Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Please show your work
![### Implicit Differentiation and Equation of Tangent Line
**Problem Statement:**
4. Use implicit differentiation to find the equation of the tangent line of the following function at the given point:
\[ 2x^2 - 3y - 3 = 2y^4 \]
At the point \((2, 1)\).
**Solution Outline:**
To find the equation of the tangent line using implicit differentiation, follow these steps:
1. **Differentiate Both Sides Implicitly:**
- Differentiate each term with respect to \(x\).
- Apply the chain rule for the terms involving \(y\) since \(y\) is a function of \(x\).
2. **Apply the Chain Rule:**
- For terms like \(2y^4\), you'll need to multiply by \(\frac{dy}{dx}\) because \(y\) is implicitly a function of \(x\).
3. **Solve for \(\frac{dy}{dx}\):**
- Collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out.
- Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line.
4. **Substitute the Given Point \((2, 1)\):**
- Plug the coordinates of the point into your equation for \(\frac{dy}{dx}\) to find the slope at that point.
5. **Equation of the Tangent Line:**
- Use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Where \((x_1, y_1)\) is the given point and \(m\) is the slope found from \(\frac{dy}{dx}\).
**Diagram Explanation:**
Although this task does not include a diagram, typically you would graph the curve represented by the equation and draw the tangent line at the point \((2, 1)\) to visually confirm your result. The slope of the tangent should match the slope you've calculated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc777eb6c-5f00-442c-a5ae-30c50b2299ca%2Fd5aae783-cdf1-4266-8d31-f7e2d0f9e636%2F71vpbi8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation and Equation of Tangent Line
**Problem Statement:**
4. Use implicit differentiation to find the equation of the tangent line of the following function at the given point:
\[ 2x^2 - 3y - 3 = 2y^4 \]
At the point \((2, 1)\).
**Solution Outline:**
To find the equation of the tangent line using implicit differentiation, follow these steps:
1. **Differentiate Both Sides Implicitly:**
- Differentiate each term with respect to \(x\).
- Apply the chain rule for the terms involving \(y\) since \(y\) is a function of \(x\).
2. **Apply the Chain Rule:**
- For terms like \(2y^4\), you'll need to multiply by \(\frac{dy}{dx}\) because \(y\) is implicitly a function of \(x\).
3. **Solve for \(\frac{dy}{dx}\):**
- Collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out.
- Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line.
4. **Substitute the Given Point \((2, 1)\):**
- Plug the coordinates of the point into your equation for \(\frac{dy}{dx}\) to find the slope at that point.
5. **Equation of the Tangent Line:**
- Use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Where \((x_1, y_1)\) is the given point and \(m\) is the slope found from \(\frac{dy}{dx}\).
**Diagram Explanation:**
Although this task does not include a diagram, typically you would graph the curve represented by the equation and draw the tangent line at the point \((2, 1)\) to visually confirm your result. The slope of the tangent should match the slope you've calculated.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning