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The problem asks for the length of the curve \( C \) determined by the vector function:

\[
r(t) = \left(e^t, \frac{21}{2} e^{-t}, \sqrt{21t} \right)
\]

We are asked to find the length of the curve for \( t \in [-1, 1] \).

The formula for the length \( L \) of a curve given by a vector function \( r(t) = (x(t), y(t), z(t)) \) over an interval \( [a, b] \) is:

\[
L = \int_a^b \| r'(t) \| \, dt
\]

where \( r'(t) \) is the derivative of the vector function \( r(t) \), and \( \| r'(t) \| \) is the magnitude of the derivative vector.

Let's go step by step:

1. **Find the derivative \( r'(t) \):**
   - The derivative of each component of \( r(t) \):
     - \( \frac{d}{dt} e^t = e^t \)
     - \( \frac{d}{dt} \frac{21}{2} e^{-t} = -\frac{21}{2} e^{-t} \)
     - \( \frac{d}{dt} \sqrt{21t} = \frac{\sqrt{21}}{2\sqrt{t}} \)

   Thus,

   \[
   r'(t) = \left( e^t, -\frac{21}{2} e^{-t}, \frac{\sqrt{21}}{2\sqrt{t}} \right)
   \]

2. **Calculate the magnitude \( \| r'(t) \| \):**

   The magnitude is given by:

   \[
   \| r'(t) \| = \sqrt{(e^t)^2 + \left(-\frac{21}{2} e^{-t}\right)^2 + \left(\frac{\sqrt{21}}{2\sqrt{t}}\right)^2}
   \]

   Simplifying each term:

   - \( (e^t)^2 = e^{2t} \)
   - \( \left(-\frac{21}{2} e^{-t}\right)^2 = \frac{441}{4} e^{-2t} \)
   - \( \left(\frac{\sqrt{21}}{2\sqrt{t}}\right)^2 = \frac{21}{4t} \)

   Thus,

   \[
   \| r'(t) \| = \sqrt{e^{2t} + \frac{441}{4} e^{-2t} + \frac{21}{4t}}
   \]

3. **Set up the integral:**

   The length of the curve is:

   \[
   L = \int_{-1}^{1} \sqrt{e^{2t} + \frac{441}{4} e^{-2t} + \frac{21}{4t}} \, dt
   \]

This integral is quite complex and may require numerical methods to evaluate. Let me calculate the numerical value of this integral next.It seems there's a division by zero in the term \( \frac{21}{4t} \) at \( t = 0 \), which causes a problem with the integral. This suggests a potential singularity at \( t = 0 \). I will address this by splitting the integral into two parts: one from \( t = -1 \) to just before \( t = 0 \), and another from just after \( t = 0 \) to \( t = 1 \).

Let me proceed with this approach to avoid the singularity.It seems that the term \( \frac{21}{4t} \) introduces further numerical instability near \( t = 0 \), and round-off errors are being detected during the integration.

An alternative approach would be to regularize or simplify the expression in the vicinity of \( t = 0 \), or exclude the singularity and focus on the behavior away from it. Would you prefer trying another method, such as approximating the problematic term or applying a different numerical strategy?

Feel free to ask questions or let me know how you would like to proceed.

Here are 5 related questions to consider:
1. What are the geometric interpretations of curve length?
2. How do singularities affect curve length calculations?
3. What other methods exist for numerically integrating problematic integrals?
4. Can parametric curves with singularities have finite lengths?
5. What real-world applications use curve length calculations?

**Tip**: Handling singularities in integrals often requires breaking down the domain into more manageable parts or using regularization techniques.

La longitud de la curva C determinada por r(t) = (e^t, (21/2)e^(-t), sqrt(21t)) para t en [-1, 1]. De ser necesario redondee su respuesta con dos cifras decimales.

This problem asks for the length of a parametric curve r(t) = (e^t, (21/2)e^(-t), sqrt(21t)) over the interval t in [-1, 1]. It involves applying the curve length formula using vector calculus and integration techniques, suitable for university-level calculus students. The result is to be rounded to two decimal places.

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