(a)
To find: The equation of the tangent line to the curve at the point.
(a)
Answer to Problem 46E
The equation of the tangent line to the curve
Explanation of Solution
Given:
The function is
Result used:
The Power Rule combined with the Chain Rule:
If n is any real number and
If
Formula used:
The equation of the tangent line at
where, m is the slope of the tangent line at
Calculation:
For
The derivative of
Apply the quotient rule as shown in equation (2),
Apply the power rule combined with the chain rule as shown in equation (1),
On further simplification, the derivative of the function becomes,
Therefore, the derivative of
The slope of the tangent line at
Thus, the slope of the tangent line is
Substitute
Therefore, the equation of the tangent line is
(b)
To sketch: The graph of the curve and the tangent line.
(b)
Explanation of Solution
Given:
The equation of the curve is
The equation of the tangent line is
Graph:
Use the online graphing calculator to draw the graph of the functions as shown below in Figure 1.
From Figure 1, it is observed that the equation of the tangent line touches on the curve
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