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Problem 3 Use series expansion to find the slope of the following function at 0. f(x)...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Problem 6: Find the cosine series for the symmetric (even) extension (or "cosine half-range expansion") f...
Problem 6: Find the cosine series for the symmetric (even) extension (or "cosine half-range expansion") f (t) of the function g(t) by using the complex Fourier series and the method of jumps f(t) = g(t) = sin t , g(-t) =-sin t , 0<t<π [Vol.III-Ch.1, 6 -r < t < 0
Question 4. Calculate the Fourier sine series and the Fourier cosine series of the function f(x)...
Question 4. Calculate the Fourier sine series and the Fourier cosine series of the function f(x) = sin(x) on the interval [0, 1]. Hint: For the cosine series, it is easiest to use the complex exponential version of Fourier series. Question 5. Solve the following boundary value problem: Ut – 3Uzx = 0, u(0,t) = u(2,t) = 0, u(x,0) = –2? + 22 Question 6. Solve the following boundary value problem: Ut – Uxx = 0, Uz(-7,t) = uz (77,t)...
5. A function f has Taylor series (at 0) f(x)=0+2x+ 4x2/2! + 3x3/3!+... Assume f−1 exists....
5. A function f has Taylor series (at 0) f(x)=0+2x+ 4x2/2! + 3x3/3!+... Assume f−1 exists. Find as much of the Taylor series of f−1 (at 0) as you can. (Since you only know the first few terms of the Taylor series for f, you can only figure out f−1. (Hint: There are two ways of doing this problem. One is get the derivatives of f−1 from knowing the derivatives of f; we talked about the first derivative in January...
[3] 4. Find a power series for the function f(z) = z2 of the form f(x)...
[3] 4. Find a power series for the function f(z) = z2 of the form f(x) = {mco bn(z – m)”. I.e. you must tell me exactly what each bn is.
5) Write down a power series expansion (centered at r = 0) for the function f(x)
5) Write down a power series expansion (centered at r = 0) for the function f(x)
Find the first four nonzero terms in a power series expansion about x = 0 for...
Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. W -3x?w'+w=0
7. (a) Use the well known Maclaurin series expansion for the cosine function: f (x )...
7. (a) Use the well known Maclaurin series expansion for the cosine function: f (x ) = cos x = 1 x? 2! + 4! х 6! + (-1)" (2n)! . * 8! 0 and a substitution to obtain the Maclaurin series expansion for g(x) = cos (x²). Express your formula using sigma notation. (b) Use the Term-by-Term Integration Theorem to obtain an infinite series which converges to: cos(x) dx . y = cos(x²) (c) Use the remainder theorem associated...
Problem 11.5. Find the Fourier cosine series of the function f(x): f(x) = 1 +X, 0...
Problem 11.5. Find the Fourier cosine series of the function f(x): f(x) = 1 +X, 0 < x < .
Find the first four nonzero terms in a power series expansion about x = 0 for...
Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation w" - x?w'+w=0 w(x)= ... (Type an expression in terms of a, and a, that includes all terms up to order 3
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...
Problem 6: Find the cosine series for the symmetric (even) extension (or "cosine half-range expansion") f (t) of the function g(t) by using the complex Fourier series and the method of jumps f(t) = g(t) = sin t , g(-t) =-sin t , 0<t<π [Vol.III-Ch.1, 6 -r < t < 0
Question 4. Calculate the Fourier sine series and the Fourier cosine series of the function f(x) = sin(x) on the interval [0, 1]. Hint: For the cosine series, it is easiest to use the complex exponential version of Fourier series. Question 5. Solve the following boundary value problem: Ut – 3Uzx = 0, u(0,t) = u(2,t) = 0, u(x,0) = –2? + 22 Question 6. Solve the following boundary value problem: Ut – Uxx = 0, Uz(-7,t) = uz (77,t)...
[3] 4. Find a power series for the function f(z) = z2 of the form f(x) = {mco bn(z – m)”. I.e. you must tell me exactly what each bn is.
5) Write down a power series expansion (centered at r = 0) for the function f(x)
Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. W -3x?w'+w=0
7. (a) Use the well known Maclaurin series expansion for the cosine function: f (x ) = cos x = 1 x? 2! + 4! х 6! + (-1)" (2n)! . * 8! 0 and a substitution to obtain the Maclaurin series expansion for g(x) = cos (x²). Express your formula using sigma notation. (b) Use the Term-by-Term Integration Theorem to obtain an infinite series which converges to: cos(x) dx . y = cos(x²) (c) Use the remainder theorem associated...
Problem 11.5. Find the Fourier cosine series of the function f(x): f(x) = 1 +X, 0 < x < .
Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation w" - x?w'+w=0 w(x)= ... (Type an expression in terms of a, and a, that includes all terms up to order 3
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