6.2 Solving IVT using LT

Laplace transfrom of the first derivitive:

• Assume $\mathcal{L}(f(t))=F(s)$

  • 6.1 Laplace Transform this is the alternate way of writing the function
  • but what is its derivitive
    • $\mathcal{L}(f^{\prime }(t))={\int }_0^{\infty }{e}^{-st}{f}^{\prime }(t)dt$
      • using integration by parts we can rewrite the integral as
        • $e^{-st}f(t){|}_0^{\infty }-{\int }_0^{\infty }-s{e}^{-st}f(t)dt$
          • ⇒ $SF(s)-f(0)$ we are using the fact that the integral is the Laplace transform, so we turned the funciton with derivitives to a function with out one