Given z = 3x²y³, when x=4, y=1, the values of the second-order cross partial derivatives are,
Understand the Problem
The question involves calculating second-order cross partial derivatives of the function z = 3x²y³ given specific values for x and y. It provides multiple choice answers to determine the correct values of the derivatives z_xy and z_yx at the given point.
Answer
\( z_{xy} = 72, z_{yx} = 72 \)
Answer for screen readers
The values of the second-order cross partial derivatives at the point ( (4, 1) ) are ( z_{xy} = 72 ) and ( z_{yx} = 72 ).
Steps to Solve
- Define the function and variables
The given function is ( z = 3x^2y^3 ). We need to calculate the second-order partial derivatives ( z_{xy} ) and ( z_{yx} ) at the point ( (x, y) = (4, 1) ).
- Calculate the first-order partial derivatives
First, we find the first-order partial derivatives with respect to ( x ) and ( y ).
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For ( z_x ): $$ z_x = \frac{\partial}{\partial x}(3x^2y^3) = 6xy^3 $$
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For ( z_y ): $$ z_y = \frac{\partial}{\partial y}(3x^2y^3) = 9x^2y^2 $$
- Evaluate first-order derivatives at the point (4, 1)
Substituting ( x = 4 ) and ( y = 1 ):
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For ( z_x ): $$ z_x(4, 1) = 6(4)(1^3) = 24 $$
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For ( z_y ): $$ z_y(4, 1) = 9(4^2)(1^2) = 144 $$
- Calculate the second-order partial derivatives
Now, we compute the second-order derivatives.
- For ( z_{xy} ): $$ z_{xy} = \frac{\partial}{\partial y}(z_x) = \frac{\partial}{\partial y}(6xy^3) = 18xy^2 $$
Evaluate at ( (4, 1) ): $$ z_{xy}(4, 1) = 18(4)(1^2) = 72 $$
- For ( z_{yx} ): $$ z_{yx} = \frac{\partial}{\partial x}(z_y) = \frac{\partial}{\partial x}(9x^2y^2) = 18xy^2 $$
Evaluate at ( (4, 1) ): $$ z_{yx}(4, 1) = 18(4)(1^2) = 72 $$
- Summarize the results
Thus, the values of the second-order cross partial derivatives are: $$ z_{xy} = 72 $$ $$ z_{yx} = 72 $$
The values of the second-order cross partial derivatives at the point ( (4, 1) ) are ( z_{xy} = 72 ) and ( z_{yx} = 72 ).
More Information
The equality of mixed partial derivatives, ( z_{xy} = z_{yx} ), follows from Clairaut's theorem, which states that if the second partial derivatives are continuous, the mixed partial derivatives are equal.
Tips
- Forgetting to evaluate the first-order derivatives at the specified point before calculating the second-order derivatives.
- Miscomputing derivatives or using incorrect formulas while differentiating.
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