IDZ 10.2 – Option 1. Solutions Ryabushko A.P.

1. Let us find the equations of the tangent plane and the normal to the surface S at the point M0(2, 1, –1).

We are looking for the gradient of the surface S: grad(S) = (2x-4, 2y, 2z+6). At the point M0(2, 1, –1) we have: grad(S) = (0, 2, 4). Since the tangent plane to the surface S at point M0 is parallel to the gradient of the surface, the equation of the tangent plane has the form: 0(x-2) + 2(y-1) + 4(z+1) = 0, that is, y + 2z - 1 = 0. The equation of the normal to the surface S at point M0 has the form: 2(x-2) + 2(y-1) + 4(z+1) = 0, that is, x + y + 2z - 8 = 0.

1. Let's find the second partial derivatives of the function z=ex2-y2 and make sure that z''xy = z''yx.

We calculate the first partial derivatives: z'x=2xe^(x^2-y^2), z'y=-2ye^(x^2-y^2). Next we find the second partial derivatives: z''xy=2e^(x^2-y^2)-4x^2e^(x^2-y^2), z''yx=2e^(x^2-y ^2)-4y^2e^(x^2-y^2). Note that z''xy = z''yx, which means that the function z=ex2-y2 satisfies the condition of equality of mixed derivatives.

1. Let's check whether the function u(x,y,z) = 2x^2+3y^2+z^2-4xy-6xz+8yz satisfies the Laplace equation.

Let us calculate Laplace from the function u(x,y,z): Δu = u''xx + u''yy + u''zz = 4 + 6 + 2 = 12. Since Δu is not equal to zero, then the function u(x ,y,z) does not satisfy the Laplace equation.

1. Let us examine the function z=y√x – 2y^2 – x + 14y for an extremum.

We calculate the partial derivatives: z'x= y/(2√x) - 1, z'y= √x - 4y + 14. Find the stationary points: z'x=0 => y/(2√x) = 1 = > y=2√x, z'y=0 => √x - 4y + 14 = 0 => √x - 8√x + 14 = 0 => x = 4, y = 4. Check the sufficient conditions for the extremum: z ''xx= -y/(4x^(3/2)) 0. Since z''xx

1. Let's find the largest and smallest values ​​of the function z=3x+y-xy in the area D, bounded by the lines y=x, y=4, x=0.

We express y in terms of x in the equation y=4: y=4. Substitute y=x into the function z=3x+y-xy: z=4x-2x^2. We calculate the derivatives: z'x=4-4x, z''xx=-4. Find the critical point: z'x=0 => x=1, then y=1. We check the sufficient conditions for the extremum: z''xx=-4

IDZ 10.2 – Option 1. Solutions Ryabushko A.P. is a digital product, which is a collection of solutions to mathematics tasks developed by A.P. Ryabushko. It contains detailed and clear solutions to assignments that will help students better understand the material and prepare for exams.

This product is available for purchase in a digital goods store in a beautifully designed html format. Beautiful design makes it easy to read and study the material, as well as quickly find the information you need. This product is suitable for both students and teachers who want to check the correctness of solving tasks.

Acquisition of IDZ 10.2 – Option 1. Decisions of Ryabushko A.P. in the digital goods store is a quick and convenient way to get quality material to prepare for exams and improve your knowledge in mathematics.

***

IDZ 10.2 – Option 1. Solutions Ryabushko A.P. is a set of problems in mathematical analysis, which covers the following tasks:

1. It is necessary to find the equations of the tangent plane and normal to a given surface S at the point M0(x0, y0, z0). The surface S is given by the equation x2 + y2 + z2 + 6z – 4x + 8 = 0, and the point M0 has coordinates (2, 1, – 1).

2. It is necessary to find the second partial derivatives of the indicated functions and check that z''xy = z''yx. The function z(x,y) is given by the equation z = ex2-y2.

3. It is necessary to check whether the function u satisfies the specified equation.

4. It is necessary to examine the function z(x,y) = y√x – 2y2 – x + 14y for its extremum.

5. It is necessary to find the largest and smallest values ​​of the function z(x,y) = 3x + y – xy in the region D, limited by the given lines y = x, y = 4, x = 0.

The set of problems is designed in Microsoft Word 2003 using the formula editor. Solutions to problems are presented in detailed form.

***

1. A very convenient digital product for exam preparation.
2. The solutions to the problems in IPD 10.2 – Option 1 are well written and easy to understand.
3. A large selection of problems, allowing you to practice solving different types of tasks.
4. The digital format allows you to quickly and conveniently check your answers.
5. Decisions Ryabushko A.P. contain detailed explanations, which helps to better understand the material.
6. Having answers to each task significantly saves time when checking your solutions.
7. IDZ 10.2 – Option 1 is perfect for those preparing for an exam in mathematics or physics.
8. A very useful digital product for exam preparation.
9. Solutions to problems in IPD 10.2 – Option 1 are well structured and easy to understand.
10. With the help of this digital product, I was able to significantly improve my level of knowledge in mathematics.
11. Decisions Ryabushko A.P. in IDZ 10.2 – Option 1 helped me better understand the theory and consolidate practical skills.
12. A very convenient format for presenting the material, which allows you to quickly and effectively prepare for the exam.
13. Solutions to problems in IDZ 10.2 – Option 1 are well structured and at a high level.
14. An excellent digital product that helps with exam preparation and improves understanding of mathematical theory.
15. Decisions Ryabushko A.P. in IDZ 10.2 - Option 1 really help in mastering tasks and improving academic performance.
16. I highly recommend this digital product to anyone who wants to pass their math exam.
17. Many thanks to the author for a high-quality digital product that helped me prepare for the exam and get an excellent grade.

Peculiarities:

Great problem solving! With the help of this IDZ, I successfully completed the exam.

I am grateful to the author for detailed and understandable explanations of problem solutions.

IDZ 10.2 - Option 1 is an excellent tool for preparing for the math exam.

With the help of this IDZ, I have greatly improved my knowledge of differential equations.

Solutions Ryabushko A.P. in IPD 10.2 - Option 1 are very clear and easy to understand.

Thank you for IDZ 10.2 - Option 1! He helped me prepare for the exam and get a high grade.

I recommend this digital product to anyone who wants to improve their math skills and pass the exam successfully.