{"id":183540,"date":"2025-01-16T15:26:02","date_gmt":"2025-01-16T15:26:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183540"},"modified":"2025-01-16T15:26:10","modified_gmt":"2025-01-16T15:26:10","slug":"find-the-laplace-transform-of-the-piecewise-defined-functions-ft-illustrated-below-by-expressing-the-functions-in-terms-of-the-piecewise-function-and-the-heaviside-step-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/find-the-laplace-transform-of-the-piecewise-defined-functions-ft-illustrated-below-by-expressing-the-functions-in-terms-of-the-piecewise-function-and-the-heaviside-step-function\/","title":{"rendered":"Find the Laplace transform of the piecewise defined functions f(t) (illustrated below) by expressing the functions in terms of the piecewise function and the Heaviside step function"},"content":{"rendered":"\n

Find the Laplace transform of the piecewise defined functions f(t) (illustrated below) by expressing the functions in terms of the piecewise function and the Heaviside step function, H(t). (a) Find L[f(t)]. Assume that 0 < a=””>< c=”” always=”” holds.=”” f(t)=”” (b,=”” 2)=”” (c,=”” 2)=”” (a,=”” 0)=”” (c,=”” 0)=”” t=”” none=””><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To solve for the Laplace transform of the piecewise function given, we need to express it in terms of Heaviside step functions ( H(t) ) to handle the transitions in the piecewise definition. However, the link you provided does not display the actual graph or the piecewise function. Based on your description, I will guide you through the general process of solving such a problem.<\/p>\n\n\n\n

Steps for Solving the Laplace Transform of a Piecewise Function<\/h3>\n\n\n\n
    \n
  1. Understanding Piecewise Function<\/strong>: A piecewise function is typically defined for different intervals of ( t ). We need to express each interval using the Heaviside step function, ( H(t – t_0) ), where ( t_0 ) is the time at which the function starts to change. The Heaviside function is 0 before ( t = t_0 ) and 1 after.<\/li>\n\n\n\n
  2. Rewriting the Function Using Heaviside<\/strong>:
    For example, suppose we have a function ( f(t) ) as:
    [
    f(t) =
    \\begin{cases}
    0 & \\text{for } t < a \\
    c & \\text{for } a \\leq t < b \\
    b & \\text{for } t \\geq c
    \\end{cases}
    ]
    We would rewrite it using Heaviside functions as:
    [
    f(t) = c \\left[ H(t – a) – H(t – b) \\right] + b H(t – c)
    ]<\/li>\n\n\n\n
  3. Laplace Transform of Each Term<\/strong>:<\/li>\n<\/ol>\n\n\n\n
      \n
    • The Laplace transform of ( H(t – t_0) ) is ( \\frac{e^{-t_0 s}}{s} ).<\/li>\n\n\n\n
    • The Laplace transform of ( f(t) = A \\cdot H(t – t_0) ) is ( \\frac{A e^{-t_0 s}}{s} ).<\/li>\n<\/ul>\n\n\n\n
        \n
      1. Taking the Laplace Transform<\/strong>: Now, for each part of the piecewise function, apply the Laplace transform. For example:<\/li>\n<\/ol>\n\n\n\n
          \n
        • The Laplace transform of ( c \\left[ H(t – a) – H(t – b) \\right] ) is:
          [
          c \\left( \\frac{e^{-a s}}{s} – \\frac{e^{-b s}}{s} \\right)
          ]<\/li>\n\n\n\n
        • The Laplace transform of ( b H(t – c) ) is:
          [
          \\frac{b e^{-c s}}{s}
          ]<\/li>\n<\/ul>\n\n\n\n
            \n
          1. Final Laplace Transform<\/strong>: Adding all parts together gives the overall Laplace transform of ( f(t) ).<\/li>\n<\/ol>\n\n\n\n

            Example of a Full Solution<\/h3>\n\n\n\n

            If ( f(t) ) were defined as follows:
            [
            f(t) =
            \\begin{cases}
            0 & \\text{for } t < 2 \\
            2 & \\text{for } 2 \\leq t < 4 \\
            5 & \\text{for } t \\geq 4
            \\end{cases}
            ]
            We would rewrite it as:
            [
            f(t) = 2 \\left[ H(t – 2) – H(t – 4) \\right] + 5 H(t – 4)
            ]
            Then, the Laplace transform is:
            [
            \\mathcal{L}[f(t)] = 2 \\left( \\frac{e^{-2s}}{s} – \\frac{e^{-4s}}{s} \\right) + \\frac{5 e^{-4s}}{s}
            ]<\/p>\n\n\n\n

            This process involves breaking the function into parts, applying the Laplace transform to each, and then combining the results.<\/p>\n\n\n\n

            If you have more specific details about the function or its intervals, feel free to share, and I can refine the solution further.<\/p>\n","protected":false},"excerpt":{"rendered":"

            Find the Laplace transform of the piecewise defined functions f(t) (illustrated below) by expressing the functions in terms of the piecewise function and the Heaviside step function, H(t). (a) Find L[f(t)]. Assume that 0 < a=””>< c=”” always=”” holds.=”” f(t)=”” (b,=”” 2)=”” (c,=”” 2)=”” (a,=”” 0)=”” (c,=”” 0)=”” t=”” none=””> The Correct Answer and Explanation […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-183540","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=183540"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183540\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}