The given vector function is:

$\mathbf{r}(t) = 2 \cos(t^2) \, \hat{i} + 2 \sin(t^2) \, \hat{j} + \frac{t}{t + 4} \, \hat{k}, \quad 0 \leq t < \infty$

Let's analyze each component of the curve to understand its shape:

1. x-component: $x(t) = 2 \cos(t^2)$

   • This component oscillates between -2 and 2 as $t^2$ increases. It forms a periodic behavior in the $x$-direction.

2. y-component: $y(t) = 2 \sin(t^2)$

   • Similar to the $x$-component, this oscillates between -2 and 2 but with a sine function, meaning it will be 90° out of phase with the cosine term.

3. z-component: $z(t) = \frac{t}{t + 4}$

   • As $t \to \infty$, $z(t)$ approaches 1 (since $\frac{t}{t + 4} \to 1$). At $t = 0$, the value of $z$ is 0. This implies that the curve will slowly rise from $z = 0$ to $z = 1$ as $t$ increases.

Overall Shape:

• The $x(t)$ and $y(t)$ components describe a circular motion with radius 2 in the $xy$-plane (as $\cos^2(t^2) + \sin^2(t^2) = 1$).
• The $z(t)$ component increases from 0 to 1 as $t$ increases.
• This suggests a spiral shape that winds around the $z$-axis, starting from the $xy$-plane and gradually moving upward.

Matching the Graph:

• Option B depicts a spiraling curve around the $z$-axis, which fits the behavior described by the function. The other graphs either show purely circular motion (without any vertical progression) or different behaviors not fitting the analysis.

Thus, the correct answer is B.

Would you like further clarification on any of the steps? Here are some questions to consider:

1. How does the $z(t)$ component influence the shape of the curve?
2. What would happen if the z-component were a constant instead of $\frac{t}{t + 4}$?
3. How does the phase difference between the sine and cosine components affect the curve?
4. What is the general form of a spiral in three-dimensional space?
5. How does the growth rate of $z(t)$ affect the vertical progression of the curve?

Tip: Visualizing parametric curves using software or graphing utilities can help solidify understanding of their behavior.