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The problem asks for the tangent vector to the curve \( \mathbf{f}(t) = (4 \cos t, 3 \sin 2t) \), where \( t \in [0, 2\pi] \), at the point \( t_0 = 0 \).

### Step 1: Find the derivative of the parametric curve
The tangent vector is the derivative of the curve \( \mathbf{f}(t) \) with respect to \( t \). So, let's compute the derivative \( \mathbf{f}'(t) \).

Given \( \mathbf{f}(t) = (4 \cos t, 3 \sin 2t) \):

- The derivative of \( 4 \cos t \) is \( -4 \sin t \).
- The derivative of \( 3 \sin 2t \) is \( 3 \cdot 2 \cos 2t = 6 \cos 2t \).

So, \( \mathbf{f}'(t) = (-4 \sin t, 6 \cos 2t) \).

### Step 2: Evaluate the derivative at \( t_0 = 0 \)
Now, substitute \( t_0 = 0 \) into the derivative:

- \( \sin(0) = 0 \), so the first component becomes \( -4 \sin(0) = 0 \).
- \( \cos(2 \times 0) = \cos(0) = 1 \), so the second component becomes \( 6 \cdot 1 = 6 \).

Thus, the tangent vector at \( t_0 = 0 \) is \( \mathbf{f}'(0) = (0, 6) \).

### Step 3: Check the options
From the given options:

a) (-4, 4)  
b) (4, 0)  
c) (-4, 0)  
d) (0, 6)  

The correct answer is **d) (0, 6)**.

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Do you have any questions or want further details? Here are some related questions to explore:

1. How is the tangent vector related to the derivative of a parametric curve?
2. Can you compute the unit tangent vector for the given curve at \( t_0 = 0 \)?
3. What is the geometric meaning of the tangent vector at a specific point on the curve?
4. How would the tangent vector change if the parametric curve had more complex trigonometric functions?
5. Can we compute the curvature of the curve \( \mathbf{f}(t) \) at \( t_0 = 0 \)?

**Tip:** When dealing with parametric curves, always remember that the tangent vector is given by the derivative of the position vector with respect to the parameter.

O vetor tangente à curva f(t) = (4 cos t, 3 sin 2t), t ∈ [0, 2π], no ponto t0 = 0 é

Discover how to calculate the tangent vector to the parametric curve f(t) = (4 cos t, 3 sin 2t) at the point t0 = 0. This detailed explanation covers key concepts in calculus and trigonometry, including the derivative of a parametric curve. Ideal for students in Grades 11-12.

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