University Exam Questions: Laplace Transform of Derivatives and Initial Conditions

📚 Quick Study Guide

• 🔎 The Laplace Transform of a derivative is given by: $L{f'(t)} = sF(s) - f(0)$, where $F(s)$ is the Laplace transform of $f(t)$.
• 💡 For the second derivative: $L{f''(t)} = s^2F(s) - sf(0) - f'(0)$. Notice the pattern!
• 📝 In general, for the nth derivative: $L{f^{(n)}(t)} = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) - ... - f^{(n-1)}(0)$.
• 🧮 Initial conditions, like $f(0)$ and $f'(0)$, are CRUCIAL. They determine the particular solution. Make sure to sub them in correctly!
• 🧠 Remember linearity: $L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}$, where a and b are constants. This is very useful for breaking down complex problems.
• ⏱️ Practice identifying the function $f(t)$ and its derivatives. Knowing the basic Laplace transforms of common functions (like $e^{at}$, $\sin(at)$, $\cos(at)$) is essential.
• ✅ Don't forget partial fraction decomposition. You'll often need it to find the inverse Laplace transform after solving for $F(s)$.

🧪 Practice Quiz

1. What is the Laplace transform of $f'(t)$ given $f(0) = 2$?
1. A. $sF(s)$
2. B. $sF(s) - 2$
3. C. $F(s) - 2$
4. D. $2sF(s)$
2. Find the Laplace transform of $f''(t)$ if $f(0) = 1$ and $f'(0) = -1$.
1. A. $s^2F(s) - s + 1$
2. B. $s^2F(s) + s - 1$
3. C. $s^2F(s) - s - 1$
4. D. $s^2F(s) + s + 1$
3. What is $L{f'(t)}$ if $F(s) = \frac{1}{s+2}$ and $f(0) = 0$?
1. A. $\frac{s}{s+2}$
2. B. $\frac{1}{s+2}$
3. C. $\frac{s}{s+2} - 0$
4. D. $\frac{1}{s+2} - s$
4. Calculate the Laplace transform of $f'(t) + 2f(t)$, given $f(0) = 1$ and $L{f(t)} = F(s)$.
1. A. $sF(s) + 2F(s)$
2. B. $sF(s) - 1 + 2F(s)$
3. C. $sF(s) + 1 + 2F(s)$
4. D. $F(s) - 1$
5. If $L{f'(t)} = \frac{1}{s}$, what is $f(0)$ assuming $L{f(t)} = F(s)$ and $F(s) = \frac{1}{s^2}$?
1. A. 0
2. B. 1
3. C. -1
4. D. 2
6. Determine the Laplace transform of $3f''(t)$ given $f(0) = 0$ and $f'(0) = 1$.
1. A. $3s^2F(s)$
2. B. $3s^2F(s) - 1$
3. C. $3s^2F(s) - 3$
4. D. $3s^2F(s) - 3s$
7. Find $L{f'(t)}$ when $f(t) = e^{-t}$ and thus $F(s) = \frac{1}{s+1}$.
1. A. $\frac{s}{s+1}$
2. B. $\frac{1}{s+1} - 1$
3. C. $\frac{s}{s+1} - 1$
4. D. $\frac{1}{s+1}$