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...
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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.
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