Option 13 IDZ 6.2

IDZ - 6.2

Below are the solutions to the problems:

1. Find y´ and y":

   Equation: x² + y² = sin y

   Answer:

   y´= (-2xy + cosy) / (2y - siny)

   y"= (-2x(2y - siny) - (cosy - 2y + siny)(-2xy + siny)) / (2y - siny)²

2. Find x and y:

   Equations: x = 5cos t; y = 4sin t

   Answer:

   Substitute t = arccos(x/5) into the equation y = 4sin t and get:

   y = 4sin(arccos(x/5)) = 4 * sqrt(1 - (x/5)²)

3. Find y‴(x0):

   Equation: y = x sinx; x0 = π/2

   Answer:

   y" = (2cosx - xsinx) / x²

   y‴ = [(x² - 4)cosx - 2xsinx] / x³

   Substitute x0 = π/2 and get y‴(x0) = -4/π³

4. Write down the formula for the nth order derivative:

   Equation: y = exp(4x)

   Answer:

   yⁿ = 4ⁿ * exp(4x)

5. Write down the normal equation:

   Equation: y = 3tg(2x) + 1; x = π/2

   Answer:

   Angular coefficient of the normal k = -1/k`, where k` is the angular coefficient of the tangent.

   k` = y' = 6cos(2x)/cos²(2x) = 6tan(2x)

   k = -1/(6tan(2x)) = -tan(x/2)

   Normal equation: y - y₀ = k(x - x₀), where (x₀, y₀) are the coordinates of the tangent point.

   Substitute x₀ = π/2 and get y - 1 = -cot(π/4)(x - π/2) => y + x = 2 + π/2

6. Find the speed of movement of the material point:

   Law of motion: S = 5t³/3 - 2t + 7; t = 4 s

   Answer:

   We obtain the speed of movement by calculating the derivative of the law of motion:

   v = S' = 5t² - 2

   Substitute t = 4 s and get v = 78 m/s

The product is a task for solving mathematical problems, included in the Individual Homework (IH) for the mathematics course with number 6.2 and option 13.

The task consists of six tasks:

1. Find y´ and y" for the equation x² + y² = sin y.
2. Find x and y for the system of equations x = 5cos t, y = 4sin t.
3. Find y‴(x0) for the function y = x sinx for x0 = π/2.
4. Write down the formula for the nth order derivative of the function y = exp(4x).
5. Write down the equation of the normal to the curve y = 3tg(2x) + 1 at the point with the abscissa x = π/2.
6. Find the speed of motion of a material point according to the law of motion S = 5t³/3 - 2t + 7 at time t = 4 s.

For each problem a corresponding solution is given.

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Option 13 IDZ 6.2 is a set of problems in mathematics, which includes the following tasks:

1. Find the derivatives y' and y'' of the function y if x² + y² = sin y.
2. The parametric equations of the curve are given: x = 5cos(t), y = 4sin(t). It is necessary to find the derivatives of y' and y'' with respect to the parameter t.
3. For the function y = x*sin(x), it is necessary to calculate the third derivative y‴(x0) at the point x0 = π/2.
4. Write the nth order derivative formula for the function y = exp(4x).
5. Find the equation of the normal to the curve y = 3tg(2x) + 1 at the point with abscissa x = π/2.
6. The law of motion of a material point is given: S = 5t³/3 - 2t + 7. It is necessary to find the speed of its movement at the moment of time t = 4 s.

These problems relate to various areas of mathematics, such as differential calculus, parametric equations, higher order derivatives, normals and velocities. Solving these problems will help improve your skills in working with functions and their derivatives, as well as your understanding of the geometric properties of curves and the movement of material points.

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