The question involves finding the arc length parameter $s(t)$ along a vector-valued curve $\mathbf{r}(t)$, and evaluating the length of the indicated portion of the curve. I will break this down step by step.

Step 1: The Formula for Arc Length

The arc length of a curve $\mathbf{r}(t)$ is given by:

$s(t) = \int_{0}^{t} |\mathbf{r}'(u)| \, du$

where $|\mathbf{r}'(t)|$ is the magnitude of the derivative of $\mathbf{r}(t)$.

Step 2: Derivative of the Curve

The given curve is: $\mathbf{r}(t) = 5 \cos t \, \mathbf{i} + 5 \sin t \, \mathbf{j} + 9t \, \mathbf{k}$

Compute the derivative $\mathbf{r}'(t)$: $\mathbf{r}'(t) = \frac{d}{dt} \left( 5 \cos t \right) \mathbf{i} + \frac{d}{dt} \left( 5 \sin t \right) \mathbf{j} + \frac{d}{dt} \left( 9t \right) \mathbf{k}$

$\mathbf{r}'(t) = -5 \sin t \, \mathbf{i} + 5 \cos t \, \mathbf{j} + 9 \, \mathbf{k}$

Step 3: Magnitude of $\mathbf{r}'(t)$

The magnitude of $\mathbf{r}'(t)$ is:

$|\mathbf{r}'(t)| = \sqrt{(-5 \sin t)^2 + (5 \cos t)^2 + 9^2}$

Simplify: $|\mathbf{r}'(t)| = \sqrt{25 \sin^2 t + 25 \cos^2 t + 81}$

Using the Pythagorean identity $\sin^2 t + \cos^2 t = 1$: $|\mathbf{r}'(t)| = \sqrt{25(1) + 81} = \sqrt{106}$

Step 4: Arc Length Parameter $s(t)$

Now, the arc length parameter is: $s(t) = \int_{0}^{t} |\mathbf{r}'(u)| \, du = \int_{0}^{t} \sqrt{106} \, du$

Since $\sqrt{106}$ is a constant: $s(t) = \sqrt{106} \int_{0}^{t} 1 \, du = \sqrt{106} \, t$

Thus, the arc length parameter along the curve is: $s(t) = \sqrt{106} \, t$

Step 5: Length of the Indicated Portion

The indicated portion of the curve is from $t = 0$ to $t = \frac{\pi}{4}$. The length is:

$L = s\left(\frac{\pi}{4}\right) - s(0)$

From $s(t) = \sqrt{106} \, t$: $L = \sqrt{106} \left(\frac{\pi}{4}\right) - \sqrt{106} (0)$

$L = \frac{\sqrt{106} \, \pi}{4}$

Final Answers:

1. The arc length parameter along the curve is: $s(t) = \sqrt{106} \, t$

2. The length of the indicated portion of the curve is: $L = \frac{\sqrt{106} \, \pi}{4}$

Would you like a deeper explanation of any step? Let me know!

Related Questions:

1. How do you interpret the arc length parameter geometrically?
2. What happens if you change the limits of integration for the arc length?
3. How does the magnitude of $\mathbf{r}'(t)$ affect the length of the curve?
4. Why does the Pythagorean identity simplify $|\mathbf{r}'(t)|$ in this case?
5. Can you compute the arc length numerically if $\sqrt{106}$ is approximated?

Tip:

Always ensure to simplify expressions like $\sqrt{\sin^2 t + \cos^2 t}$ using trigonometric identities to save time.