Solution to problem 17.2.9 from the collection of Kepe O.E.

Problem 17.2.9 from the collection of Kepe O.?. is as follows: given a system of equations of the form Ax = b, where A is a square matrix of order n, x and b are vectors of dimension n. It is required to find a solution to this system using the Gaussian method.

To solve the problem you need to perform the following steps:

1. Reduce matrix A to triangular form using the Gaussian method. This is achieved by sequentially subtracting the rows of the matrix from each other in order to obtain zeros under the main diagonal.
2. After reducing the matrix A to triangular form, the solution to the system Ax = b is found using the reverse of the Gaussian method - first the last element of the solution vector is found, then the penultimate one, and so on until the first element.

Solution to problem 17.2.9 from the collection of Kepe O.?. allows you to understand the Gauss method and its application to solve systems of linear equations. This problem is a typical example for studying the Gauss method and its application in solving practical problems.

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Problem 17.2.9 from the collection of Kepe O.?. is formulated as follows:

"Given a function $f(x) = x^3 - 12x + a$. Study it for increasing and decreasing, find extrema and intervals of monotonicity for different values ​​of the parameter $a$."

To solve this problem, it is necessary to calculate the first and second derivatives of the function $f(x)$:

$f'(x) = 3x^2 - 12$

$f''(x) = 6x$

Next, you need to find the roots of the first order derivative:

$f'(x) = 0 \Leftrightarrow 3x^2 - 12 = 0 \Leftrightarrow x^2 = 4 \Leftrightarrow x_{1,2} = \pm 2$

Thus, the extremum points of the function $f(x)$ will be located at the points $x = -2$ and $x = 2$.

Next, it is necessary to analyze the signs of the derivatives in the intervals between the found roots and beyond them in order to determine the intervals of monotonicity of the function and its extrema.

If $x < -2$, then $f'(x) < 0$, which means that the function $f(x)$ is decreasing on this interval. If $-2 < x < 2$, then $f'(x) > 0$, which means that the function $f(x)$ is increasing on this interval. If $x > 2$, then $f'(x) < 0$, which means that the function $f(x)$ is decreasing on this interval.

Thus, the monotonicity intervals of the function $f(x)$ for different values ​​of the parameter $a$ will depend on the position of the roots of the first-order derivative and will be determined by the signs of the derivatives on these intervals.

For example, if $a < -16$, then both roots of the first-order derivative will be outside the domain of definition of the function $f(x)$, and the function $f(x)$ will decrease throughout the entire domain of definition. If $a = -16$, then one of the roots will coincide with the left end of the domain of definition of the function $f(x)$, and on this interval the function $f(x)$ will decrease, and on the rest of the domain of definition it will increase. If $-16 < a < 16$, then both roots will be inside the domain of definition of the function, and the function $f(x)$ will increase on the central monotonicity interval, and decrease on the two outer ones. If $a = 16$, then one of the roots will coincide with the right end of the domain of definition of the function $f(x)$, and on this interval the function $f(x)$ will increase, and in the rest of the domain of definition it will decrease. If $a > 16$, then both roots of the first-order derivative will be outside the domain of definition of the function $f(x)$, and the function $f(x)$ will increase throughout the entire domain of definition.

Solution to problem 17.2.9 from the collection of Kepe O.?. consists in determining the main moment of inertia of a homogeneous disk of radius r = 0.2 m with mass m = 2 kg relative to the axis of rotation O, displaced at a distance e = 0.1 m from the center of mass C. The disk rotates uniformly accelerated with angular acceleration ε = 10 rad /s^2.

The main moment of inertia forces is determined by the formula: I = I0 + md^2, where I0 is the moment of inertia of the disk relative to the axis passing through its center of mass, m is the mass of the disk, d is the distance between the axes of rotation.

To find the main moment of inertia, it is necessary to determine the moment of inertia of the disk relative to the axis passing through the displaced center of mass. The moment of inertia of the disk relative to such an axis can be found using the Steiner formula: I = I0 + md^2, where I0 is the moment of inertia of the disk relative to the axis passing through its center of mass, m is the mass of the disk, d is the distance between the axes of rotation.

For this problem, the moment of inertia of the disk relative to the axis passing through its center of mass is equal to I0 = (m*r^2)/2, where r is the radius of the disk. The distance between the axes of rotation is d = r - e.

Thus, the main moment of inertia of the disk relative to the axis displaced by a distance e from the center of mass is equal to: I = (mr^2)/2 + m(r-e)^2 = 0.6 kg*m^2.

Answer: 0.6 kg*m^2.

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