Answer
$[-1,1)$
$ \mathbf{r}(t_0) = 2\mathbf{i}$
Work Step by Step
Step 1
We first consider the domain of each component function.
Step 2
The domain of a function is the set of input or argument values for which the function is real and defined.
Step 3
Note that: \[ x(t) = 2e^{-t}, \quad y(t) = \sin^{-1}(t), \quad z(t) = \ln(1-t) \]
Step 4
The domain of the function $x(t)$ is $(-\infty, \infty)$.
The domain of the function $y(t)$ is $[-1, 1]$.
The domain of the function $z(t)$ is $(-\infty, 1)$.
Step 5
Hence, the domain of the vector-valued function $\mathbf{r}$ is $[-1, 1)$.
Step 6
If $t_0 = 0$, then: \[ \mathbf{r}(t_0) = 2\mathbf{e}^{0} \mathbf{i} + \sin^{-1}(0) \mathbf{j} + \ln(1-0) \mathbf{k} = 2\mathbf{i} \] Result \[ \mathbf{r}(t_0) = 2\mathbf{i} \]
