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.

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.

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

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?

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