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Advanced Math Archive: Questions from December 17, 2022

• #complex analysis
  \( \int_{0}^{2 \pi} \frac{d \theta}{1+k \sin \theta}=\frac{2 \pi}{\sqrt{1-k^{2}}}, \quad\left(k^{2}
  2 answers
• determine the limit, if it does not exist, explain why
  I. Determine el limite, en caso de que no exista explique por qué. a) \( \lim _{(x, y) \rightarrow(0,1)} \frac{\arccos (x / y)}{1+x y} \) b) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x}+\sq
  2 answers
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  Evaluate the definite integral. \[ =\frac{2}{6} \sqrt[4]{3 \cdot 4 r}\left(3+4 y^{\prime}\right)+C \] \[ \int_{0}^{2}(x+2)^{3} d x \] \[ \left.=\frac{\left(x+y^{4}\right.}{4}\right]_{0}^{2}=F(2)-f(0)=
  2 answers
• help
  4. (10pt) Solve equations: (a) \( \frac{d y}{d x}=(x+y+1)^{2}-(x+y-1)^{2} \) (b) \( x y^{\prime}=\sqrt{x^{2}-y^{2}}+y \).
  2 answers
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  Parte I: Halle las ecuaciones simétricas de la recta normal a la superficie en el punto dado 1. \( x^{2}+y^{2}+z^{2}=9,(3,3,3) \) 2. \( z=x^{2}-y^{2},(3,2,5) \) 3. \( x y z=10,(1,25) \) 4. \( x y-z=0
  2 answers
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  Parte II: Determine el ángulo de inclinación del plano tangente a la superficie en el punto dado. 1. \( 3 x^{2}+2 y^{2}-z-15,(2,2,5) \) 2. \( 2 x y-z^{2},(2,2,2) \) 3. \( x^{2}+y^{2}-5,(2,1,3) \)
  2 answers
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  4. (10pt) Solve equations: (a) \( \frac{d y}{d x}=(x+y+1)^{2}-(x+y-1)^{2} \) (b) \( x y^{\prime}=\sqrt{x^{2}-y^{2}}+y \).
  2 answers
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  Example 45. Find \( \frac{d^{2} y}{d x^{2}} \) for 1. \( y=x \ln x \) 2. \( y=\cos \left(x^{2}+1\right) \)
  2 answers
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  Evaluate \( \int_{39}^{80} \int_{0}^{4} \int_{y}^{4} \frac{x \cos z}{z} d z d y d x \)
  2 answers
• 1. Area of ​​a Triangle: Find the area of ​​a triangle whose vertices are (0,0), (2,0) and (0, 3) (15 points) 2. Collinear Points: Determine if the following points are collinear: (-2, 5), (0
  Parte II: Aplicaciones de Determinantes 1. Area de un Triángulo: Halle el área de un triángulo cuyos vértices son \( (0,0),(2,0) \) y \( (0,3) \quad(15 \) puntos) 2. Puntos Colineales: Determine s
  2 answers
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  \[ P(x, y)=42+3 x-y \] subject to : \[ \left\{\begin{array}{l} x+y \geq 4 \\ 2 x+3 y \leq 18 \\ x \geq-1 \\ y \geq 2 \\ y \leq 6 \end{array}\right. \]
  2 answers

• 

  
  Solve \( F_{y y}+3 F_{x y}+2 F_{x x}=2 x, F(x, 0)=-x, \partial_{y} F(x, 0)=0 \) \[ F(x, y)= \]
  2 answers
• ​​​​​​​
  Solve \( F_{\mathrm{yy}}+3 F_{\mathrm{xy}}+2 F_{\mathrm{xx}}=-2 x, F(x, 0)=-x, \partial_{y} F(x, 0)=0 \) \[ F(x, y)= \]
  2 answers
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  Find the numerical value if \( x=\frac{1}{2} \) and \( y=\frac{-1}{3} \). \[ \frac{2 x+y^{2}}{5 y-x^{2}} \]
  2 answers
• Solve the following DE using Laplace transform/inverse transform
  \( t^{2} y^{\prime \prime}+t y^{\prime}+t^{2} y=0 \) con \( y(0)=1 \) y \( y^{\prime}(0)=0 \)
  2 answers