Heaviside Function Calculator

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Example

Created on 2024-06-20Asked by William Nguyen (Solvelet student)

Express the piecewise function

f(x)

as a combination of Heaviside functions:

f(x) = \begin{cases} 0 & \text{if } x < 1 \\ 1 & \text{if } 1 \le x < 2 \\ 2 & \text{if } x \ge 2 \end{cases}

Solution

To express the piecewise function

f(x)

as a combination of Heaviside functions: The Heaviside function

H(x)

is defined as:

H(x) = \begin{cases} 0 & \text{if } x < 0 \ 1 & \text{if } x \ge 0 \end{cases}

We can represent

f(x)

using the Heaviside function

H(x)

as follows:

f(x) = 1 \cdot H(x - 1) + 1 \cdot H(x - 2)

= H(x - 1) + H(x - 2)

Where: -

H(x - 1)

is 0 when

x < 1

and 1 when

x \ge 1

. -

H(x - 2)

is 0 when

x < 2

and 1 when

x \ge 2

. Therefore, the piecewise function can be written as:

f(x) = H(x - 1) - H(x - 2) + 2 H(x - 2)

Simplifying:

f(x) = H(x - 1) + H(x - 2)

This combination of Heaviside functions correctly represents the piecewise function. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Ava Gonzalez on Solvelet

1. Graph the Heaviside function

H(t)

and identify its properties,2. Calculate the Laplace transform of the function

f(t) = 2 - 3H(t - 1)

DefinitionThe Heaviside function, besides its many other names, which is also called the unit step function, is one of a few discontinuous functions, discontinuous due to the jump from 0 to 1 at some point. Conducts switch logic, may perform signal processing, also used in control. The Heaviside function H(x)=0 for x<0 and H(x)=1 for x≥0. Examples -- Instantly turning on via the Heaviside function The Heaviside function itself gives a behaviour of switching from H(t=0) = 0 to H(t>0) = 1 at exactly t=0. Solve Heaviside-Function

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