https://r2.erweima.ai/mathbot/image/1725760358665dvy0vy2o.jpg

The problem in the image is asking for the tangent vector to the curve \( f(\theta) = (\cos(\theta), 3\sin(\theta)) \) at the point \( \theta_0 = 0 \).

### Step-by-step solution:

1. **Given function:**
   \[
   f(\theta) = (\cos(\theta), 3\sin(\theta))
   \]
   This is a parametric curve.

2. **Tangent vector:**
   The tangent vector to the curve at any point \( \theta \) is given by the derivative of \( f(\theta) \) with respect to \( \theta \), i.e., \( f'(\theta) \).

   Let's compute the derivative component-wise:
   
   \[
   f'(\theta) = \left( \frac{d}{d\theta}[\cos(\theta)], \frac{d}{d\theta}[3\sin(\theta)] \right)
   \]
   Derivatives:
   \[
   \frac{d}{d\theta}[\cos(\theta)] = -\sin(\theta)
   \]
   \[
   \frac{d}{d\theta}[3\sin(\theta)] = 3\cos(\theta)
   \]

   Therefore:
   \[
   f'(\theta) = (-\sin(\theta), 3\cos(\theta))
   \]

3. **Evaluate at \( \theta_0 = 0 \):**
   \[
   f'(0) = (-\sin(0), 3\cos(0)) = (0, 3)
   \]

   So the tangent vector at \( \theta_0 = 0 \) is \( (0, 3) \).

### Conclusion:
The correct answer is \( \mathbf{d.} \ (0, 3) \).

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Would you like further details or have any questions about this solution? Here are some additional related questions:

1. How do you find tangent vectors for parametric curves in 3D space?
2. What is the significance of the tangent vector in applications like physics?
3. Can we calculate the normal vector of the curve at \( \theta_0 = 0 \)?
4. How do we generalize this process for more complex curves?
5. What is the geometric interpretation of a tangent vector?

**Tip:** Tangent vectors are useful in understanding the direction and rate of change of curves at specific points, and they play a crucial role in fields like physics and engineering.

O vetor tangente à curva f(θ) = (cos(θ), 3sin(θ)) para θ ∈ [0, 2π] no ponto θ0 = 0.

Learn how to find the tangent vector to the parametric curve f(θ) = (cos(θ), 3sin(θ)) at θ0 = 0 using derivatives. This problem involves calculus concepts, such as parametric equations and tangent vectors.

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