To determine To prove: The initial value theorem lim s → ∞ s X ( s ) = x ( 0 + ) . Explanation Let us assume a continuous right-sided function x ( t ) such that x ( t ) = 0 ∀ t < 0 Now taking Laplace transform of the derivative of this function such that L ( d x ( t ) d t ) = ∫ − ∞ ∞ d x ( t ) d t . e − s t d t ∵ L ( x ( t ) ) = X ( s ) = ∫ − ∞ ∞ x ( t ) . e − s t d t ⇒ ∫ − ∞ ∞ d x ( t ) d t . e − s t d t = ∫ 0 ∞ d x ( t ) . e − s t ∵ x ( t ) is the right-sided function. Using integration of product rule: ∫ a b u ⋅ d v = u ⋅ v | a b − ∫ a b v ⋅ d u ∫ − ∞ ∞ d x ( t ) d t ⋅ e − s t d t = ∫ 0 ∞ d x ( t ) ⋅ e − s t = x ( t ) ⋅ e − s t | 0 ∞ − ∫ 0 ∞ x ( t ) ⋅ d ( e − s t ) d t d t ⇒ ∫ − ∞ ∞ d x ( t ) d t ⋅ e − s t d t = x ( ∞ ) ⋅ e − ∞ − x ( 0 ) ⋅ e 0 + s ∫ 0 ∞ x ( t ) ⋅ e − s t d t ⇒ ∫ − ∞ ∞ d x ( t ) d t ⋅ e − s t d t = s X ( s ) − x ( 0 ) ∵ e − ∞ ≈ 0 , e 0 ≈ 1 , X ( s ) = ∫ 0 ∞ x ( t )

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Chapter 2, Problem 2.55P

To determine

To prove:

The initial value theorem lims→∞sX(s)=x(0+).

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Chapter 2, Problem 2.54P

Chapter 2, Problem 2.56P

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EBK SYSTEM DYNAMICS

Ch. 2 - Prob. 2.1P Ch. 2 - Solve each of the following problems by direct...Ch. 2 - Solve each of the following problems by separation...Ch. 2 - Prob. 2.4P Ch. 2 - Prob. 2.5P Ch. 2 - Obtain the Laplace transform of the following...Ch. 2 - Obtain the Laplace transform of the function shown...Ch. 2 - Prob. 2.8P Ch. 2 - Prob. 2.9P Ch. 2 - Prob. 2.10P

Ch. 2 - Prob. 2.11P Ch. 2 - Obtain the inverse Laplace transform xt for each...Ch. 2 - Solve the following problems: 5x=7tx0=3...Ch. 2 - Solve the following: 5x+7x=0x0=4 5x+7x=15x0=0...Ch. 2 - Solve the following problems: x+10x+21x=0x0=4x0=3...Ch. 2 - Solve the following problems: x+7x+10x=20x0=5x0=3...Ch. 2 - Solve the following problems: 3x+30x+63x=5x0=x0=0...Ch. 2 - Solve the following problems where x0=x0=0 ....Ch. 2 - Invert the following transforms: 6ss+5 4s+3s+8...Ch. 2 - Invert the following transforms: 3s+2s2s+10...Ch. 2 - Prob. 2.21P Ch. 2 - Compare the LCD method with equation (2.4.4) for...Ch. 2 - Prob. 2.23P Ch. 2 - Prob. 2.24P Ch. 2 - (a) Prove that the second-order system whose...Ch. 2 - For each of the following models, compute the time...Ch. 2 - Prob. 2.27P Ch. 2 - Prob. 2.28P Ch. 2 - Prob. 2.29P Ch. 2 - If applicable, compute , , n , and d for the...Ch. 2 - Prob. 2.31P Ch. 2 - For each of the following equations, determine the...Ch. 2 - Prob. 2.33P Ch. 2 - Obtain the transfer functions Xs/Fs and Ys/Fs for...Ch. 2 - a. Obtain the transfer functions Xs/Fs and Ys/Fs...Ch. 2 - Prob. 2.36P Ch. 2 - Solve the following problems for xt . Compare the...Ch. 2 - Prob. 2.38P Ch. 2 - Prob. 2.39P Ch. 2 - Prob. 2.40P Ch. 2 - Determine the general form of the solution of the...Ch. 2 - a. Use the Laplace transform to obtain the form of...Ch. 2 - Prob. 2.43P Ch. 2 - Prob. 2.44P Ch. 2 - Obtain the inverse transform in the form xt=Asint+...Ch. 2 - Use the Laplace transform to solve the following...Ch. 2 - Express the oscillatory part of the solution of...Ch. 2 - Prob. 2.48P Ch. 2 - Prob. 2.49P Ch. 2 - Prob. 2.50P Ch. 2 - Prob. 2.51P Ch. 2 - Prob. 2.52P Ch. 2 - Prob. 2.53P Ch. 2 - 2.54 The Taylor series expansion for tan t...Ch. 2 - 2.55 Derive the initial value theorem: Ch. 2 - Prob. 2.56P Ch. 2 - Prob. 2.57P Ch. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to solve for and plot the unit-step...Ch. 2 - Use MATLAB to solve for and plot the unit-impulse...Ch. 2 - Use MATLAB to solve for and plot the impulse...Ch. 2 - Use MATLAB to solve for and plot the response of...

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