Calculus Archive: Questions from July 02, 2023
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11. \( \iiint_{E} \frac{z}{x^{2}+z^{2}} d V \), where \[ E=\{(x, y, z) \mid 1 \leqslant y \leqslant 4, y \leqslant z \leqslant 4,0 \leqslant x \leqslant z\} \]2 answers -
Find \( y^{\prime} \) if \( \tan ^{-1}\left(3 x^{2} y\right)=x+5 x y^{2} \). \[ y^{\prime}=\frac{1+5 y^{2}+9 x^{2} y^{2}+45 x^{4} y^{4}-6 x y}{3 x^{2}-10 x y-90 x^{5} y^{3}} \]2 answers -
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Find the general solution of \[ y^{\prime \prime}+y=2 \sec ^{3} x \] \[ \begin{array}{l} C_{1} \cos x+C_{2} \sin x-\frac{\cos (2 x)}{\cos x} \\ C_{1} \cos x+C_{2} \sin x-\frac{1}{\sin x} \\ C_{1} \cos2 answers -
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Find the partial derivatives of the function \[ f(x, y)=x y e^{3 y} \] \[ \begin{aligned} f_{x}(x, y) & =x y e^{3 y} \\ f_{y}(x, y) & =y e^{3 y} \\ f_{x y}(x, y) & =x e^{-3 y}-3 x y e^{3 y} \\ f_{y x}2 answers -
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Given \( f(x, y)=3 x y^{2}-9 x^{4} y \). Compute: \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}= \\ \frac{\partial^{2} f}{\partial y^{2}}= \end{array} \]2 answers -
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Answer Part 1,2,3
Find the absolute maxima and minima of \( f(x, y) \) on the given regions (4) \[ \begin{array}{l} f(x, y)=x^{2}+y^{2} \\ R=\{(x, y) \mid 0 \leq x \leq 1, \quad 0 \leq y \leq 2-2 x\} \end{array} \] \[2 answers -
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