Laplace Transform Quiz: Test!

1. L(f ') = sL(f) – f(0)

True

False

The given equation L(f') = sL(f) - f(0) is a valid equation in Laplace transform theory. It states that the Laplace transform of the derivative of a function f is equal to s times the Laplace transform of f minus the value of f at 0. This equation holds true and is commonly used in solving differential equations using Laplace transforms.

Explanation

The given equation L(f') = sL(f) - f(0) is a valid equation in Laplace transform theory. It states that the Laplace transform of the derivative of a function f is equal to s times the Laplace transform of f minus the value of f at 0. This equation holds true and is commonly used in solving differential equations using Laplace transforms.

2. $(f*g)(t)=\int ^{t}_{0} {f(v)g(t-v)dv}$

True

False

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Explanation

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3. L(f * g) = L(f).L(g)

True

False

The given statement is the multiplication property of the Laplace transform. According to this property, the Laplace transform of the product of two functions is equal to the product of their individual Laplace transforms. Therefore, the statement is true.

Explanation

The given statement is the multiplication property of the Laplace transform. According to this property, the Laplace transform of the product of two functions is equal to the product of their individual Laplace transforms. Therefore, the statement is true.

4. ${L}^{-1}\left \{ {F(s)G(s)} \right \} =\int ^{t}_{0} {f(v)g(t-v)dv}$

True

False

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Explanation

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5. L^-1(F(s) + G(s)) = f(t)+ g(t)

True

False

The given equation represents the Laplace transform of the sum of two functions, F(s) and G(s), which is equal to the sum of the time-domain representations of those functions, f(t) and g(t). This is a fundamental property of the Laplace transform, known as the linearity property. Therefore, the statement is true.

Explanation

The given equation represents the Laplace transform of the sum of two functions, F(s) and G(s), which is equal to the sum of the time-domain representations of those functions, f(t) and g(t). This is a fundamental property of the Laplace transform, known as the linearity property. Therefore, the statement is true.

6. $(f*g)(t)=\int ^{t}_{0} {f(t-v)g(v)dv}$

True

False

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Explanation

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7. L(f(t)/t) = -F'(s)

True

False

The given equation L(f(t)/t) = -F'(s) is not true. This is because the left-hand side of the equation represents the Laplace transform of the function f(t)/t, while the right-hand side represents the derivative of the Laplace transform of the function F(s). These two expressions are not equivalent, so the equation is false.

Explanation

The given equation L(f(t)/t) = -F'(s) is not true. This is because the left-hand side of the equation represents the Laplace transform of the function f(t)/t, while the right-hand side represents the derivative of the Laplace transform of the function F(s). These two expressions are not equivalent, so the equation is false.

8. $(f*g)(t)=\int ^{t}_{0} {f(t-v)g(t-v)dv}$

True

False

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Explanation

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9. $(f*g)(t)=\int ^{t}_{0} {f(v)g(v)dv}$

True

False

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Explanation

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10. L^-1(F(s) G(s)) = f(t) g(t)

True

False

The given equation L-1(F(s) G(s)) = f(t) g(t) is false. The Laplace inverse of the product of two Laplace transforms is not equal to the product of their inverse transforms. Therefore, the statement is incorrect.

Explanation

The given equation L-1(F(s) G(s)) = f(t) g(t) is false. The Laplace inverse of the product of two Laplace transforms is not equal to the product of their inverse transforms. Therefore, the statement is incorrect.