Calculus Archive: Questions from August 06, 2022
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18. Find \( \frac{d x}{d y} \) when \( y=4 x^{2} \) (assuming \( x \geq 0 \) ). \[ \begin{array}{l} \frac{2}{\sqrt{y}} \\ \frac{1}{4 \sqrt{y}} \\ 8 x \\ \frac{1}{2 \sqrt{y}} \\ \frac{4}{\sqrt{y}} \end1 answer -
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9. Find \( f^{\prime}(0) \) if \[ f(x)=\left\{\begin{array}{ll} \frac{x^{3} \sin \frac{1}{x}}{\sin x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. \]1 answer -
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Find the derivative of the function. \[ y=\left(3 x^{2}+5 x+1\right)^{3 / 2} \] A. \( y^{\prime}=\left(3 x^{2}+5 x+1\right)^{1 / 2} \) B. \( y^{\prime}=(6 x+5)\left(3 x^{2}+5 x+1\right)^{1 / 2} \) C.1 answer -
Calculate \( \iint_{S} f(x, y, z) d S \) For \[ x^{2}+y^{2}=9, \quad 0 \leq z \leq 5 ; \quad f(x, y, z)=e^{-z} \] \( \iint_{S} f(x, y, z) d S= \)3 answers -
Find the first partial derivatives of the function. \[ f(x, y, z)=5 x \sqrt{y z} \] \( f_{x}(x, y, z)= \) \[ f_{y}(x, y, z)= \] \[ f_{z}(x, y, z)= \]1 answer -
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Find the values of the function. \[ f(x, y)=x^{2} /\left(1+y^{2}\right) \] (a) \( f(-5,0) \) (b) \( f(12,1) \) (c) \( f\left(\frac{1}{2},-\frac{1}{2}\right) \) (d) \( f(-7, y) \)3 answers -
Find the first partial derivatives of the function. \[ g(x, y)=7 \ln (4 x+\ln y) \] \[ g_{x}(x, y)= \] \[ g_{y}(x, y)= \]1 answer -
5.) show all work
5) Find the eigenvalues of \( A=\left[\begin{array}{ccc}1 & 3 & 0 \\ 3 & 1 & 0 \\ 0 & 0 & -2\end{array}\right] \) (15 pts)1 answer