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
![**Problem Statement:**
Consider the function \( f(x, t) = (x - ct)^6 + (x + ct)^6 \) where \( c \) is a constant. Calculate \( \frac{\partial^2 f}{\partial x^2} \) and \( \frac{\partial^2 f}{\partial t^2} \).
---
**Calculations:**
1. **Second Partial Derivative with respect to \( x \):**
\[
\frac{\partial^2 f}{\partial x^2} = \text{[Expression to be calculated]}
\]
2. **Second Partial Derivative with respect to \( t \):**
\[
\frac{\partial^2 f}{\partial t^2} = \text{[Expression to be calculated]}
\]
---
**Key Equations:**
- **One-dimensional wave equation:**
\[
\frac{\partial^2 f}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}
\]
- **One-dimensional heat equation:**
\[
\frac{\partial f}{\partial t} = c^2 \frac{\partial^2 f}{\partial x^2}
\]
---
**Question:**
What can be said about \( f \)?
- \( f \) satisfies the one-dimensional wave equation.
- \( f \) neither satisfies the one-dimensional wave equation nor the one-dimensional heat equation.
- \( f \) satisfies the one-dimensional heat equation.
- \( f \) satisfies both the one-dimensional wave equation and the one-dimensional heat equation.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b5ef7d0-c879-48bd-8064-864fb9c6c6dd%2Fd744a5a8-cc8e-499b-b0e8-a2c21a941761%2Fdp3y0wf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider the function \( f(x, t) = (x - ct)^6 + (x + ct)^6 \) where \( c \) is a constant. Calculate \( \frac{\partial^2 f}{\partial x^2} \) and \( \frac{\partial^2 f}{\partial t^2} \).
---
**Calculations:**
1. **Second Partial Derivative with respect to \( x \):**
\[
\frac{\partial^2 f}{\partial x^2} = \text{[Expression to be calculated]}
\]
2. **Second Partial Derivative with respect to \( t \):**
\[
\frac{\partial^2 f}{\partial t^2} = \text{[Expression to be calculated]}
\]
---
**Key Equations:**
- **One-dimensional wave equation:**
\[
\frac{\partial^2 f}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}
\]
- **One-dimensional heat equation:**
\[
\frac{\partial f}{\partial t} = c^2 \frac{\partial^2 f}{\partial x^2}
\]
---
**Question:**
What can be said about \( f \)?
- \( f \) satisfies the one-dimensional wave equation.
- \( f \) neither satisfies the one-dimensional wave equation nor the one-dimensional heat equation.
- \( f \) satisfies the one-dimensional heat equation.
- \( f \) satisfies both the one-dimensional wave equation and the one-dimensional heat equation.
---
![A function \( u = f(x, y) \) with continuous second partial derivatives satisfying Laplace's equation
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\]
is called a harmonic function.
Calculate the indicated derivatives and determine if the function \( u(x, y) = x^3 - 3xy^2 \) is harmonic.
\[
\frac{\partial^2 u}{\partial x^2} = \underline{\hspace{5cm}}
\]
\[
\frac{\partial^2 u}{\partial y^2} = \underline{\hspace{5cm}}
\]
Is the function \( u = x^3 - 3xy^2 \) harmonic?
- ☐ no
- ☐ yes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b5ef7d0-c879-48bd-8064-864fb9c6c6dd%2Fd744a5a8-cc8e-499b-b0e8-a2c21a941761%2F9vgcqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A function \( u = f(x, y) \) with continuous second partial derivatives satisfying Laplace's equation
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\]
is called a harmonic function.
Calculate the indicated derivatives and determine if the function \( u(x, y) = x^3 - 3xy^2 \) is harmonic.
\[
\frac{\partial^2 u}{\partial x^2} = \underline{\hspace{5cm}}
\]
\[
\frac{\partial^2 u}{\partial y^2} = \underline{\hspace{5cm}}
\]
Is the function \( u = x^3 - 3xy^2 \) harmonic?
- ☐ no
- ☐ yes
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