Find the domain of ~r(t) and the value of ~r(t0).
r(t) = <3t + 1, t2> ; t0 = π.

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. **\( \mathbf{r}(\pi) \)**: \( \langle 3\pi + 1, \pi^2 \rangle \).

---

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!

Find the domain of ~r(t) and the value of ~r(t0). r(t) = <3t + 1, t^2>; t0 = π.

Determine the domain of the vector function r(t) = <3t + 1, t^2> and calculate the value of r(t0) when t0 = π. This solution explores vector functions and their domains, suitable for students in Grades 10-12.

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