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Problem: For the piecewise-constant functions \( f(x) \) and \( g(x) \) of the form \[ y(x)=\left\{\begin{array}{ll} 0, & x

11.3.5 Convolution of functions

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Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes the value, equal to zero, when \( x \notin\left[x_{1} ; x_{4}\right] \), and when \( x \in\left[x_{1} ; x_{4}\right] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A\left(x_{1} ; a\right), \quad B\left(x_{2} ; a\right), \quad C\left(x_{3} ; b\right), \quad D\left(x_{4} ; 0\right) . \\ x_{1}=-1, \quad x_{2}=1, \quad x_{3}=4, \quad x_{4}=6, \quad a=-1, \quad b=2 . \end{array} \] The function \( g(x) \) has the form \[ g(x)=\left\{\begin{array}{ll} 0, & x<0 ; \\ 1, & 0 \leq x<1 ; \\ 0, & x \geq 0 . \end{array}\right. \]

11.3.6 Convolution of functions

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Problem: For piecewise-constant functions \( f(x) \) and \( g(x) \) of the form \[ y(x)=\left\{\begin{array}{ll} 0, & x

11.3.7 Convolution of functions

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Problem: Determine the order of smallness \( \alpha(x)=\sqrt{x^{2}+9}-3 \) with respect to \( \beta(x)=x \) when \( x \rightarrow 0 \).

2.12.11 Asymptotic analysis

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Problem: Determine the order of smallness of \( \alpha(x)=\ln \left(x^{3}+3 x^{2}+3 x+2\right) \) with respect to \( \beta(x)=x+1 \) when \( x \rightarrow-1 \).

2.12.12 Asymptotic analysis

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Problem: Determine the order of smallness of \( \alpha(x)=\sqrt{\cos \ln x}-1 \) with respect to \( \beta(x)=x-1 \) when \( x \rightarrow 1 \).

2.12.13 Asymptotic analysis

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Problem: Given vectors \( \vec{a}=\overrightarrow{O A}, \vec{b}=\overrightarrow{O B}, \vec{c}=\overrightarrow{O C}, \vec{d}=\overrightarrow{O D} \). Beams \( O A, O B \) and \( O C \) are edges of e trihedral angle \( T \). 1) Prove that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are linearly independent. 2) Decompose vector \( \vec{d} \) into vectors \( \vec{a}, \vec{b}, \vec{c} \) (solve the resulting system of equations using the inverse matrix). 3) Determine whether point \( D \) lies inside \( T \), outside \( T \), on one of the boundaries of \( T \) (which one?). 4) Determine for what values of the real parameter \( \lambda \) the vector \( \vec{d}+\lambda \vec{a} \), plotted from the point \( O \), lies inside the trihedral angle \( T \). \[ \vec{a}=\overrightarrow{\{3 ; 2 ; 1\}}, \quad \vec{b}=\overrightarrow{\{1 ;-1 ;-2\}}, \quad \vec{c}=\overrightarrow{\{-2 ; 3 ; 5\}}, \quad \vec{d}=\overrightarrow{\{7 ; 4 ; 1} . \]

1.1.4 Vector Algebra

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Problem: Calculate the surface of the area \( D \) : \[ D:\left(\frac{x}{a}+\frac{y}{b}\right)^{3}=\frac{x y}{c^{2}} \text {. } \]

9.1.12 Double integrals

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Problem: Find the double integral of the given function \( f(x, y) \) in the given area \( G \) : \[ \begin{array}{l} f(x, y)=e^{-\left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}} \\ G:\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2} \geq 1, \quad \iint_{G} f(x, y) d x d y-? \end{array} \]

9.1.13 Double integrals

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Problem: Change the order of integration in iterated integral: \[ \int_{-1}^{1} d x \int_{x^{2}}^{2-x^{2}} f(x, y) d y \]

9.1.14 Double integrals

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Problem: Find the double integral over the given area: \[ I=\iint_{D} e^{-x^{2}-y^{2}} d x d y, \quad D:\left\{\begin{array}{l} x=\sqrt{1-y^{2}} \\ x=y \\ x=-y \sqrt{3} \end{array}\right. \]

9.1.15 Double integrals

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Problem: Find the particular solution of the equation with separable variables, satisfying the initial condition: \[ \left(1+y^{2}\right) d x-x y d y=0, \quad y_{0}=1 \text { when } x_{0}=2 . \]

8.1.1.27 First order differential equations

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Problem: Find the general solution of the first order inhomogeneous differential equation: \[ x y^{\prime}+y=\ln x+1 \text {. } \]

8.1.1.28 First order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation: \[ y^{\prime \prime}+4 y^{\prime}=0 \text {. } \] clip2net.com

8.1.2.29 Second order differential equations

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Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} x=16 \sqrt{2 y}, \quad x=\sqrt{2 y} \\ z+y=2, \quad z=0 \end{array}\right. \]

9.5.6 Volume of a solid

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