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