Calculus Archive: Questions from August 03, 2022
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Id \( \nabla \cdot(\nabla \times \mathbf{F}) \), if \( \mathbf{F}(x, y, z)=7 e^{x z} \mathbf{i}+2 x e^{y} \mathbf{j}-5 e^{y z} \mathbf{k} \) \[ \nabla \cdot(\nabla \times \mathbf{F})=[ \]1 answer -
ind \( \nabla \times(\nabla \times \mathbf{F}) \) if \( \mathbf{F}(x, y, z)=8 x y \mathbf{j}+10 x y z \mathbf{k} \) \[ \nabla \times(\nabla \times \mathbf{F})=[ \]3 answers -
Find \( \nabla \cdot(\nabla \times \mathbf{F}) \) if \( \mathbf{F}(x, y, z)=8 \sin x \mathbf{i}+5 \cos (10 x-11 y) \mathbf{j}+8 z \mathbf{k} \) \[ \nabla \cdot(\nabla \times \mathbf{F})= \]3 answers -
ad \( \nabla \cdot(\nabla \times \mathbf{F}) \), if \( \mathbf{F}(x, y, z)=e^{x z} \mathbf{i}+3 x e^{y} \mathbf{j}-5 e^{y z} \mathbf{k} \) \[ \nabla \cdot(\nabla \times \mathbf{F})=1 \]1 answer -
Find the derivative \[ f(x)=\left(2 x^{3}+x^{2}-\underline{7}\left(3 x^{2}+2 x-3\right)\right. \] a) \( f^{\prime}(x)=\left(3 x^{2}+2\right)\left(3 x^{2}+2 x-3\right)+\left(6 x^{2}+2\right)\left(2 x^{1 answer -
If \( f(x, y)=\ln \left(3 x y^{3}+x^{2} y\right) \), compute \( f_{x x}(1,1) \) A. \( -\frac{5}{16} \) B. \( \frac{33}{16} \) C. \( -\frac{33}{16} \) D. \( \frac{17}{16} \) E. \( -\frac{17}{16} \)1 answer -
determine when the series are absolutely convergent, condicionally convergent or divergent
Determinar cuando la serie es absolutamente convergente, condicionalmente convergente o divergente. 1. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2 n^{2}} \) 2. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\s1 answer -
1 answer
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If \( R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\} \), then find \( \iint_{R}\left(\frac{x^{2}+1}{y^{2}+2}\right) d A \).1 answer -
2) Integrate the function \( F(x, y, z)=z-x \) over the cone \[ z=\sqrt{x^{2}+y^{2}}, \quad 0 \leq z \leq 1 \]1 answer -
1 answer
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1.which of these integrals represents the area of the surface generated when spinning the curve r=, throughout the line 0=pi/2 2.reasons why the others arent correct
1. ¿Cuál de estas integrales representa el área de la superficie generada al girar la curva \( r=e^{2 \theta}, 0 \leq \theta \leq \frac{\pi}{2} \), alrededor de la línea \( \theta=\frac{\pi}{2} \)1 answer -
3.4
Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ \begin{array}{c} y=\sqrt{\sin (x)} \\ y^{\prime}=\frac{\csc (x)\left(\frac{1}{2}\right)_{\cos (x)}}{2} \end{array} \] \[ y^{\prime \prime}=-\frac{1 answer -
19)
Evaluate the integral using the given substitution. \[ \int x \cos \left(8 x^{2}\right) d x, u=8 x^{2} \] A. \( \frac{1}{16} \sin \left(8 x^{2}\right)+C \) B. \( \frac{1}{u} \sin (u)+C \) C. \( \sin \3 answers -
1 answer
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If \( f(x, y)=x^{3} y+5 x y^{4} \), find the followin \[ \left.\frac{\partial^{2}}{\partial x^{2}} f(x, y)\right|_{(1,-1)}= \]1 answer -
help
8. Solve the initial value problem: \( y^{\prime}+y \cot x=x \csc x, y\left(\frac{\pi}{2}\right)=1 \) \( \frac{x^{2}-2+\frac{\pi^{2}}{3}}{2 \sin x} \) b) \( \frac{x^{2}+4-\frac{\pi^{2}}{2}}{2 \cos x}3 answers -
Find the derivative. a. \( y=\frac{1}{3} x^{3}-x^{2}+4 x+2 \) b. \( f(x)=2 \sqrt[3]{x}+3 \sqrt{x} \) c. \( y=\left(x^{5}-4 x^{3}+x\right)^{-4} \) d. \( f(x)=\left(x^{2}+5\right)^{3}\left(x^{2}-1\right1 answer -
Find the Jacobian of the transformation. \[ x=8 v+8 w^{2}, \quad y=7 w+7 u^{2}, \quad z=4 u+4 v^{2} \] \[ \frac{\partial(x, y, z)}{\partial(u, v, w)}= \]1 answer -
Problem 5 (18 points). Solve the following IVP: \[ y^{\prime \prime}-2 y^{\prime}+y=3 t+e^{t}, \quad y(0)=1, \quad y^{\prime}(0)=1 . \]1 answer -
1 answer
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Find \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) \), if \( \mathbf{F}(x, y, z)=5 x \mathbf{i}+\mathbf{j}+7 y \mathbf{k} \) and \( \mathbf{G}(x, y, z)=x \mathbf{i}+y \mathbf{j}-z \mathbf{k} \). \[ \n1 answer -
1 answer
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Let \( f(x, y)=x^{2}-y^{2} \) and \( g(x, y)=x-4 y+45 \) Now, define a new function \( F \) as \[ \begin{aligned} F(x, y, \lambda) &=f(x, y)-\lambda g(x, y) \\ &=x^{2}-y^{2}-\lambda(x-4 y+45) \end{ali1 answer -
1 answer
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Let \( f(x, y)=x^{2}-y^{2} \) and \( g(x, y)=x-4 y+45 \) Now, define a new function \( F \) as \[ \begin{aligned} F(x, y, \lambda) &=f(x, y)-\lambda g(x, y) \\ &=x^{2}-y^{2}-\lambda(x-4 y+45) . \end{a3 answers -
Use Laplace transform to solve the IVP \[ y^{\prime \prime}-y=e^{t} \cos t, \quad y(0)=y^{\prime}(0)=0 \]1 answer -
19. If \( g(x, y)=\sin \left(x y^{2}\right) \), then \( \frac{\partial^{2} g}{\partial x \partial y}= \)1 answer -
20. Let \( z=\sqrt{x^{2}+y}, x=2 s+3 t, y=s t \). Then \( \left.\frac{\partial z}{\partial t}\right|_{s=-2, t=3}= \)1 answer