Find the unit tangent vector 
T(t)
 for 
r(t) = 4ti + 8t2j + 2tk.
T(t) =

To find the unit tangent vector \( \mathbf{T}(t) \) for the vector function \( \mathbf{r}(t) = 4t \mathbf{i} + 8t^2 \mathbf{j} + 2t \mathbf{k} \), follow these steps:

### Step 1: Compute the derivative \( \mathbf{r}'(t) \)
The derivative \( \mathbf{r}'(t) \) is the velocity vector, and it is obtained by differentiating \( \mathbf{r}(t) \) component-wise:

\[
\mathbf{r}'(t) = \frac{d}{dt} \left( 4t \mathbf{i} + 8t^2 \mathbf{j} + 2t \mathbf{k} \right)
\]

\[
\mathbf{r}'(t) = 4 \mathbf{i} + 16t \mathbf{j} + 2 \mathbf{k}
\]

### Step 2: Find the magnitude \( |\mathbf{r}'(t)| \)
The magnitude of \( \mathbf{r}'(t) \) is the length of the velocity vector:

\[
|\mathbf{r}'(t)| = \sqrt{(4)^2 + (16t)^2 + (2)^2}
\]

\[
|\mathbf{r}'(t)| = \sqrt{16 + 256t^2 + 4}
\]

\[
|\mathbf{r}'(t)| = \sqrt{20 + 256t^2}
\]

### Step 3: Compute the unit tangent vector \( \mathbf{T}(t) \)
The unit tangent vector \( \mathbf{T}(t) \) is given by:

\[
\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}
\]

Substitute \( \mathbf{r}'(t) = 4 \mathbf{i} + 16t \mathbf{j} + 2 \mathbf{k} \) and \( |\mathbf{r}'(t)| = \sqrt{20 + 256t^2} \):

\[
\mathbf{T}(t) = \frac{4 \mathbf{i} + 16t \mathbf{j} + 2 \mathbf{k}}{\sqrt{20 + 256t^2}}
\]

This is the unit tangent vector.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore further:
1. How do you find the unit normal vector \( \mathbf{N}(t) \)?
2. What is the formula for the curvature \( \kappa(t) \) of a curve?
3. How do you compute the torsion of a space curve?
4. What are the differences between tangent and normal vectors?
5. How does the concept of arc length relate to unit tangent vectors?

**Tip:** Always simplify your vector expressions as much as possible to avoid computational errors in later steps.

Find the unit tangent vector T(t) for r(t) = 4ti + 8t^2j + 2tk.

Explore a step-by-step solution to find the unit tangent vector T(t) for the vector function r(t) = 4ti + 8t^2j + 2tk. Learn about key concepts in vector calculus, including differentiation, unit tangent vectors, and their applications. Suitable for undergraduate Calculus I students.

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