https://file.aiquickdraw.com/mathbot/file/1729219895844cb058voa.jpg

Let's solve the problem in the image! The task asks for the first and second derivatives (\(\vec{r}'(t)\), \(\vec{r}''(t)\)), and to evaluate them at \(t = 0\). 

We will handle each part (a, b, and c) step-by-step.

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### **Part a:**  
\[
\vec{r}(t) = \left( e^t + e^{-t} \right) \hat{i} - e^{3t} \hat{j}
\]
#### 1st derivative:  
Differentiate each component.
\[
\vec{r}'(t) = \left( e^t - e^{-t} \right) \hat{i} - 3e^{3t} \hat{j}
\]

Evaluate at \(t = 0\):
\[
\vec{r}'(0) = \left( e^0 - e^0 \right) \hat{i} - 3e^0 \hat{j} = 0 \hat{i} - 3 \hat{j}
\]

#### 2nd derivative:  
Differentiate again.
\[
\vec{r}''(t) = \left( e^t + e^{-t} \right) \hat{i} - 9e^{3t} \hat{j}
\]

Evaluate at \(t = 0\):
\[
\vec{r}''(0) = \left( e^0 + e^0 \right) \hat{i} - 9e^0 \hat{j} = 2 \hat{i} - 9 \hat{j}
\]

---

### **Part b:**  
\[
\vec{r}(t) = \tan(t) \hat{i} - 2t^{5/3} \hat{j}
\]
#### 1st derivative:
\[
\vec{r}'(t) = \sec^2(t) \hat{i} - \frac{10}{3} t^{2/3} \hat{j}
\]

Evaluate at \(t = 0\):
\[
\vec{r}'(0) = \sec^2(0) \hat{i} - \frac{10}{3} \cdot 0^{2/3} \hat{j} = 1 \hat{i} - 0 \hat{j}
\]

#### 2nd derivative:
\[
\vec{r}''(t) = 2 \sec^2(t) \tan(t) \hat{i} - \frac{20}{9} t^{-1/3} \hat{j}
\]

Evaluate at \(t = 0\):
\(\tan(0) = 0\), so:
\[
\vec{r}''(0) = 2 \cdot 1 \cdot 0 \hat{i} - \frac{20}{9} \cdot \infty \hat{j}
\]
The second derivative is undefined at \(t = 0\) due to the \(\frac{20}{9} t^{-1/3}\) term.

---

### **Part c:**  
\[
\vec{r}(t) = e^{2t} \hat{i} + \ln(t^3) \hat{j} + t^2 \hat{k}
\]
#### 1st derivative:
\[
\vec{r}'(t) = 2e^{2t} \hat{i} + \frac{3}{t} \hat{j} + 2t \hat{k}
\]

Evaluate at \(t = 0\):
The term \(\frac{3}{t}\) is undefined at \(t = 0\), so the derivative does not exist at this point.

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Would you like a deeper dive into any part of the calculations? 

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### **5 Related Questions:**
1. How do you differentiate vector-valued functions component-wise?
2. What are the conditions for differentiability at a point?
3. How does the chain rule apply in the context of vector functions?
4. What happens when a derivative is undefined at a specific point?
5. How do higher-order derivatives affect the behavior of curves?

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**Tip:** When differentiating functions with logarithmic terms, always check their domains to avoid undefined values.

Tentukan \(\vec{r}'(t)\), \(\vec{r}''(t)\), \(\vec{r}'(0)\), dan \(\vec{r}''(0)\) untuk fungsi-fungsi berikut: \(\vec{r}(t)\) pada bagian a, b, dan c.

This problem focuses on finding the first and second derivatives of vector-valued functions. Detailed steps include differentiation of exponential, trigonometric, and logarithmic functions. Solutions include evaluating derivatives at specific points for each given function.

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