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Problem: Find the equation of the tangent plane to the surface \( x^{3}-2 x y+2 z^{2}=8 \) at the point \( A(2 ; 2 ; 2) \).

3.4.2 Tangents and normals

1.53 $

Problem: Find the points of intersection of the straight line \( z=2 x=2 y \) and the surface \( x y=y+z \), as well as the angles between the straight line and the surface.

3.4.3 Tangents and normals

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Problem: Find admissible extremals of the functional: \[ \begin{array}{l} F(x)=\int_{0}^{\frac{\pi}{2}}\left(\dot{x}^{2}-2 \cos t x\right) d t \rightarrow \text { extr }, \\ x(0)=1, \quad x\left(\frac{\pi}{2}\right)=0 . \end{array} \]

4.1 Variational calculus

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Problem: Find admissible extremals of the functional: \[ \begin{array}{l} F(x)=\int_{-1}^{1}\left(\dot{x}^{2}+x^{2}-2 t x\right) d t \rightarrow \text { extr }, \\ x(-1)=-1, \quad x(1)=1 . \end{array} \]

4.2 Variational calculus

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Problem: Find admissible extremals in the problem with free endpoints: \[ F(x)=\int_{1}^{2}\left(\dot{x}^{2}+4 x\right) d t \rightarrow \text { extr }, x(1)=1 . \]

4.3 Variational calculus

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Problem: Write a parametric and algebraic affine span of sets. \[ \{(1,1,1,1,1),(2,2,2,2,2),(1,-1,2,-2,3)\} \text {. } \]

5.1.1 Affine transformations

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Problem: The plane \( P: x_{1}+2 x_{2}+3 x_{3}+a x_{4}=5 \) is parallel to the plane \( Q: x_{1}+2 x_{2}=1, x_{3}+x_{4}=2 \) for \( a=\cdots \).

5.1.2 Affine transformations

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Problem: Affine span of a set \( \left\{(1,1,1,1),\left(1, \frac{3}{2}, 1, \frac{3}{2}\right),(2,1,2,1)\right\} \) parallel to the line passing through the points \( (1,0,0,0) \) and \( (3, a, 2,1) \), for \( a=\cdots \).

5.1.3 Affine transformations

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Problem: Prove that all translations to vectors collinear to the vector \( \vec{a},(\vec{a} \neq 0) \), and all symmetries about axes, perpendicular to \( \vec{a} \), form a group.

5.2.1.1 Transformations on the plane

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Problem: Prove that if a figure has two and only two axes of symmetry, these axes are perpendicular.

5.2.1.2 Transformations on the plane

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Problem: Inside the triangle \( A B C \) an arbitrary point \( M \) is chosen. Find the perimeter of the triangle with vertices at the centroids of the triangles \( A B M, B C M \) and \( A C M \), if the perimeter of the triangle \( A B C \) is equal to \( 2 p \).

5.2.1.3 Transformations on the plane

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Problem: Prove that if the line segment connecting the midpoints of the bases of the trapezoid forms equal angles with its sides, the trapezoid is isosceles.

5.2.1.4 Transformations on the plane

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Problem: An elementary random function has the form \( Y(t)=a X+t \), where \( X \) is a random variable, distributed in accordance with the normal law with the parameters \( m, \sigma\left(p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-m)^{2}}{2 \sigma^{2}}}\right) \), \( a \) is a non-random value. Find the characteristics of the elementary random function \( Y(t) \).

15.1.1 Theory of random processes

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Problem: Let \( S_{0}=0, S_{k}=\xi_{1}+\cdots+\xi_{k}, 1 \leq k \leq n \), where \( \xi_{1}, \ldots, \xi_{k} \) are independent normally distributed, \( \mathcal{N}(0,1) \), random variables. Let \( \phi(x)=P\left\{\xi_{1} \leq x\right\} \), \( \mathcal{F}_{k}=\sigma\left(\xi_{1}, \ldots, \xi_{k}\right), 1 \leq k \leq n, \mathcal{F}_{0}=\{\emptyset, \Omega\} \). Show that for any \( a \in R \) the sequence \( X=\left(X_{k}, \mathcal{F}_{k}\right)_{0 \leq k \leq n} \) with \( X_{k}=\phi\left(\frac{a-S_{k}}{\sqrt{n-k}}\right) \quad \) is a martingale.

15.1.2 Theory of random processes

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Problem: A random process \( X(t)=u \cos t+v e^{t}+t \), is given, where \( u \) and \( v \) are random variables with \( M(u)=M(v)=2 ; \quad D(u)=D(v=0.2) \), \( \operatorname{cov}(u, v)=0,1 \). Find the characteristics of the random process \( Y(t)=2 X^{\prime}(t)-2 \).

15.1.3 Theory of random processes

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