Understanding the Unit Step Function and Its Applications

This video tutorial reviews the unit step function, also known as the Heaviside function, and its applications in creating and modifying functions. It explains how the function interacts with the Laplace transform, including the shifting property. An example problem involving a mass-spring system is solved using these concepts, demonstrating the use of the inverse Laplace transform to find the solution.

What is the value of the unit step function U(t) when t is less than zero?

• 1

• 0

• t

• -1

How does the unit step function U(t-a) affect the graph of U(t)?

• Stretches it vertically

• Reflects it over the y-axis

• Shifts it to the right by a units

• Shifts it to the left by a units

In the context of piecewise functions, what does the expression U(t - pi) * sin(t) represent?

• A constant function

• A sine wave starting at t = 0

• A linear function

• A sine wave starting at t = pi

How can the step function that is one on the interval [1, 2) and zero elsewhere be expressed?

• U(t+1) - U(t+2)

• U(t-1) - U(t-2)

• U(t+1) + U(t+2)

• U(t-1) + U(t-2)

What is the Laplace transform of U(t-a)?

• s / e^(-as)

• e^(as) / s

• e^(-as) / s

• s / e^(as)

What does the shifting property of the Laplace transform state?

• It scales the function

• It shifts the function vertically

• It reflects the function

• It shifts the function horizontally

In the mass-spring system example, what is the forcing function F(t) on the interval [1, 5)?

• 0

• 1

• t

• sin(t)

What is the initial condition for X'(0) in the mass-spring system example?

• 1

• 0

• -1

• t

What mathematical technique is used to solve the differential equation in the example?

• Integration by parts

• Partial fraction decomposition

• Taylor series expansion

• Completing the square

What is the inverse Laplace transform of 1/s?

• t

• 1

• sin(t)

• cos(t)