Calculus Archive: Questions from July 23, 2022
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Minimize \( Q=4 x^{2}+6 y^{2} \), where \( x+y=10 \) A. \( x=4 ; y=6 \) B. \( x=6 ; y=4 \) C. \( x=10 ; y=0 \) D. \( x=0 ; y=10 \)3 answers -
Find the Jacobian of the transformation. \[ x=5 v+5 w^{2}, \quad y=8 w+8 u^{2}, \quad z=2 u+2 v^{2} \] \[ \frac{\partial(x, y, z)}{\partial(u, v, w)}= \]1 answer -
3 answers
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Given \( f(x, y)=-4 x^{5}+6 x y^{4}-2 y^{2} \), find the following numerical values: \[ \begin{array}{l} f_{x}(3,4)= \\ f_{y}(3,4)= \end{array} \]1 answer -
\( S_{1}=\left\{(x, y, z) \mid x^{2}+y^{2}=8^{2}, 0 \leq z \leq 8\right\}, S_{2}=\left\{(x, y, z) \mid x^{2}+y^{2}+(z-8)^{2}=8^{2}, z \geq 8\right\} \) \( S=S_{1} \cup S_{2} \) \( \mathbf{F}(x, y, z)=1 answer -
Find the Jacobian of the transformation. \[ x=5 v+5 w^{2}, \quad y=8 w+8 u^{2}, \quad z=2 u+2 v^{2} \] \[ \frac{\partial(x, y, z)}{\partial(u, v, w)}= \]1 answer -
Find the Taylor polynomial \( T_{3}(x) \) for the function \( f \) centered at the number \( a \). \[ f(x)=\arcsin (9 x), \quad a=0 \] \[ T_{3}(x)= \] Graph \( f \) and \( T_{3} \) on the same screen.1 answer -
3 answers
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Evaluate \( \iiint_{E} 3 x z d V \) where \( E=\{(x, y, z) \mid 1 \leq x \leq 2, x \leq y \leq 2 x, 0 \leq z \leq x+3 y\} \)1 answer -
\( f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\sin (x) \cos \left(\frac{x}{2}\right)^{2} \). Determine the Fourier series of \( \mathbf{f} \). Hint: For \( x, y \in R \) holds \[ \begin{array}{l} \sin1 answer -
Problem 1. Solve the following first-order DEs: 1.a. ( 7 pts \( ) \) \[ \left(e^{2 y}-y \cos (x y)\right) d x+\left(2 x e^{2 y}-x \cos (x y)+2 y\right) d y=0 \] \( \left(8\right. \) pts) \( \quad x\l1 answer -
roblem 2a. (6 pts) Solve the following DE: \( y^{\prime \prime \prime}+2 y^{\prime \prime}=4 \cos (2 x) \)3 answers