Find laplace transform of f(t)=0,t<4 t^2-8t+22,t>=4

To find the Laplace transform of the given piecewise function:

\[
f(t) =
\begin{cases}
0, & t < 4 \\
t^2 - 8t + 22, & t \geq 4
\end{cases}
\]

We can express \(f(t)\) using the Heaviside step function \(u(t)\):

\[
f(t) = (t^2 - 8t + 22)u(t-4)
\]

### Step 1: Compute the Laplace Transform

The Laplace transform of a function multiplied by the Heaviside step function can be calculated using the formula:

\[
\mathcal{L}\{f(t)u(t-a)\} = e^{-as}F(s)
\]

where \(F(s)\) is the Laplace transform of \(f(t)\) without the step function.

### Step 2: Calculate \(F(s)\)

First, we need to compute the Laplace transform of \(g(t) = t^2 - 8t + 22\):

\[
F(s) = \mathcal{L}\{t^2 - 8t + 22\} = \mathcal{L}\{t^2\} - 8\mathcal{L}\{t\} + 22\mathcal{L}\{1\}
\]

Using the standard transforms:

- \(\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}\)
- \(\mathcal{L}\{1\} = \frac{1}{s}\)

Calculating each term:

1. \(\mathcal{L}\{t^2\} = \frac{2!}{s^3} = \frac{2}{s^3}\)
2. \(\mathcal{L}\{t\} = \frac{1}{s^2}\)
3. \(\mathcal{L}\{1\} = \frac{1}{s}\)

Now substituting these back into \(F(s)\):

\[
F(s) = \frac{2}{s^3} - 8\left(\frac{1}{s^2}\right) + 22\left(\frac{1}{s}\right) = \frac{2}{s^3} - \frac{8}{s^2} + \frac{22}{s}
\]

### Step 3: Laplace Transform of \(f(t)\)

Now applying the Heaviside step function:

\[
\mathcal{L}\{f(t)\} = e^{-4s}F(s) = e^{-4s}\left(\frac{2}{s^3} - \frac{8}{s^2} + \frac{22}{s}\right)
\]

### Final Result

Thus, the Laplace transform of \(f(t)\) is:

\[
\mathcal{L}\{f(t)\} = e^{-4s} \left( \frac{2}{s^3} - \frac{8}{s^2} + \frac{22}{s} \right)
\]

Would you like more details on any specific step or further questions about Laplace transforms? Here are some related questions to consider:

1. What are the properties of the Laplace transform?
2. How does the Heaviside step function affect the Laplace transform?
3. Can you give examples of other piecewise functions and their Laplace transforms?
4. What applications do Laplace transforms have in solving differential equations?
5. How do you find the inverse Laplace transform? 

Tip: Always remember to express piecewise functions using the Heaviside function for easier Laplace transformation!

Discover how to find the Laplace transform of the piecewise function f(t) defined as 0 for t < 4 and t^2 - 8t + 22 for t ≥ 4. This solution involves the Heaviside step function and standard Laplace transform formulas. Ideal for high school students studying advanced calculus.

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