Given the vector function $\mathbf{r}(t) = \langle 3t + 1, t^2 \rangle$, we need to find:

1. The domain of $\mathbf{r}(t)$.
2. The value of $\mathbf{r}(t_0)$ when $t_0 = \pi$.

1. Finding the domain of $\mathbf{r}(t)$

The domain of a vector function is determined by the set of values of $t$ for which each component of the vector is defined.

• The first component of $\mathbf{r}(t)$ is $3t + 1$, which is defined for all real numbers $t$.
• The second component of $\mathbf{r}(t)$ is $t^2$, which is also defined for all real numbers $t$.

Since both components are defined for all real numbers, the domain of $\mathbf{r}(t)$ is: $\text{Domain of } \mathbf{r}(t): (-\infty, \infty)$

2. Finding $\mathbf{r}(t_0)$ at $t_0 = \pi$

We substitute $t_0 = \pi$ into the function $\mathbf{r}(t)$.

$\mathbf{r}(\pi) = \langle 3(\pi) + 1, \pi^2 \rangle$ $\mathbf{r}(\pi) = \langle 3\pi + 1, \pi^2 \rangle$

Thus, the value of $\mathbf{r}(\pi)$ is: $\mathbf{r}(\pi) = \langle 3\pi + 1, \pi^2 \rangle$

Final Answer:

1. Domain: $(-\infty, \infty)$.
2. $\mathbf{r}(\pi)$: $\langle 3\pi + 1, \pi^2 \rangle$.

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Would you like more details or further explanation? Here are 5 related questions to explore further:

1. How do you find the domain of more complex vector functions?
2. What if one of the components had a restriction like $\sqrt{t}$ or $\frac{1}{t}$?
3. How do you compute the derivative of a vector function like $\mathbf{r}(t)$?
4. How would you interpret the vector $\mathbf{r}(t)$ geometrically?
5. What is the significance of the value $\mathbf{r}(t_0)$ in applications?

Tip: When solving for vector functions, always ensure that each component is well-defined for the given domain!