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dw when dt Use the Chain Rule to find W = xey/z and t2, y = 3 – t, z = 3+4t. x =

dw when dt Use the Chain Rule to find W = xey/z and t2, y = 3 – t, z = 3+4t. x =

Calculus: Early Transcendentals
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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**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.
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.
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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