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The problems listed are about finding the Laplace Transform \(\mathcal{L}\{f(t)\}\) of the given functions using Theorem 7.2, which likely includes standard formulas for Laplace transforms. Below are detailed solutions for some of the problems:

### Example Solutions:
1. **Problem 19**: \( f(t) = 2t^4 \)
   \[
   \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}, \quad \mathcal{L}\{2t^4\} = 2 \cdot \frac{4!}{s^5} = \frac{48}{s^5}
   \]

2. **Problem 20**: \( f(t) = t^5 \)
   \[
   \mathcal{L}\{t^5\} = \frac{5!}{s^6} = \frac{120}{s^6}
   \]

3. **Problem 21**: \( f(t) = 4t - 10 \)
   \[
   \mathcal{L}\{4t\} = 4 \cdot \frac{1}{s^2} = \frac{4}{s^2}, \quad \mathcal{L}\{10\} = \frac{10}{s}
   \]
   \[
   \mathcal{L}\{4t - 10\} = \frac{4}{s^2} - \frac{10}{s}
   \]

4. **Problem 22**: \( f(t) = 7t + 3 \)
   \[
   \mathcal{L}\{7t\} = 7 \cdot \frac{1}{s^2} = \frac{7}{s^2}, \quad \mathcal{L}\{3\} = \frac{3}{s}
   \]
   \[
   \mathcal{L}\{7t + 3\} = \frac{7}{s^2} + \frac{3}{s}
   \]

5. **Problem 23**: \( f(t) = t^2 + 6t - 3 \)
   \[
   \mathcal{L}\{t^2\} = \frac{2!}{s^3} = \frac{2}{s^3}, \quad \mathcal{L}\{6t\} = \frac{6}{s^2}, \quad \mathcal{L}\{-3\} = -\frac{3}{s}
   \]
   \[
   \mathcal{L}\{t^2 + 6t - 3\} = \frac{2}{s^3} + \frac{6}{s^2} - \frac{3}{s}
   \]

These examples show the application of standard Laplace Transform formulas for polynomial and constant functions. 

Would you like more details on these or help with a specific problem from the list? 

### Follow-up Questions:
1. What is the Laplace Transform of \( \cosh(kt) \)?
2. How would you solve the Laplace Transform for a function involving \( e^{at} \)?
3. Can you derive the Laplace Transform of \( \sin(bt) \)?
4. What theorem might be used for transforming \( (t + 1)^3 \)?
5. How do we apply the shifting property in the Laplace Transform?

**Tip:** The Laplace Transform of a polynomial \( t^n \) is \(\frac{n!}{s^{n+1}}\).

Find the Laplace Transform L{f(t)} for each of the functions in problems 19-38 using Theorem 7.2.

This page provides detailed solutions for finding the Laplace Transform L{f(t)} of functions, including polynomials, exponentials, and hyperbolic functions, based on Theorem 7.2. Learn key formulas like L{t^n}, L{e^(at)}, and L{sinh(at)}.

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