Calculus Archive: Questions from April 24, 2023
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\( \left.\begin{array}{c}u+v^{2}-2 x+5 y^{2}=0 \\ u-v+x+y^{3}=0\end{array}\right\} \quad u_{x}=? \quad v_{y}=? \) 4)2 answers -
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Calculate the following limit value: \[ \lim _{x \rightarrow 0} \frac{\sin 3 x+\sin \left(x^{3}\right)-3 \sin x}{(\sin x)^{3}} \]2 answers -
derivar por regla de cadena
\( \frac{\partial \mathrm{z}}{\partial \mathrm{s}} \mathrm{y} \frac{\partial \mathrm{z}}{\partial \mathrm{s}} \) siendo \( \mathrm{z}=\mathrm{x}^{2} \mathrm{y}^{3}, \quad \mathrm{x}= \) scost,\( \quad2 answers -
\[ f(x)=\frac{x^{2}}{x-2} \] smaller \( x \)-value \( (x, y)= \) larger \( x \)-value \[ (x, y)=( \]1 answer -
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Encuentre la derivada direccional de la función en el punto dado en la dirección del vector \( v \). \[ f(x, y)=1+2 x \sqrt{y}, \quad(3,4), \quad \mathbf{v}=\langle 4,-3\rangle \]2 answers -
Solve the system of differential equations. \[ \begin{array}{l} \left\{\begin{array}{l} x^{\prime}=13 x-12 y \\ y^{\prime}=18 x-17 y \end{array}\right. \\ x(0)=7, y(0)=9 \end{array} \]2 answers -
(1) Utilice el teorema de Green para evaluar la integral de línea a lo largo de la curva de orientación positiva dada. \[ \int_{C} y e^{x} d x+2 e^{x} d y \] donde \( C \) es el rectángulo con vér2 answers -
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Considere la siguiente integral de línea. \[ \int_{C} x y d x+x^{2} y^{3} d y \] \( C \) gira en sentido contrario a las agujas del reloj alrededor del triángulo con vértices \( (0,0),(1,0) \) y \(2 answers -
(3) Determine cuáles de los siguientes campos vectoriales son conservativos: 1. \( \overrightarrow{\mathbf{F}}(x, y)=\left(2 x y, x^{2}\right) \) 2. \( \overrightarrow{\mathbf{F}}(x, y)=(\sin (x y),2 answers -
(5) Halla un campo vectorial conservativo que tenga la función potencial dada en cada uno de los siguientes incisos: 1. \( f(x, y)=2 x^{3}-3 x^{2} y+x y^{2}-4 y^{3} \) 2. \( f(x, y, z)=x^{2} y e^{-42 answers -
\( \begin{array}{l}\mathbf{r i}(\mathbf{F} \times \mathbf{G})=\nabla \times(\mathbf{F} \times \mathbf{G}) \\ \mathbf{F}(x, y, z)=\mathbf{i}+4 x \mathbf{j}+2 y \mathbf{k} \\ \mathbf{G}(x, y, z)=x \math2 answers -
Evaluate the double integral. D 10y x2 − y2 dA, D = (x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ x
Evaluate the double integral. \[ \iint_{D} 10 y \sqrt{x^{2}-y^{2}} d A, \quad D=\{(x, y) \mid 0 \leq x \leq 4,0 \leq y \leq x\} \]2 answers -
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6. Find \( \operatorname{div} \mathbf{F} \) and \( \operatorname{curl} \mathbf{F} \). (a) \( \mathbf{F}(x, y, z)=7 y^{3} z^{2} \mathbf{i}-8 x^{2} z^{5} \mathbf{j}-3 x y^{4} \mathbf{k} \) (b) \( \mathb2 answers -
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Find the total differential for the function \( w=12 z^{3} y \sin x \). \[ \begin{array}{l} d w=24 z^{3} y \sin x d x+12 z^{3} \sin x d y+36 z^{2} y \sin x d z \\ d w=12 z^{3} y \cos x d x+24 z^{3} \s2 answers -
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Given \( f(x, y)=x^{5} y-9 x y^{3} \) \[ \begin{array}{l} \frac{\partial^{2} f}{\partial x^{2}}= \\ \frac{\partial^{2} f}{\partial y^{2}}= \end{array} \]2 answers -
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resolver es resolver cada integral
(1) Compute los siguientes integrales. a. \( \int_{0}^{2} \int_{y}^{2} e^{x^{2}} d x d y= \) b. \( \int_{1}^{2} \int_{1}^{x^{2}} \frac{x}{4} d y d x= \) (2) La masa total \( M(5) \) de nargión sólid2 answers -
2. The change of variable \( u=y / x \) in the equation \( y^{\prime}=\tan \left(\frac{x^{2}+y^{2}}{x^{2}}\right) \) leads to the equation: A. \( x u^{\prime}=\tan \left(u^{2}+1\right) \) B. \( x u^{\2 answers -
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18] 2. Find the derivative for each of the following (do not simplify): (a) \( y=8 x^{3}+7 x^{6}+12 \) (b) \( y=\frac{2}{3} x^{-5}-5 \sqrt{x}+5 \) (c) \( y=\left(2 x^{2}+3\right)\left(x^{3}+x-2\right)2 answers -
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Find the potential function \( f \) for the field \( F \). \[ F=\frac{1}{z} i-9 j-\frac{x}{z^{2}} k \] A) \( f(x, y, z)=\frac{x}{z}+C \) B) \( f(x, y, z)=\frac{x}{z}-9 y+C \) C) \( f(x, y, z)=\frac{x}2 answers -
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