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Problem: Find the point on the parabola \( y^{2}=8 x \), the distance to which from the directrix is 4 .

3.1.8 Curves of the 2-nd order

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Problem: Identify the surfaces defined by the following equations and depict them schematically: \[ 3 x^{2}+3 x y+7 y^{2}=60 \text {. } \]

3.1.9 Curves of the 2-nd order

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Problem: Calculate the derivative of the function, given parametrically: \[ \left\{\begin{array}{l} x=\cot t \\ y=\cos t \end{array}\right. \]

2.2.22 Derivatives and differentials

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Problem: Find the limit of the function using L'Hopital's rule: \[ \lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2}{1-\cos x} \text {. } \]

2.1.18 Calculation of limits

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Problem: Conduct a complete examination of the function by methods of differential calculus and plot the graph: \[ y=\frac{x^{2}-6 x+10}{x-3} \text {. } \]

2.4.8 Graphing functions using derivatives

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Problem: Let \( M \) - be a set of polynomials \( p \in P_{n} \) with real coefficients satisfying the specified conditions. Prove that \( M \) - is a linear subspace in \( P_{n} \), find its basis and dimension. Complement the basis of \( M \) to the basis of the entire space \( P_{n} \). Find the transition matrix from the canonical basis of the space \( P_{n} \) to the constructed basis. \[ n=3, M=\left\{p \in P_{3} \mid p^{\prime \prime}(a)+p^{\prime}(0)=0\right\} \text {. } \]

1.9.1 Linear spaces

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Problem: Prove that the set of matrices \( M \) is a subspace in the space of all matrices of a given size. Construct a basis and find the dimension of the subspace \( M \). Check that the matrix \( \mathrm{B} \) belongs to \( \mathrm{M} \) and decompose it according to the found basis. \( M=\left\{A \in M_{3 \times 3} \mid A=A^{T}\right. \) (symmetrical), the sums of the elements in the columns are the same, sums of elements in rows alternate\}, \[ B=\left(\begin{array}{ccc} 0 & 1 & -1 \\ 1 & -1 & 0 \\ -1 & 0 & 1 \end{array}\right) \]

1.9.2 Linear spaces

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Problem: Find the derivative of the function: \[ f(z)=z^{2}+2 i z-1 . \]

10.6.1 Analytic functions

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Problem: For what values of the parameter \( a \) is this function the real (imaginary) part of some analytic function. Find this function. \[ u=x^{3}+6 x^{2} y-3 x y^{2}-a y^{3} . \]

10.6.2 Analytic functions

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Problem: According to the results of a selective study the distribution of average milk yields from one cow per day in a farm (liters) was found. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline Interval & \begin{tabular}{c} \( 7,5- \) \\ 10,5 \end{tabular} & \begin{tabular}{c} \( 10,5- \) \\ 13,5 \end{tabular} & \begin{tabular}{c} \( 13,5- \) \\ 16,5 \end{tabular} & \begin{tabular}{c} \( 16,5- \) \\ 19,5 \end{tabular} & \begin{tabular}{c} \( 19,5- \) \\ 22,5 \end{tabular} & \begin{tabular}{c} \( 22,5- \) \\ 25,5 \end{tabular} & \begin{tabular}{c} \( 25,5- \) \\ 28,5 \end{tabular} & \begin{tabular}{c} \( 28,5- \) \\ 31,5 \end{tabular} & \begin{tabular}{c} \( 31,5- \) \\ 34,5 \end{tabular} \\ \hline \begin{tabular}{l} Number \\ of cows \end{tabular} & 2 & 6 & 10 & 17 & 33 & 11 & 9 & 7 & 5 \\ \hline \end{tabular} Find the probability \( P(15,4

20.5 Mathematical statistics

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Problem: Let \( \mu(A)<\infty \). Prove that the non-negative function \( f \) is integrable with respect to \( A \) only when the following series converges: \[ \sum_{n=1}^{\infty} 2^{n} \mu\left(A \cap\left\{x: f(x) \geq 2^{n}\right\}\right) . \]

19.6.1.1 Lebesgue measure and integration

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Problem: Calculate the integral: \[ I=\int_{0}^{1} k(x) d k^{2}(x) \text {, } \] where \( k(x) \) is the Cantor function.

19.6.1.3 Lebesgue measure and integration

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Problem: The signed measure of \( v_{F} \), constructed from the function \( F(x)=\left\{\begin{array}{ll}e^{x}, & 0 \leq x \leq 2 \\ x^{2}, & 2

19.6.1.4 Lebesgue measure and integration

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Problem: Is the set of functions \[ M=\{x \in C([0,1]):|x(t)| \leq t \forall t \in[0,1]\} \] precompact in the space \( C([0,1]) \) ?

19.5.1 Compactness

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Problem: The densities of uniformly distributed independent random variables \( X \) and \( Y \) are given: \( f_{1}(x)=1 \) in the interval \( (0 ; 1) \), outside this interval \( f_{1}(x)=0, f_{2}(y)=1 \) in the interval \( (0 ; 1) \), outside this interval \( f_{2}(y)=0 \). Find the distribution function and the distribution density of the random variable \( Z=X++Y \). Plot the graph of the distribution density \( g(z) \).

15.5.1 Two-dimensional random variables and their characteristics

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