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Problem: Expand function \[ f(x)=\frac{\pi}{4}-\frac{x}{2}, \] given in interval \( (0 ; \pi) \), into Fourier series in sines. Plot the graph of the sum of Fourier series.

2.6.2.10 Trigonometric Fourier series

1.78 $

Problem: Expand function, given in interval \( x \in[-\pi ; \pi] \) into Fourier series. \[ f(x)=\left\{\begin{array}{cc} x^{2}, & 0

2.6.2.11 Trigonometric Fourier series

2.55 $

Problem: Expand the function, given in interval \( x \in[0 ; \pi] \), by cosines and sines of multiple arcs. \[ f(x)=\frac{\pi}{4}-\frac{x}{2} \text {. } \]

2.6.2.12 Trigonometric Fourier series

2.55 $

Problem: Expand function \( f(x) \) into the trigonometric Fourier series in \( [-\pi ; \pi] \), continuing function in \( [-\pi ; 0] \) evenly. \[ f(x)=\left\{\begin{array}{ll} x-\frac{\pi}{2}, & 0 \leq x \leq \frac{\pi}{2} \\ 2 x-\pi, & \frac{\pi}{2} \leq x \leq \pi \end{array}\right. \]

2.6.2.13 Trigonometric Fourier series

2.55 $

Problem: a) Expand function \( y=f(x) \), given on the halfcycle \( (0 ; l) \), into Fourier series in cosines. Plot the graphs of the 2nd and 3rd partial sums. Write Parseval's equality for the resulting series. b) Expand function \( y=f(x) \), given on the halfcycle \( (0 ; l) \), into Fourier series in cosines. Plot the graphs of the 2nd and 3rd partial sums. c) Expand function \( y=f(x) \) into Fourier series, continuing it on the half-cycle \( (0 ; l) \) with a function, equal to zero. Construct the graphs of the second and the fourth partial sums. \[ y=1-x, \quad l=4 \]

2.6.2.15 Trigonometric Fourier series

7.64 $

Problem: Find the domain of the function: \[ y=\frac{\sqrt{x^{2}-16}}{\log _{2}(x+5)} . \]

2.7.1 Function properties

0 $

Problem: Find the domain of function \( y=\sqrt{35-|3 x-11|} \).

2.7.2 Function properties

0 $

Problem: \( y=f(x) \) is a function with various analytical expressions for different domains of change of the independent variable. Find the discontinuity points of the function, if they exist. Plot the graph. \[ y=\left\{\begin{array}{cc} x^{2}, & x \leq 0 \\ x, & 02 \end{array} .\right. \]

2.7.3 Function properties

0.76 $

Problem: Find the intervals of monotonicity, the extremum points of the function: \[ y=\frac{1}{x}+\frac{2 x}{x^{2}-1} \text {. } \]

2.7.4 Function properties

0.76 $

Problem: Find the discontinuity of the function and determine their types: \[ f(x)=\frac{x^{2}+2 x-8}{|x-2|\left(x^{2}-2 x-24\right)} . \]

2.7.5 Function properties

1.27 $

Problem: Find the discontinuity points of the function and determine their types: \[ f(x)=\left\{\begin{array}{cc} \frac{4}{x+3}, & x \in(-\infty ;-4) \\ 2 x^{2}+5 x-3, & x \in(-4 ; 2) . \\ \frac{-2}{x-1}, & x \in[2 ;+\infty) \end{array}\right. \]

2.7.6 Function properties

0.76 $

Problem: Find asymptotes of the graph of the function: \[ f(x)=\frac{-2 x^{3}+9 x^{2}+4 x-7}{x^{2}-3 x-4} \text {. } \]

2.7.7 Function properties

1.02 $

Problem: Find asymptotes of the graph of the function: \[ f(x)=\frac{3 \cdot 2^{x}-5 \cdot 5^{x}}{-2 \cdot 2^{x}-4 \cdot 5^{x}} . \]

2.7.8 Function properties

1.02 $

Problem: Find the intervals of convexity and concavity, as well as the inflection points of the function: \[ f(x)=x^{4}-6 x^{2}+5 \text {. } \]

2.7.9 Function properties

0.51 $

Problem: Calculate the value of the integral with the given accuracy: \[ \int_{0}^{1} \frac{x-\sin x}{x^{3}} d x, \quad \varepsilon=0,001 . \]

2.8.1 Power series

0.76 $

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