(a)
To find: The slope of the tangent line to the parabola
(a)
Answer to Problem 3E
The slope of the tangent line to the parabola
(i) Using Definition 1 is
(ii) Using Equation 2 is
Explanation of Solution
Given:
The slope of the tangent line to the parabola
Formula used:
Definition 1: The slope of the tangent curve
Equation 2: The slope of the tangent line in definition 1 becomes,
Section (i)
Obtain the slope of the tangent line to the parabola at the point (1, 3) by using Definition 1.
Substitute
The factors of
Therefore, the slope of the tangent line to the parabola becomes,
Since the limit x approaches 1 but not equal to 1, cancel the common term
Thus, the slope of the tangent line to the parabola at the point (1, 3) is
Section (ii)
Obtain the slope of the tangent line to the parabola at the point (1, 3) by using Equation 2.
Substitute
Simplify further and obtain the value of m.
Since the limit h tends to 0 but not equal to 0, cancel the common term
Thus, the slope of the tangent line becomes,
Thus, the slope of the tangent line to the parabola at the point (1, 3) by using equation 2 is
(b)
To find: The equation of the tangent line in part(a).
(b)
Answer to Problem 3E
The equation of the tangent line in part (a) is
Explanation of Solution
Equation of the tangent line:
The equation of the tangent line to the curve
Since the tangent line to the curve
Substitute
Isolate y as shown below.
Thus, the equation of the tangent line is
(c)
To sketch: The graph of the function is the tangent line,
(c)
Explanation of Solution
Calculation:
The equation of the tangent line is
The equation of the given curve is
Use the online graphing calculator to draw the graph of the functions as shown below in Figure 1.
Graph:
Use the online graphing calculator to zoom toward the point (1,3) as shown below in Figure 2.
From Figure 2, it is observed that when zoom in the graph toward the point (1, 3), the graph of the tangent line and the curve looks likes almost identical.
Hence, it is verified that the graph of the tangent line and the parabola zoom in toward the point (1, 3) until the tangent line are indistinguishable.
Chapter 2 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