Example 1. Form the equation of a straight line using the least squares method based on the data provided in the table.

| $x$ | 1 | 2 | 3 | 5 |
| :---: | :---: | :---: | :---: | :---: |
| $y$ | 3 | 4 | 2.5 | 0.5 |
Solution. Let's determine the coefficients of the system of equations:

$$
\begin{aligned}
& A_{1}=\sum x_{i}^{2}=39, B_{1}=\sum x_{i}=11, C_{1}=\sum x_{i} y_{i} \\
& A_{2}=11, B_{2}=4, C_{2}=\sum y_{i}=10
\end{aligned}
$$

From the system of equations

$$
\left\{\begin{array}{l}
39 k+11 p=21 \\
11 k+4 p=10
\end{array}\right.
$$

we find: $k=-\frac{26}{35}, p=\frac{159}{35}$.

Answer. $y=-\frac{26}{35} x+\frac{159}{35}$.

## Exercises

Using the method of least squares, derive the equation of the line $y=$ $=k x+p$ from the tabular data.

1. 

| $x$ | 0.5 | 0.1 | 2.0 | 2.5 | 3.0 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $y$ | 0.62 | 1.64 | 3.7 | 5.02 | 6.04 |

2. 

| $x$ | 1 | 2 | 3 | 5 |
| :---: | :---: | :---: | :---: | :---: |
| $y$ | 0 | 1 | -0.5 | -2.5 |

## Answers

1. $y=2.08 x-0.5$. 2. $y=-\frac{26}{35} x+\frac{54}{35}$.

## Control Tasks

1. Find and plot the domains of definition of the given functions:
1) $z=\sqrt{16-x^{2}-y^{2}}$; 2) $z=\sqrt{9-2 x^{2}+3 y^{2}}$; 3) $z=x-\arcsin y$;
2) $z=\frac{1}{\sqrt{x y}}$; 5) $z=\sqrt{x^{2}-y}$; 6) $z=\sqrt{25-5 x^{2}-15 y^{2}}$.
2. Find the total differential of the functions at the given points:
1) $z=\arcsin (\sqrt{x}+2 y), \quad(4,1) ; \quad 2) z=x^{y}, \quad(4,2)$;
2) $\left.z=\sqrt{\frac{x^{2}-y^{2}}{x^{2}+y^{2}}}, \quad(5,4) ; \quad 4\right) z=\arccos \sqrt{\frac{x}{y}}, \quad(1,4)$;
3) $z=\ln \left(x^{2}+y^{2}\right), \quad(3,4)$.
3. Prove that the given functions satisfy Laplace's equation $\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0$
1) $z=\ln \left(x^{2}+y^{2}\right) ;$ 2) $z=\frac{1}{\sqrt{x^{2}+y^{2}}}$; 3) $z=\operatorname{arctg} \frac{y}{x}$.
4. Find the derivative of the given function at the given point in the direction of the given vector. Also find the gradient of each function at the given point:
1) $z=x^{2}+y^{2}+x y, M_{0}(1,2), \vec{l}=2 \vec{i}+\vec{j}$
2) $z=\operatorname{arctg} \sqrt{\frac{y}{x}}, M_{0}(3,3), \vec{l}=(1,-2)$;
3) $z=\arcsin \sqrt{x-y}, M_{0}\left(1, \frac{1}{2}\right), \vec{l}=(-2,1)$;
4) $z=\sqrt{\frac{2 x+3 y}{x-2 y}}, M_{0}(3,1), \vec{l}=\vec{i}-\vec{j}$
5. Find the extremum of the given function:
1) $z=4 x^{2}+8 x y+2 y^{2}-10 x+3 y+1$; 2) $z=x y+\frac{10}{x}+\frac{20}{y}$
2) $z=x^{3}+3 x y^{2}+5 x+8 y$; 4) $z=x^{2} y(4-2 x-y)$.
6. Find the maximum and minimum values of the function in the closed domain $D$ :
1) $z=x^{2}+2 x y, D-$ triangle $A B C: A(0,0), B(4,0), C(0,6)$;
2) $z=x^{2}-2 x y+y^{2}, D$ - rectangle $A B C D: A(0,0), B(4,0)$, $C(4,6), D(0,6)$
3) $z=x^{2}+y^{2}-3 x y-2 x-2 y$, for $x \leqslant 0, y \leqslant 0, x+y \leqslant 5$;
7. On the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ find the point closest to the line $3 x-4 y=42$.

## List of Literature

1. Bugrov Ya.S., Nikolsky S.M. Elements of Linear Algebra and Analytic Geometry. - M.: Nauka, 1988.
2. Bugrov Ya.S., Nikolsky S.M. Differential and Integral Calculus. - M.: Nauka, 1988.
3. Piskunov N.S. Differential and Integral Calculus. Vol. 1 and 2. M.: Nauka, 1988.
4. Kletenik D.V. Collection of Problems in Analytic Geometry. - M.: Nauka, 1988.
5. Collection of Problems in Mathematics for Technical Universities. Linear Algebra and Fundamentals of Mathematical Analysis / Ed. by A.V. Efimov and B.P. Demidovich. M.: Nauka, 1988.
6. Collection of Problems in Mathematical Analysis / Ed. by B.P. Demidovich. M.: Nauka, 1983.
Educational Publication

LUNGU Konstantin Nikitovich MAKAROV Evgeny Vasilyevich

## HIGHER MATHEMATICS. GUIDANCE FOR SOLVING PROBLEMS

Part 1

Editor V.S. Aralovich Proofreader N.A. Likhachyova Original layout: Ya.V. Zhabytsky Layout cover: A.Yu. Alyokhina

Publishing firm "Physical and Mathematical Literature" MAIK "Nauka/Interperiodica"

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[^0]:    ${ }^{1}$ ) Subtraction in the formulas of point $3^{\circ}$ is replaced by addition for arithmetic reasons (subtraction is addition of the opposite number).

[^1]:    5 K. N. Lungu, E. V. Makarov

[^2]:    1) Here we indicate not all variants of the answers.