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Advanced Math Archive: Questions from January 30, 2023

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  Q2 (i)Are the following function linear Transformation. \[ \begin{array}{l} \text { T1: R } 3 \rightarrow R 2, T 1(x, y, z)=(x+y+z, x-z+y) \\ \text { T2: R } 3 \rightarrow \text { R 2, T2 }(x, y, z)=(
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• Find the general solution
  \( y^{\prime \prime} y=2 x\left(y^{\prime}\right)^{2} \)
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  If \( \int_{a}^{x} f(y) d y=3 \sin \left(\frac{x^{2}}{2 a^{2}} \pi\right)+b \) then \( b= \) Answer:
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  2. Determine el valor de \( b \) tal que la gráfica de la función \( f(x)=b^{x} \) pase por el punto \( \left(3, \frac{27}{8}\right) \).
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  Let \( Q(x, y) \) be the propositional function: " \( y
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  Solve the separable initial value problem. 1. \( y^{\prime}=\ln (x)\left(1+y^{2}\right), y(1)=0 \Rightarrow y= \) 2. \( y^{\prime}=6 x \sqrt{1+x^{2}}\left(1+y^{2}\right), y(0)=0 \Rightarrow y= \)
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  Match the function with its graph. \[ f(x, y)=|x|+|y| \] \[ f(x, y)=\sin (|x|+|y|) \] B. \[ f(x, y)=\frac{1}{1+x^{2}+y^{2}} \] \[ f(x, y)=|x y| \] D. \[ f(x, y)=(x-y)^{2} \] E. \[ f(x, y)=\left(x^{2}-
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• a)Prove that if x and y are real numbers and |x − y| < ε for all ε > 0 then x = y b)Let x, y, z, g ∈ R with g≥ z > y ≥ x. Show g − x ≥ z − y.
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  Find \( f_{x x}, f_{x y}, f_{y x}, f_{y y} \) if (a) \( f(x, y)=\ln (x+y) \). (b) \( f(x, y)=x e^{y}+y+1 \).
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  For each of the following functions, (a) find \( f_{x}\left(\frac{1}{5}, \frac{2}{5}\right) \), (b) \( f_{y}\left(\frac{1}{5}, \frac{2}{5}\right) \), (c) \( f_{x x}\left(\frac{1}{5}, \frac{2}{5}\right
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• Let x, y, z, g ∈ R with g≥ z > y ≥ x. Show(Prove) g − x ≥ z − y
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  \( \begin{array}{ccc}2 u_{t t}+4 u_{t}+2 u=9 u_{x x} & -\infty
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