Calculus Archive: Questions from August 07, 2022
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Let \( y(x)=\frac{8}{x^{2}} \). Find \( y^{\prime}(-3) \). Enter the exact value, i.e., no decimals. \[ y^{\prime}(-3)= \] ģ3 answers -
Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For \[ y=5-z^{2}, \quad 0 \leq x, z \leq 7 ; \quad f(x, y, z)=z \] \[ \iint_{\mathcal{S}} f(x, y, z) d S= \]3 answers -
Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For \[ x^{2}+y^{2}=25, \quad 0 \leq z \leq 2 ; \quad f(x, y, z)=e^{-z} \] \[ \iint_{\mathcal{S}} f(x, y, z) d S= \]1 answer -
3 answers
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\( \mathbf{I}(\mathbf{F} \times \mathbf{G})=\nabla \times(\mathbf{F} \times \mathbf{G}) \) \( \mathbf{F}(x, y, z)=\mathbf{i}+5 x \mathbf{j}+4 y \mathbf{k} \) \( \mathbf{G}(x, y, z)=x \mathbf{i}-y \mat1 answer -
\( \mathbf{v}(\mathbf{F} \times \mathbf{G})=\nabla \cdot(\mathbf{F} \times \mathbf{G}) \) \( \mathbf{F}(x, y, z)=\mathbf{i}+9 x \mathbf{j}+9 y \mathbf{k} \) \( \mathbf{G}(x, y, z)=x \mathbf{i}-y \math1 answer -
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30-33 Find the extreme values of \( f \) subject to both constraints. 30. \( f(x, y, z)=z ; \quad x^{2}+y^{2}=z^{2}, \quad x+y+z=24 \) 31. \( f(x, y, z)=x+y+z ; \quad x^{2}+z^{2}=2, \quad x+y=1 \) 32.1 answer -
27-34 Calculate the double integral. 27. \( \iint_{R} x \sec ^{2} y d A, \quad R=\{(x, y) \mid 0 \leqslant x \leqslant 2,0 \leqslant y \leqslant \pi / 4\} \) 28. \( \iint_{R}\left(y+x y^{-2}\right) d1 answer -
15. \( f(x, y)=x^{4}-2 x^{2}+y^{3}-3 y \) 16. \( f(x, y)=x^{2}+y^{4}+2 x y \) 17. \( f(x, y)=x y-x^{2} y-x y^{2} \) 18. \( f(x, y)=\left(6 x-x^{2}\right)\left(4 y-y^{2}\right) \) 19. \( f(x, y)=e^{x}3 answers -
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