Advanced Math Archive: Questions from November 15, 2022
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\( \left(3 x y^{2}+2 \sin y-4 x^{2}\right) d x+\left(2 x^{2} y+x \cos y\right) d y=0, \quad y(1)=0 \)2 answers -
(40 points) Solve the initial value problem \[ y^{\prime \prime}+3 x y^{\prime}-12 y=0, y(0)=2, y^{\prime}(0)=0 \]2 answers -
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Sketch the vector field \( \mathbf{F}(x, y)=\langle 0,-y\rangle \) Sketch the vector field \( \mathbf{F}(x, y)=\langle-y, x\rangle \).2 answers -
Please solve and show all work. Thank You!
B. \( y^{\prime \prime}-5 y^{\prime}=0, \quad y(0)=1, y^{\prime}(0)=2 \)2 answers -
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i) Show that when \( z=e^{-t} \sin \theta, \frac{\partial^{2} z}{\partial t^{2}}=-\frac{\partial^{2} z}{\partial \theta^{2}} \) ii) If \( z=\frac{x}{y} \ln y \), evaluate \( \frac{\partial^{2} z}{\par2 answers -
Use Lagrange multipliers to find the maximum and minimum values of the Functionf(x,y,z)=8x−4zsubjectaax# +10y# +z# =5.
1) [8 pts.] Utiliza multiplicadores de Lagrange para hallar los valores máximos y minimos de la función \( f(x, y, z)=8 x-4 z \) sujeta a \( x^{2}+10 y^{2}+z^{2}=5 \).2 answers -
3. Solve the differential equations: a) \( x^{2}+y-x y^{\prime}=0 \); b) \( x y^{2}+y-x y^{\prime}=0 \); c) \( y^{2}+(x y-1) y^{\prime}=0 \); d) \( x \sin y+y \cos y+(x \cos y-y \sin y) y^{\prime}=0 \2 answers -
1. Solve the differential equations: a) \( x y^{\prime}=\operatorname{tg} y \); b) \( y^{\prime}=(x \sin x-\cos x) \cdot y \); c) \( (2 x-y) y^{\prime}+2 y=0 \); d) \( y^{\prime}+1=e^{-y} \); e) \( y^2 answers -
Find the volume of the solid bounded by the planes z = x, y = x, x + y = 2 y, z = 0.
2) [7 pts.] Halla el volumen del sólido acotado por los planos \( z=x, y=x, x+y=2 \) \[ y, z=0 \text {. } \]2 answers -
A pool of 20 feet by 30 feet is filled with water. The depth is measured at intervals of 5 feet, starting at a corner of the pool and the values were placed on the table. Estimate the volume of water
[7 pts.] Una piscina de 20 pies por 30 pies se llena de agua. La profundidad se mide en intervalos de 5 pies, comenzando en una esquina de la piscina y los valores se colocaron en la tabla. Estima el2 answers -
Evaluate the integral ∫ ∫ ∫ x#e& dV, where it is limited by the parabolic cylinder #, Z = 1 − and and the planes z = 0, x = 1 and x = −1. Note: You do not have to solve the last integra
4) [7 pts.] Evalúa la integral \( \iiint_{E} x^{2} e^{y} d V \), donde Eestá acotada por el cilindro paraboloide \( z=1-y^{2} y \operatorname{los} \) planos \( z=0, x=1 \) y \( x=-1 \). Nota: No tie2 answers -
Find ∫ xyz#ds, where C is the line segment from (−1, 5, 0) to (1, 6, 4).
6) [7 pts.] Halla \( \int_{C} x y z^{2} d s \), donde \( C \) es el segmento de línea desde \( (-1,5,0) \) hasta \( (1,6,4) \).2 answers -
To be able to evaluate the integral ∫* ∫# $ √) &!' % % dydx we need to reverse the integration limits. The integral that would result from such a reversal is
2. Para poder evaluar la integral \( \int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{y^{x}+1} d y d x \) necesitamos revertir los límites de integración. La integral que resultaria de dicha reversión es:2 answers -
Evaluating the integral ∫# ∫* x#e& dydx is equivalent to evaluating:
Evaluar la integral \( \int_{1}^{2} \int_{3}^{4} x^{2} e^{y} d y d x \) es equivalente a evaluar: a) \( \int_{3}^{4} \int_{1}^{2} x^{2} e^{y} d y d x \) b) \( \int_{1}^{2} x^{2} d x \int_{3}^{4} e^{y}2 answers -
(1 point) Calculate all four second-order partial derivatives of \( f(x, y)=\sin \left(\frac{5 x}{y}\right) \). \[ \begin{array}{l} f_{x x}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \\ f_{y y}(x, y)2 answers -
Consider the following equation and determine \( (\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}): y=f(x)=x^{3 / 2}+1 \) a) The arc length between \( \mathrm{x}=0 \) and \( \mathrm{x}=2 \) uses the ex1 answer -
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4. Solve the IVP. (Similar questions 4.3: \( 9-12 \) ) \[ \begin{aligned} 4 y^{\prime \prime \prime}-y^{\prime} &=4 t+12 e^{2 t} \\ y(0) &=0 \\ y^{\prime}(0) &=0 \\ y^{\prime \prime}(0) &=0 \end{align2 answers -
Find the marginal-product functions for the Cobb-Douglas production function \[ y=A x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}} x_{3}^{\alpha_{3}} x_{4}^{\alpha_{4}}, \quad A>0,02 answers -
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Solve the initial value problem
\( y^{\prime \prime}-y=\left\{\begin{array}{ll}1, & t3\end{array} \quad y(0)=1, y^{\prime}(0)=2\right. \)2 answers -
Solve the initial value problem:
\( y^{\prime \prime}+4 y=\left\{\begin{array}{lll}3 \sin t, & 0 \leq t \leq 2 \pi \\ 0, & t>2 \pi\end{array} \quad y(0)=1, y^{\prime}(0)=3\right. \)2 answers -
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\[ \left.\begin{array}{r} x+y+z=5 \\ y+2 z=8 \\ y-z=-4 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 5 \\ 8 \\ -4 \end{array}\right] \] Solve by invert2 answers -
please do 3 and 5, show all steps
To do: Use LT to solve each initial-value \( \begin{aligned} \text { 1. } x^{\prime} &=3 x-2 y ; & & x(0)=1, \\ y^{\prime} &=3 y-2 x ; & & y(0)=1 \\ \text { 2. } x^{\prime} &=x-y ; & & x(0)=-1 \\ y^{\1 answer -
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