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
Topic Video
Question
Need help with this please answer all four parts
![**Problem Statement:**
**Find all the second-order partial derivatives of the following function.**
\[ w = 6x \sin \left( 7x^2y \right) \]
**Solution and Explanation:**
We are given the function \( w = 6x \sin(7x^2 y) \), and we need to find all the second-order partial derivatives.
Define:
- \( w \) as the dependent variable.
- \( x \) and \( y \) as the independent variables.
We need to compute the four second-order partial derivatives:
1. \( \frac{\partial^2 w}{\partial x^2} \)
2. \( \frac{\partial^2 w}{\partial y^2} \)
3. \( \frac{\partial^2 w}{\partial x \partial y} \)
4. \( \frac{\partial^2 w}{\partial y \partial x} \)
**Step-by-Step Solution:**
1. Find the first-order partial derivatives:
- \( w_x = \frac{\partial w}{\partial x} \)
- \( w_y = \frac{\partial w}{\partial y} \)
2. Differentiate \( w_x \) with respect to \( x \) and \( y \) to find \( w_{xx} \) and \( w_{xy} \).
3. Differentiate \( w_y \) with respect to \( x \) and \( y \) to find \( w_{yy} \) and \( w_{yx} \).
Note that due to the symmetry of mixed partial derivatives (Clairaut's theorem), \( w_{xy} \) should equal \( w_{yx} \) if the function is continuous and differentiable.
**First-Order Partial Derivatives:**
\[ w = 6x \sin(7x^2y) \]
1. **First Partial Derivative with respect to \( x \):**
\[ w_x = \frac{\partial w}{\partial x} = 6 \sin(7x^2 y) + 6x \cos(7x^2 y) \cdot \frac{\partial}{\partial x}(7x^2 y) \]
\[ w_x = 6 \sin(7x^2 y) + 6x \cos(7x^2 y) \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc13a662a-8d65-4c56-a285-fa9dd6b59b44%2F0f50b014-060c-41d0-ab21-623247467304%2Fbsptfw_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**Find all the second-order partial derivatives of the following function.**
\[ w = 6x \sin \left( 7x^2y \right) \]
**Solution and Explanation:**
We are given the function \( w = 6x \sin(7x^2 y) \), and we need to find all the second-order partial derivatives.
Define:
- \( w \) as the dependent variable.
- \( x \) and \( y \) as the independent variables.
We need to compute the four second-order partial derivatives:
1. \( \frac{\partial^2 w}{\partial x^2} \)
2. \( \frac{\partial^2 w}{\partial y^2} \)
3. \( \frac{\partial^2 w}{\partial x \partial y} \)
4. \( \frac{\partial^2 w}{\partial y \partial x} \)
**Step-by-Step Solution:**
1. Find the first-order partial derivatives:
- \( w_x = \frac{\partial w}{\partial x} \)
- \( w_y = \frac{\partial w}{\partial y} \)
2. Differentiate \( w_x \) with respect to \( x \) and \( y \) to find \( w_{xx} \) and \( w_{xy} \).
3. Differentiate \( w_y \) with respect to \( x \) and \( y \) to find \( w_{yy} \) and \( w_{yx} \).
Note that due to the symmetry of mixed partial derivatives (Clairaut's theorem), \( w_{xy} \) should equal \( w_{yx} \) if the function is continuous and differentiable.
**First-Order Partial Derivatives:**
\[ w = 6x \sin(7x^2y) \]
1. **First Partial Derivative with respect to \( x \):**
\[ w_x = \frac{\partial w}{\partial x} = 6 \sin(7x^2 y) + 6x \cos(7x^2 y) \cdot \frac{\partial}{\partial x}(7x^2 y) \]
\[ w_x = 6 \sin(7x^2 y) + 6x \cos(7x^2 y) \cdot
![The image shows partial differential equations involving a function \( w \). Here's the transcription:
\[
\frac{\partial^2 w}{\partial x^2}
\]
\[
\frac{\partial^2 w}{\partial y^2}
\]
\[
\frac{\partial^2 w}{\partial y \partial x}
\]
\[
\frac{\partial^2 w}{\partial x \partial y}
\]
### Explanation:
- **\(\frac{\partial^2 w}{\partial x^2}\)** represents the second partial derivative of \( w \) with respect to \( x \). It measures the rate at which the first derivative \(\frac{\partial w}{\partial x}\) changes as \( x \) changes.
- **\(\frac{\partial^2 w}{\partial y^2}\)** represents the second partial derivative of \( w \) with respect to \( y \). It measures the rate at which the first derivative \(\frac{\partial w}{\partial y}\) changes as \( y \) changes.
- **\(\frac{\partial^2 w}{\partial y \partial x}\)** and **\(\frac{\partial^2 w}{\partial x \partial y}\)** represent the mixed partial derivatives of \( w \). These measure how the function \( w \) changes as both \( x \) and \( y \) vary.
For educational context, these partial derivatives are crucial in fields such as multivariable calculus, differential equations, and mathematical physics, where they help in understanding how functions change in multiple dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc13a662a-8d65-4c56-a285-fa9dd6b59b44%2F0f50b014-060c-41d0-ab21-623247467304%2Fel7areo_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows partial differential equations involving a function \( w \). Here's the transcription:
\[
\frac{\partial^2 w}{\partial x^2}
\]
\[
\frac{\partial^2 w}{\partial y^2}
\]
\[
\frac{\partial^2 w}{\partial y \partial x}
\]
\[
\frac{\partial^2 w}{\partial x \partial y}
\]
### Explanation:
- **\(\frac{\partial^2 w}{\partial x^2}\)** represents the second partial derivative of \( w \) with respect to \( x \). It measures the rate at which the first derivative \(\frac{\partial w}{\partial x}\) changes as \( x \) changes.
- **\(\frac{\partial^2 w}{\partial y^2}\)** represents the second partial derivative of \( w \) with respect to \( y \). It measures the rate at which the first derivative \(\frac{\partial w}{\partial y}\) changes as \( y \) changes.
- **\(\frac{\partial^2 w}{\partial y \partial x}\)** and **\(\frac{\partial^2 w}{\partial x \partial y}\)** represent the mixed partial derivatives of \( w \). These measure how the function \( w \) changes as both \( x \) and \( y \) vary.
For educational context, these partial derivatives are crucial in fields such as multivariable calculus, differential equations, and mathematical physics, where they help in understanding how functions change in multiple dimensions.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning