Step functions

Step functions

Lessons

A Heaviside Step Function (also just called a “Step Function”) is a function that has a value of 0 from 0 to some constant, and then at that constant switches to 1.

The Heaviside Step Function is defined as,

Heaviside Step Function Equation

unit step function

The Laplace Transform of the Step Function:

LL{uc(t)u_{c}(t) f(tc)f(t - c)} = esce^{-sc}LL{f(t)f(t)}

LL{uc(t)u_{c}(t)} = escs\frac{e^{-sc}}{s}

These Formulae might be necessary for shifting functions:

sin(a+b)=sin(a)cos(b)+cos(a)sin(b)\sin{(a + b)} = \sin(a)\cos(b) + \cos(a)\sin(b)

cos(a+b)=cos(a)cos(b)sin(a)sin(b)\cos{(a + b)} = \cos(a)\cos(b) - \sin(a)\sin(b)

(a+b)2=a2+2ab+b2(a + b)^{2} = a^{2} + 2ab +b^{2}

  • 1.
    a)
    What is the Heaviside Step Function?

    b)
    What are some uses of the Heaviside Step Function and what is the Laplace Transform of a Heaviside Step Function?


  • 2.
    Determining Heaviside Step Functions

    Write the following graph in terms of a Heaviside Step Function

    graph of a Heaviside Step Function


  • 3.
    Determining the Laplace Transform of a Heaviside Step Function

    Find the Laplace Transform of each of the following Step Functions:

    a)
    f(t)=6u3(t)e3t15u5(t)+3(t7)2u7(t)f(t) = 6u_{3}(t) - e^{3t - 15}u_{5}(t) + 3(t - 7)^{2}u_{7}(t)

    b)
    g(t)=sin(t)uπ(t)+2t2u4(t)g(t) = -\sin{(t)}u_{\pi}(t) + 2t^{2}u_{4}(t)


  • 4.
    Determining the Inverse Laplace Transform of a Heaviside Step Function

    Find the inverse Laplace Transform of the following function:

    F(s)=4e3s(s2)(s+3)F(s) = \frac{4e^{-3s}}{(s - 2)(s + 3)}