Initial Value Theorem Quiz

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Questions and Answers

According to the initial value theorem, what is the value of x(0+), given the transform X(s)?

x(0+) = lims→∞[sX(s)] (correct)

x(0+) = limt→0+x(t)

x(0+) = limt→∞[x(t)]

x(0+) = lims→0[sX(s)]

Under what conditions is the initial value theorem valid?

The limit s → ∞ exists

The transform of x(t) exists

The transform of dx/dt exists

All of the above (correct)

What does the final value theorem state?

x(∞) = lims→∞ sX(s)

x(∞) = lims→0 sX(s) (correct)

x(∞) = limt→0 x(t)

x(∞) = limt→∞ x(t)

Which of the following equations correctly represents the inverse transform of $X(s) = 1/s$?

$x(t) = 1$ Signup and view all the answers

What is the inverse transform of $X(s) = 7(s + 4)^2 + 49$?

$x(t) = e^{-4t} \sin(7t)$ Signup and view all the answers

Which of the following statements is true about the transform $X(s) = 9s + \frac{2},{s(s-8)}$?

The inverse transform of $X(s)$ is a periodic function. Signup and view all the answers

What is the final value theorem for the Laplace transform?

If a function f(t) in continuous time has the Laplace transform F(s), then the final value theorem states that under certain conditions, the limit of f(t) as t approaches infinity is equal to the limit of sF(s) as s approaches 0. Signup and view all the answers

What is the final value theorem for the Z-transform?

If a function f[k] in discrete time has the Z-transform F(z), then the final value theorem states that under certain conditions, the limit of f[k] as k approaches infinity is equal to the limit of (z-1)F(z) as z approaches 1. Signup and view all the answers

What is the difference between an Abelian final value theorem and a Tauberian final value theorem?

An Abelian final value theorem makes assumptions about the time-domain behavior of f(t) (or f[k]) to calculate the limit of sF(s), while a Tauberian final value theorem makes assumptions about the frequency-domain behavior of F(s) to calculate the limit of f(t) (or f[k]). Signup and view all the answers

What is the standard final value theorem?

The standard final value theorem states that if every pole of $F(s)$ is either in the open left half plane or at the origin, and $F(s)$ has at most a single pole at the origin, then $sF(s) \to L \in \mathbb{R}$ as $s \to 0$, and $\lim_{t \to \infty} f(t) = L$. Signup and view all the answers

What is the final value theorem using Laplace transform of the derivative?

The final value theorem using Laplace transform of the derivative states that if $f(t)$ and $f'(t)$ both have Laplace transforms that exist for all $s > 0$, and if $\lim_{t \to \infty} f(t)$ exists and $\lim_{s \to 0} sF(s)$ exists, then $\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s)$. Signup and view all the answers

What is the improved Tauberian converse final value theorem?

The improved Tauberian converse final value theorem states that if $f: (0, \infty) \to \mathbb{C}$ is bounded and differentiable, and $tf'(t)$ is also bounded on $(0, \infty)$, and if $sF(s) \to L \in \mathbb{C}$ as $s \to 0$, then $\lim_{t \to \infty} f(t) = L$. Signup and view all the answers

What is the extended final value theorem?

The extended final value theorem states that if every pole of $F(s)$ is either in the open left half-plane or at the origin, then one of the following occurs: $sF(s) \to L \in \mathbb{R}$ as $s \downarrow 0$, and $\lim_{t \to \infty} f(t) = L$; $sF(s) \to +\infty \in \mathbb{R}$ as $s \downarrow 0$, and $f(t) \to +\infty$ as $t \to \infty$; $sF(s) \to -\infty \in \mathbb{R}$ as $s \downarrow 0$, and $f(t) \to -\infty$ as $t \to \infty$. Signup and view all the answers

Study Notes

Laplace Transform Theorems

The initial value theorem states the value of x(0+) can be obtained from the Laplace transform X(s).
The initial value theorem is valid under certain conditions.

Final Value Theorem

The final value theorem states the final value of a signal can be obtained from the Laplace transform X(s).
The final value theorem for the Laplace transform states that the final value of a signal x(t) is x(∞) = lim(s → 0){sX(s)}.

Inverse Laplace Transform

The inverse transform of X(s) = 1/s is u(t), where u(t) is the unit step function.
The inverse transform of X(s) = 7(s + 4)^2 + 49 is still to be determined.

Laplace Transform Analysis

X(s) = 9s + 2/{s(s-8)} has a pole at s=8 and a zero at s=0.

Final Value Theorem Variants

The standard final value theorem states that the final value of a signal x(t) is x(∞) = lim(s → 0){sX(s)}.
The final value theorem using the Laplace transform of the derivative is x(∞) = lim(s → 0){s^2X(s)}.
The Abelian final value theorem and Tauberian final value theorem are two different classes of final value theorems.
The Tauberian final value theorem is an improvement over the Abelian final value theorem.
The improved Tauberian converse final value theorem is a stronger version of the Tauberian final value theorem.
The extended final value theorem is a generalization of the standard final value theorem.

Z-Transform

The final value theorem for the Z-transform states that the final value of a discrete-time signal x[k] is x[∞] = lim(z → 1){(z-1)X(z)}.

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Description

Test your knowledge on the Initial Value Theorem in calculus and its applications in finding the value of a function at t = 0+. Explore the conditions for which the theorem is valid and understand the limit s → ∞ along the real axis.