Please give detail as much as possible. 6. Let \( C \) be the curve in \( \mathbb{R}^5 \) given by the vector-valued function \( \mathbf{r} : \mathbb{R} \to \mathbb{R}^5 \) defined as \( \mathbf{r}(t) = \langle \cos(e - t), \ln(\ln(t)), t^t, t^{2/3}, 0 \rangle \).(a) Determine whether the curve \( C \) passes through the point \( (1, 0, e^e, \sqrt[3]{e^2}, 0) \).(b) Compute the tangent vector to \( C \) when \( t = e \) (Note: This does not need to have unit length.). Where is this curve differentiable? Give your answer in interval notation.(c) What is the domain of \( \mathbf{r} \)? The curve \( C \) belongs to a particular subset of \( \mathbb{R}^5 \). What is this set? Explain. (Note: A basic element/point of \( \mathbb{R}^5 \) should have the form \((x_1, x_2, x_3, x_4, x_5)\). For the subset question, think in terms of higher-dimensional "planes").

Name: Please give detail as much as possible. 6. Let \( C \) be the curve in
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please give detail as much as possible. 6. Let \( C \) be the curve in \( \mathbb{R}^5 \) given by the vector-valued function \( \mathbf{r} : \mathbb{R} \to \mathbb{R}^5 \) defined as \( \mathbf{r}(t) = \langle \cos(e - t), \ln(\ln(t)), t^t, t^{2/3},