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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Question
![**Equation 6: Differential Equation**
The equation presented is a first-order differential equation that describes the rate of change of a variable \( w \) with respect to time \( t \). Specifically, the equation is given as:
\[
\frac{dw}{dt} = \left(2t + \frac{x}{z} + \frac{4xy}{z^2}\right) e^{y/z}
\]
### Explanation:
- **\(\frac{dw}{dt}\)**: This represents the derivative of \( w \) with respect to time \( t \), indicating how \( w \) changes as time progresses.
- **Expression \(\left(2t + \frac{x}{z} + \frac{4xy}{z^2}\right)\)**: This part of the equation consists of three terms:
- \(2t\): A term directly proportional to time.
- \(\frac{x}{z}\): A term representing the ratio of \( x \) to \( z \).
- \(\frac{4xy}{z^2}\): A term representing the product of \( x \) and \( y \) scaled by the square of \( z \).
- **Exponential \( e^{y/z} \)**: The equation is multiplied by an exponential term, where the exponent is the ratio of \( y \) to \( z \). This implies that the change in \( w \) is exponentially affected by the relationship between \( y \) and \( z \).
This equation could be used in various applications, such as modeling physical phenomena where these variables interact under the influence of time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F476a7a93-3242-4b81-b338-34642657d3b8%2F3bac3990-e28e-41b5-aae6-5983e6fc588f%2Fbsdcvfv_processed.png&w=3840&q=75)
Transcribed Image Text:**Equation 6: Differential Equation**
The equation presented is a first-order differential equation that describes the rate of change of a variable \( w \) with respect to time \( t \). Specifically, the equation is given as:
\[
\frac{dw}{dt} = \left(2t + \frac{x}{z} + \frac{4xy}{z^2}\right) e^{y/z}
\]
### Explanation:
- **\(\frac{dw}{dt}\)**: This represents the derivative of \( w \) with respect to time \( t \), indicating how \( w \) changes as time progresses.
- **Expression \(\left(2t + \frac{x}{z} + \frac{4xy}{z^2}\right)\)**: This part of the equation consists of three terms:
- \(2t\): A term directly proportional to time.
- \(\frac{x}{z}\): A term representing the ratio of \( x \) to \( z \).
- \(\frac{4xy}{z^2}\): A term representing the product of \( x \) and \( y \) scaled by the square of \( z \).
- **Exponential \( e^{y/z} \)**: The equation is multiplied by an exponential term, where the exponent is the ratio of \( y \) to \( z \). This implies that the change in \( w \) is exponentially affected by the relationship between \( y \) and \( z \).
This equation could be used in various applications, such as modeling physical phenomena where these variables interact under the influence of time.
![### Using the Chain Rule to Find \(\frac{dw}{dt}\)
The objective is to find \(\frac{dw}{dt}\) using the chain rule, where:
\[
w = xe^{y/z}
\]
And:
\[
x = t^2, \quad y = 3 - t, \quad z = 3 + 4t.
\]
Below are five potential expressions for \(\frac{dw}{dt}\):
1. \[
\frac{dw}{dt} = \left( t + \frac{x}{z} + \frac{3xy}{z} \right) e^{y/z}
\]
2. \[
\frac{dw}{dt} = \left( t - \frac{x}{z} - \frac{3xy}{z} \right) e^{y/z}
\]
3. \[
\frac{dw}{dt} = \left( t + \frac{x}{z} + \frac{3xy}{z^2} \right) e^{y/z}
\]
4. \[
\frac{dw}{dt} = \left( 2t - \frac{x}{z} - \frac{4xy}{z} \right) e^{y/z}
\]
5. \[
\frac{dw}{dt} = \left( 2t - \frac{x}{z} - \frac{4xy}{z^2} \right) e^{y/z}
\]
### Explanation of Terms:
- \( x = t^2 \) represents the expression for \( x \) in terms of \( t \).
- \( y = 3 - t \) provides the relationship for \( y \) in terms of \( t \).
- \( z = 3 + 4t \) gives the equation for \( z \) as a function of \( t \).
Using these, the chain rule will differentiate \( w \) with respect to \( t \), taking into account the interdependency of \( x \), \( y \), and \( z \) on \( t \). The correct expression will satisfy the derivative as per the chain rule application.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F476a7a93-3242-4b81-b338-34642657d3b8%2F3bac3990-e28e-41b5-aae6-5983e6fc588f%2F0m0cv6_processed.png&w=3840&q=75)
Transcribed Image Text:### Using the Chain Rule to Find \(\frac{dw}{dt}\)
The objective is to find \(\frac{dw}{dt}\) using the chain rule, where:
\[
w = xe^{y/z}
\]
And:
\[
x = t^2, \quad y = 3 - t, \quad z = 3 + 4t.
\]
Below are five potential expressions for \(\frac{dw}{dt}\):
1. \[
\frac{dw}{dt} = \left( t + \frac{x}{z} + \frac{3xy}{z} \right) e^{y/z}
\]
2. \[
\frac{dw}{dt} = \left( t - \frac{x}{z} - \frac{3xy}{z} \right) e^{y/z}
\]
3. \[
\frac{dw}{dt} = \left( t + \frac{x}{z} + \frac{3xy}{z^2} \right) e^{y/z}
\]
4. \[
\frac{dw}{dt} = \left( 2t - \frac{x}{z} - \frac{4xy}{z} \right) e^{y/z}
\]
5. \[
\frac{dw}{dt} = \left( 2t - \frac{x}{z} - \frac{4xy}{z^2} \right) e^{y/z}
\]
### Explanation of Terms:
- \( x = t^2 \) represents the expression for \( x \) in terms of \( t \).
- \( y = 3 - t \) provides the relationship for \( y \) in terms of \( t \).
- \( z = 3 + 4t \) gives the equation for \( z \) as a function of \( t \).
Using these, the chain rule will differentiate \( w \) with respect to \( t \), taking into account the interdependency of \( x \), \( y \), and \( z \) on \( t \). The correct expression will satisfy the derivative as per the chain rule application.
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