1-solve the Fourier series for the function f (x) = x² + 2 ,and %3D π
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- Prove that f(x)={-3x if -pi<x<0, 3x if 0<x<pi is an even function using fourier series expansion.Given the following linear system of ODE: V₁' = 4V1 +2V2 V₂' = 10/₁-4/2 Solve the initial value problem, given y₁ (0) = 1 and y₂ (0) = 2 O y = 7/6 O y = et + O --8--8- H y [1] H y=-1 ebt = [2] e e³t+5/6 -¹ [He y = 1/6 L - 1/6 [15] ebt + 23 -6t 5 e-6t e-3t e-ot 1/6[1]eºt +5/6 [11/5] 0 e h -6tA periodic function f(x) is defined by f(x) = x² + 3 for –2d) Write the corresponding Fourier series of f (x). A periodic function of f (x) is given by [0, f(x)={-x-7, f(x+27). - T < x < 0, 0Solve for Fourier coefficient, Ssin x 02x27 f(x)= 0 T >x 22n cosx sec x + cosx csc csc² x csc? x secx cos? x cos x cos x § f(x)dx _ ? If f(x)* , then 1(b) Using the answer to Part (a) and without using Theorem 2.2.4, find the real Fourier series of the function f(x) = -2x 2r-8 if 0 < x < 2 if 2 < x < 4 and f(x+4)=f(x).Q4/ Choose the correct option represents the following Fourier series f(x) = x +n . T+2sinx-sin2x+2/3 sin3x-2/4 sin4x+... 3m + 2sinx – sin2x + sin3x -sinax + O Option 1 O Option 2 -2n + 2sinx - sin2x +sin3x –- sin4x + Option 3 O No one= 1) The function f(x) periodic on the interval [0, 2л] has complex Fourier series f(x): Σ(1/n²) einx where the sum over n goes from - infinity to infinity. Convert this to cosine and sine Fourier Series by finding the values of A's and B's in the expression Ao + ΣAn cos(nx) + Σ Bn sin(nx) where each sum goes from 1 to infinity. Hint: consider the n and -n term together in the complex Fourier Series or use Euler's identity.Consider f(t)=9+2t+6t², for -A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chessboard. On the second square, the King would place two grains of wheat, on the third square, four grains of wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining squares, how many grains of wheat should be placed on a square 16? Also, find the total number of grains of wheat on the board at this time and their total weight in pounds. (Assume that each grain of wheat weighs 1/7000 pounds.)2 Denote the Fourier series of f(z) = { -x, 0f(x) and g(x) is periodic and has a Fourier Series representation: f(x) = cos(6*pi*x) g(x)=sin(6*pi*x) z(x) = f(x)*g(x) Find a0, a1, a2, a3 for z(x). Simplify as much as possible!Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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