MATH RC EXAM 09 (to edit)

Find the Laplace transform of f(t) = 2eᵗ cost + 3eᵗ sint.

(2s + 1)/(s² - 2s + 2)

Find the Z - transform of f(n) = cos(n - 2)u(n - 2)

(2z - cos(1))/(z(z² - 2zcos(1) + 1))

Find the inverse Laplace transform of F(s) = 3/(s² + 4).

(3/2) sin(2t)

Find the Laplace transform of the integral of eᵃᵗ cos (bt) with limits from 0 to t.

(s - a)/{s[(s - a)² + b²]}

Find the Fourier transform of f(x) = xe^(-a abs(x)), a > 0

-(4awi)/(w² + a²)²

Find the Fourier transform of f(x) = xe^(-x^2)

-(iw/2sqrt(2))e^(-w²/4)

Given the 4 x 4 matrix A = [2, -6, 1, 2; 12, 3, 4, 5; 8, -1, 0, 3; 6, 5, 8, 7], determine det(A).

-708

Given the 5 x 5 matrix C = [2, 4, 5/2, -2, 7; 3, 5, -21, 5, 1; -6, 3, 0, 1, 0; 11, 2, -9, 3, 0; 2, 4, 8, -3, 4], determine det(C).

-8266.5

The radius of convergence of Σ(xⁿ, 0, ∞) is

1

Find the Laplace transform of f(t) = e³ᵗ + cos(6t) - e³ᵗ cos(6t).

1/(s - 3) + s/(s² + 36) - (s - 3)/[(s - 3)² + 36]

Find the inverse Laplace transform of F(s) = 1/[s(s + 1)(s + 2)].

1/2 - e¯ᵗ + (1/2)e¯²ᵗ

Find the Laplace transform of the integral of te¯³ᵗdt with limits from 0 to t.

1/[s(s + 3)²]

Solve for the Fourier Transform of the given function f(t) = e^(-t/2) u(t).

2/(1 + j2ω)

Given the 4 x 4 matrix B = [0, -5, 2, 11; -1, 2, 3, 9; 7/2, 11, 2, 6; 3, 1, 5, 0], determine det(B).

2300

Solve for the Fourier Transform of the given function f(t) =e^(-a|t|).

2a/(a² + ω²)

Find the inverse Laplace transform of F(s) = (2s + 2)/(s² + 2s + 5)

2e¯ᵗ cos(2t)

The radius of convergence of the power series Σ((-1)ⁿ (xⁿ)/(4ⁿ ln(n)), 2, ∞) is

4

Given the 5 x 5 matrix D = [0, 1, -5, 4, 8; 3, 4, -2, 11, 3; -1, 20, 3, -13, 1; 6, 2, -1, 2, 11; 0, 2, 3, 0, 3], determine det(D).

57,332

Find the Laplace transform of f(t) = 6e¯⁵ᵗ + e³ᵗ + 5t³ - 9.

6/(s + 5) + 1/(s - 3) + 30/s⁴ - 9/s

The interval of convergence of the power series Σ((-1)ⁿ/(2(n+1)(n+2)) x^(2n+4), 0, ∞) is

[-1, 1]

Solve for the Fourier Transform of the given function f(t) = e¯²ᵗ u(t - 3)

e^[-3(2 + jω)] / (2 + jω)

Compute the MacLaurin series of sin(x).

x - (1/3!) x³ + (1/5!) x⁵ - (1/7!) x⁷ + (1/9!) x⁹ + ...

The Z transform of a left sided signal x(n) = -aⁿ u(-n - 1) is

z/(z - a)

Find the Taylor series of f(x) = lnx centered at x = 1.

Σ((-1)^(n+1)(1/n)(x - 1)ⁿ, 1, ∞)

Find the taylor series for xsin(-x).

Σ((-1)^(n+1)x^(2n+2)/(2n+1)!, 0, ∞)