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Problem: Given the matrix of the operator \( A=\left(\begin{array}{lll}2 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1\end{array}\right) \) in basis \( \left(e_{1}, e_{2}, e_{3}\right) \). Find its matrix in basis \( \left(e_{1}^{\prime}, e_{2}^{\prime}, e_{3}{ }^{\prime}\right) \), where \( \left\{\begin{array}{c}e_{1}{ }^{\prime}=e_{1}-e_{2}+e_{3} \\ e_{2}{ }^{\prime}=-e_{1}+e_{2}-2 e_{3} \\ e_{3}{ }^{\prime}=-e_{1}+2 e_{2}+e_{3}\end{array}\right. \).

1.7.5 Linear transformations

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Problem: Given the operators \( A x=\left\{x_{2}-x_{3}, x_{1}, x_{1}+x_{3}\right\} \), \( B x=\left\{x_{2}, 2 x_{3}, x_{1}\right\} \). Find the operator \( \left(B^{2}-2 A\right) \).

1.7.6 Linear transformations

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Problem: Investigate the quadratic form for signdefiniteness, bring it to the canonical form by orthogonal transformation and write down the transition matrix. \[ \begin{array}{l} f\left(x_{1}, x_{2} x_{3}\right)=3 x_{1}{ }^{2}+x_{2}{ }^{2}-\frac{3}{2} x_{3}{ }^{2}+2 \sqrt{3} x_{1} x_{2}- \\ -x_{1} x_{3}+\sqrt{3} x_{2} x_{3} . \end{array} \]

1.8.1 Quadratic forms

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Problem: Compute the limits of the functions: 1. \( \lim _{x \rightarrow 2} \frac{2^{x}-x^{2}}{(x-2)^{2}} \), 2. \( \lim _{x \rightarrow 0} \frac{\tan ^{-1}(2 x)-\tan (2 x)}{x^{3}} \), 3. \( \lim _{x \rightarrow 0} \frac{x^{2}+2^{x}}{2 x^{3}-2} \).

2.1.1 Calculation of limits

1.43 $

Problem: Compute the limit of the function \[ \lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{\cot x}{x}\right) \text {. } \]

2.1.2 Calculation of limits

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Problem: Find the limits of functions without using L'Hopital's rule. 1. \( \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1}+\sqrt{x}}{\sqrt[5]{x^{3}+x}-2 x} \), 2. \( \lim _{x \rightarrow 0} \frac{\cos 2 x-\cos 4 x}{3 x^{2}} \) 3. \( \lim _{x \rightarrow \infty}\left(\frac{2 x+5}{1+2 x}\right)^{5 x} \).

2.1.3 Calculation of limits

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Problem: Check that function \( U=\ln \frac{1}{\sqrt{x^{2}+y^{2}}} \) is a solution to the partial differential equation \( \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \).

2.2.1 Derivatives and differentials

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Problem: Calculate the derivative of the complex function: \[ f(x)=\tan ^{-1}\left(\frac{\ln (x+2)}{\sin \sqrt{2 x}}\right), g(x)=\sqrt{\cos 4 x} . \]

2.2.2 Derivatives and differentials

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Problem: Calculate the derivative of the implicitly defined function \( y^{2}-\sin (2 x+y)=x^{3} \).

2.2.3 Derivatives and differentials

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Problem: Find the first and second order partial derivatives of the function: \[ z=\frac{1}{\tan ^{-1} \frac{y}{x}} . \]

2.2.4 Derivatives and differentials

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Problem: Find the first and second order partial derivatives of the function: \[ z=\ln \left(\frac{1}{\sqrt[3]{x}}-\frac{1}{\sqrt[3]{y}}\right) \]

2.2.5 Derivatives and differentials

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Problem: Find the first and second order total differentials of the function: \[ \omega=\sin ^{-1} \frac{x}{y} \text {. } \]

2.2.6 Derivatives and differentials

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Problem: Approximately calculate (using the differential): \[ A=(1,02)^{2} \cdot(0,97)^{2} \text {. } \]

2.2.7 Derivatives and differentials

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Problem: Find the partial derivatives and the differentials of the first and second order: \[ z=\tan ^{-1} \frac{y}{x} . \]

2.2.8 Derivatives and differentials

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Problem: A complex function is given: \[ z=\frac{y}{x} \text {, where } x=e^{t}, y=1-e^{2 t} \text {. } \] Find \( \frac{d z}{d t}, \quad \frac{d^{2} z}{d t^{2}} \).

2.2.9 Derivatives and differentials

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