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Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?
(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*
(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*
(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*
(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?
(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*
(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*
(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*
(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?
Most Upvoted Answer
Suppose that the dependent variables z and w are function of the depen...
Understanding the Problem
The problem involves two equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0, with the relationship f_z * g_w - f_w * g_z = 1, which indicates a non-degenerate system of equations. We need to find the expression for z{x}.
Using Implicit Function Theorem
The Implicit Function Theorem helps us establish relationships between the variables. The expression for z{x} can be derived using the derivatives of f and g with respect to x and their partial derivatives with respect to z and w.
Evaluating the Options
- Option (a): z{x} = f_w * g_x - f_x * g_w
- This expression does not correctly align with the relationships stated.
- Option (b): z{x} = f_x * g_w - f_w * g_x
- This expression also does not match the necessary criteria.
- Option (c): z{x} = f_z * g_x - f_x * g_z
- This matches the formula derived from the Implicit Function Theorem when considering partial derivatives.
- Option (d): z{x} = f_z * g_w - f_z * g_x
- This expression is incorrect according to the relationships of the functions.
Conclusion
The correct expression for z{x} is provided in option (c).
Final Answer
- Correct Choice: (c) z{x} = f_z * g_x - f_x * g_z
The problem involves two equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0, with the relationship f_z * g_w - f_w * g_z = 1, which indicates a non-degenerate system of equations. We need to find the expression for z{x}.
Using Implicit Function Theorem
The Implicit Function Theorem helps us establish relationships between the variables. The expression for z{x} can be derived using the derivatives of f and g with respect to x and their partial derivatives with respect to z and w.
Evaluating the Options
- Option (a): z{x} = f_w * g_x - f_x * g_w
- This expression does not correctly align with the relationships stated.
- Option (b): z{x} = f_x * g_w - f_w * g_x
- This expression also does not match the necessary criteria.
- Option (c): z{x} = f_z * g_x - f_x * g_z
- This matches the formula derived from the Implicit Function Theorem when considering partial derivatives.
- Option (d): z{x} = f_z * g_w - f_z * g_x
- This expression is incorrect according to the relationships of the functions.
Conclusion
The correct expression for z{x} is provided in option (c).
Final Answer
- Correct Choice: (c) z{x} = f_z * g_x - f_x * g_z
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Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?
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Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?.
Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*?.
Solutions for Suppose that the dependent variables z and w are function of the dependent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0 where f_{z}*g_{w}* - f_{w}*g_{z} *= 1 Which one of the following is correct?(a) ) z*{x} = f_{w}*g_{x}* - f_{x}*g_{w}*(b) z*{x}= f_{x}*g*{w}-f_{w}*g_{x}*(c) z*{x} = f_{z}*g_{x}* - f_{x}*g_{z}*(d) z*{x} = f_{z}*g_{w} *- f_{z}*g_{x}*? in English & in Hindi are available as part of our courses for UPSC.
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