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Problem: Find the envelope of the family of curves \( \left(x+C^{2}\right)^{3}=(y-C)^{2} \).

7.8 Differential geometry

2.5 $

Problem: At what \( a \) and \( b \) are the torsion and the curvature of the curve \( x=a \sinh t \), \( y=a \cosh t, z=b t \) equal?

7.9 Differential geometry

4.99 $

Problem: Find the equations of the tangent plane of the torus. \[ \begin{array}{l} x=(7+5 \cos u) \cos v, \quad y=(7+5 \cos u) \sin v, \\ z=5 \sin u, \quad \text { at the point } M(u, v) . \end{array} \]

7.10 Differential geometry

3.75 $

Problem: Find the first quadratic form of the ellipsoid \( x^{2}+2 y^{2}+4 z^{2}=1 \).

7.11 Differential geometry

4.99 $

Problem: For the curve given parametrically, find: 1. The vectors of the accompanying trihedron at the point \( t=t_{0} \); 2. The planes and lines of the accompanying trihedron at the point \( t=t_{0} \) 3. The tangent lines parallel to coordinate planes; 4. The contiguous planes perpendicular to the coordinate axes; 5. The curvature and torsion of the curve at the point \( t=t_{0} \). \[ \left\{\begin{array}{c} x(t)=2 t^{3} \\ y(t)=3 t \quad t_{0}=1 . \\ z(t)=3 t^{2} \end{array}\right. \]

7.14 Differential geometry

12.49 $

Problem: For the surface given parametrically, find: 1. The unit vector of the normal at the point \( \left(u=u_{0}, v=v_{0}\right) \); 2. The equation of the tangent plane and the normal at the point \( \left(u=u_{0}, v=v_{0}\right) \) 3. The volume of the tetrahedron formed by the tangent plane at the point ( \( u=u_{0}, v=v_{0} \) ) to the given surface and coordinate planes; 4. The normals parallel to coordinate planes; 5. The first quadratic form of the surface; 6. The second quadratic form of the surface; 7. The angle between the coordinate lines of the surface at the point \( \left(u=u_{0}, v=v_{0}\right) \); 8. The Gaussian and mean curvatures of the surface; 9. The elliptic, hyperbolic and parabolic points on the given surface. \[ \left\{\begin{array}{c} x(u, v)=3 \cos u \cos v \\ y(u, v)=3 \cos u \sin v \quad u_{0}=\frac{\pi}{3}, \quad v_{0}=\frac{\pi}{4}, \quad M_{0}\left(u_{0}, v_{0}\right) . \\ z(u, v)=5 \sin u \end{array}\right. \]

7.15 Differential geometry

24.97 $

Problem: Find the curvature and torsion of a curve given by the equations \( x^{2}-y^{2}+z^{2}=1 \) \( y^{2}+z-2 x=0 \quad \) at the point \( M(1 ; 1 ; 1) \).

7.16 Differential geometry

7.49 $

Problem: For the line \( \vec{\tau}(t)=t^{3} \vec{\imath}+(t+1)^{2} \vec{\jmath}+\sqrt{t^{2}+1} \vec{k} \) find at the point \( t=-2 \) : a) the unit vector of tangent \( \vec{\tau} \); b) the unit vector of binormal \( \vec{b} \); c) the unit vector of principal normal \( \vec{n} \); d) the curvature \( K \); e) the equation of the tangent; f) the equation of the normal plane.

7.17 Differential geometry

12.49 $

Problem: For the line \( y=-3 x^{3} \) find: a) the curvature, b) the radius of the curvature, \( c \) ) the center of the curvature at the point with abscissa 1.

7.18 Differential geometry

3.75 $

Problem: The random process \( X(t) \) is given by the canonical expansion \( \quad X(t)=(t+1) u+\sin t \cdot v+e^{-3 t} \), where \( D(u)=D(v)=3 \). Find: a) the characteristics of the random process \( X(t) \); b) the characteristics of the random process \[ Y(t)=e^{2 t} \int_{0}^{t} X(\tau) d \tau \text {. } \]

15.1.4 Theory of random processes

4.99 $

Problem: Let \( X(t) \) be a stationary random function with a correlation function \( K_{x}(\tau)=e^{-\alpha|\tau|}, \alpha>0 \). Find the spectral density \( S_{x}(\omega) \).

15.1.5 Theory of random processes

3 $

Problem: The random function \( X(t) \) has the characteristics \( m_{X}(t)=0, \quad k_{x}\left(t, t^{\prime}\right)=\frac{1}{1+\left(t-t^{\prime}\right)^{2}} \). Find the characteristics of the random function \( Y(t)=\int_{0}^{t} X(t) d t \) and find out whether \( X(t), Y(t) \) are stationary?

15.1.6 Theory of random processes

3.75 $

Problem: The random process has the form: \( X(t)=V \cdot e^{-t}+a t^{2} \), where \( V \) is a random value, distributed according to the exponential law with the parameter \( \lambda, a= \) const. Find the expected value, the mathematical expectation, the normalized autocovariance function and the dispersion of \( t^{2} X(t) \).

15.1.7 Theory of random processes

3.75 $

Problem: The characteristics of the random process \( X(t) \) are known: \( m_{X}(t)=2 t^{2}-1, R_{X}\left(t_{1}, t_{2}\right)=2 e^{-3 t_{1}-t_{2}} \). Find the expected value and the dispersion of the process \( Y(t)=\frac{d X(t)}{d t}-2 \).

15.1.8 Theory of random processes

2.5 $

Problem: A random process \( Y(t)=a+X t^{2} \) is given, where \( X \) is a random variable: \( f(x) \sim R[0 ; 3] \). 1) Draw the trajectory of \( Y(t) \), 2) Find: \( M[Y(t)], K_{Y}\left(t_{1}, t_{2}\right), D[Y(t)] \).

15.1.9 Theory of random processes

3.75 $

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