Math Problems and solutions
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2.4.1 Graphing functions using derivatives
2.55 $
2.6.1.1 Fourier integral
5.1 $
2.6.1.2 Fourier integral
2.6.1.3 Fourier integral
2.6.1.4 Fourier integral
2.6.1.5 Fourier integral
2.6.2.1 Trigonometric Fourier series
2.6.2.2 Trigonometric Fourier series
6.37 $
2.6.2.3 Trigonometric Fourier series
1.27 $
2.6.2.4 Trigonometric Fourier series
2.6.2.5 Trigonometric Fourier series
2.6.2.6 Trigonometric Fourier series
2.6.2.7 Trigonometric Fourier series
2.6.2.8 Trigonometric Fourier series
3.82 $
2.6.2.9 Trigonometric Fourier series
0.51 $
![Problem: Examine the function by methods of differential calculus and plot the graph: \[ f(x)=(x-1)^{2}(x+1)^{2} \text {. } \]](/storage/uploads/task_mls/91/en/2.4.1.png){kind=link}
![Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-2,-2) \) & \( (1,0) \) & \( (4,-2) \) & \( (6,-2) \) \\ \hline \end{tabular}](/storage/uploads/task_mls/92/en/2.6.1.1.png){kind=link}
![Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-1,-1) \) & \( (0,-1) \) & \( (1,-2) \) & \( (2,-1) \) \\ \hline \end{tabular}](/storage/uploads/task_mls/93/en/2.6.1.2.png){kind=link}
![Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (0,-1) \) & \( (2,2) \) & \( (4,-1) \) & \( (5,-1) \) \\ \hline \end{tabular}](/storage/uploads/task_mls/94/en/2.6.1.3.png){kind=link}
 \) \( =\int_{-\infty}^{+\infty} f(t) e^{-i v t} d t \) of function \( f(t) \), using equalities \( F[ \) rect \( t](v)=\operatorname{sinc} \frac{v}{2} \) and \( F[\Lambda(t)](v)=\operatorname{sinc}^{2} \frac{v}{2} \). The graph of function \( f(t) \) consists a polygonal chain, connecting points \( A, B, C \) and \( D \), as well as from parts of the \( \mathrm{x} \)-axis (to the left of \( A \) and to the right of \( D \) ). \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-1,1) \) & \( (1,-2) \) & \( (4,-2) \) & \( (6,1) \) \\ \hline \end{tabular}](/storage/uploads/task_mls/95/en/2.6.1.4.png){kind=link}
![Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-2 ;-1) \) & \( (-1 ;-1) \) & \( (0 ; 0) \) & \( (1 ;-1) \) \\ \hline \end{tabular}](/storage/uploads/task_mls/96/en/2.6.1.5.png){kind=link}
![Problem: Expand function \( f(x)=\frac{x^{2}-1}{2} \) into a trigonometric Fourier series on the line segment \( [-\pi ; \pi] \). Plot the graphs of function \( f(x) \) and the sum of the Fourier series. Write down the first three partial sums of the Fourier series and approximately plot their graphs.](/storage/uploads/task_mls/97/en/2.6.2.1.png){kind=link}
![Problem: Expand function \( f(x) \) in a Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot the amplitude and phase spectra. The values of parameters \( T, h, p \) and \( q \) are given in the table: \begin{tabular}{|c|c|c|c|} \hline\( T \) & \( h \) & \( p \) & \( q \) \\ \hline 2 & 2 & -2 & 1 \\ \hline \end{tabular} \( f(x)=\left\{\begin{array}{cc}h-\frac{2 h}{T} x, & -T / 2 \leq x<0 \\ p, & 0 \leq x](/storage/uploads/task_mls/101/en/2.6.2.5.png){kind=link}
![Problem: Expand function \( f(x) \) in a Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot: 1. Amplitude and phase spectra; 2. Graphs of partial sums of the Fourier series \( S_{3}(x), S_{10}(x), S_{20}(x), S_{100}(x) \). The values of parameters \( T, h, p \) and \( q \) are given in the table:](/storage/uploads/task_mls/102/en/2.6.2.6.png){kind=link}
![Problem: Expand function \( f(x) \) into Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot: 1. Amplitude and phase spectra; 2. Graphs of partial sums of the Fourier series \( S_{3}(x), S_{10}(x), S_{20}(x), S_{100}(x) \). The values of parameters \( T, h, p \) and \( q \) are given in the table: \begin{tabular}{|c|c|c|c|} \hline\( T \) & \( h \) & \( p \) & \( q \) \\ \hline 2 & -2 & 2 & -1 \\ \hline & \( p \), & \begin{tabular}{l} \( / 2 \leq \) \\ \( -T / 4 \) \end{tabular} & \\ \hline & \( 2-\frac{2 l}{T} \) & & \\ \hline \end{tabular}](/storage/uploads/task_mls/103/en/2.6.2.7.png){kind=link}
![Problem: Expand function \( f(x) \) into the trigonometric Fourier series in cosines. 1) \( f(x)=x^{3} \) in \( [0 ; 1] \), 2) \( f(x)=\sin 2 x \) in \( [0 ; 1] \), 3) \( f(x)=\frac{\pi}{4}-\frac{x}{2} \) in \( [0 ; 1] \).](/storage/uploads/task_mls/104/en/2.6.2.8.png){kind=link}
