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We are asked to find the curvature \( k \) of the curve \( y = 4 \cos\left(\frac{x}{2}\right) \) at the point \( (0, 4) \).

### Step 1: Curvature Formula
The curvature \( k \) for a function \( y = f(x) \) is given by the formula:
\[
k = \frac{|f''(x)|}{\left(1 + (f'(x))^2\right)^{3/2}}
\]
where \( f'(x) \) is the first derivative of \( f(x) \) and \( f''(x) \) is the second derivative.

### Step 2: Find the first derivative \( f'(x) \)
The function is:
\[
y = f(x) = 4 \cos\left(\frac{x}{2}\right)
\]
Using the chain rule, we differentiate:
\[
f'(x) = 4 \cdot \left(-\sin\left(\frac{x}{2}\right)\right) \cdot \frac{1}{2} = -2 \sin\left(\frac{x}{2}\right)
\]

### Step 3: Find the second derivative \( f''(x) \)
Now, differentiate \( f'(x) \) again:
\[
f''(x) = -2 \cdot \cos\left(\frac{x}{2}\right) \cdot \frac{1}{2} = - \cos\left(\frac{x}{2}\right)
\]

### Step 4: Evaluate at \( x = 0 \)
At \( x = 0 \), we compute the values of \( f'(0) \) and \( f''(0) \):
\[
f'(0) = -2 \sin\left(\frac{0}{2}\right) = 0
\]
\[
f''(0) = - \cos\left(\frac{0}{2}\right) = -1
\]

### Step 5: Calculate the curvature at \( x = 0 \)
Now, substitute into the curvature formula:
\[
k = \frac{|f''(0)|}{\left(1 + (f'(0))^2\right)^{3/2}} = \frac{| -1 |}{\left(1 + 0^2\right)^{3/2}} = \frac{1}{1} = 1
\]

Thus, the curvature at the point \( (0, 4) \) is \( \boxed{1} \).

Would you like further clarification or additional details?

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Here are five related questions to extend your understanding:
1. What is the geometric interpretation of curvature for a given function?
2. How does curvature change if the function was \( y = \sin(x) \)?
3. Can you find the curvature of \( y = e^x \) at \( x = 0 \)?
4. What is the relationship between curvature and the second derivative of a function?
5. How does the chain rule affect differentiation in problems involving trigonometric functions?

**Tip**: When computing the curvature of a curve, it is important to carefully handle both the first and second derivatives, as small calculation errors can impact the result significantly!

La curvatura de la gráfica y = 4cos(x/2) en el punto (0, 4) es

Calculate the curvature of the curve y = 4cos(x/2) at the point (0, 4) using key calculus concepts. This problem involves finding the first and second derivatives and applying the curvature formula, suitable for university-level calculus students.

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