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Resuelva, la función N(t) = 1000 * ((t^3/2)+t^2 ). Encuentre N'(t). Evalué N(9) y N'(t) e interprete  estos valores

The new problem you provided involves a different function \( N(t) \) defined as:

\[
N(t) = 1000 \times \left(\left(\frac{t^3}{2}\right) + t^2 \right)
\]

### Task:
1. Find \( N'(t) \), the derivative of \( N(t) \).
2. Evaluate \( N(9) \) and \( N'(9) \).
3. Interpret these values.

Let's solve it step by step.

### Step 1: Find \( N'(t) \)

The function \( N(t) = 1000 \times \left(\left(\frac{t^3}{2}\right) + t^2 \right) \) can be expanded and differentiated as follows:

First, expand the expression inside:

\[
N(t) = 1000 \times \left(\frac{t^3}{2} + t^2 \right)
\]

This is already in its expanded form. Now, differentiate with respect to \( t \):

\[
N'(t) = 1000 \times \frac{d}{dt} \left(\frac{t^3}{2} + t^2 \right)
\]

Now differentiate term by term:

\[
N'(t) = 1000 \times \left(\frac{3t^2}{2} + 2t\right)
\]

\[
N'(t) = 1000 \times \left(\frac{3t^2}{2} + 2t\right) = 1000 \times \left(\frac{3}{2} t^2 + 2t \right)
\]

So:

\[
N'(t) = 1000 \times \left(\frac{3t^2}{2} + 2t\right)
\]

### Step 2: Evaluate \( N(9) \) and \( N'(9) \)

- **\( N(9) \):**

\[
N(9) = 1000 \times \left(\frac{9^3}{2} + 9^2 \right) = 1000 \times \left(\frac{729}{2} + 81 \right)
\]

\[
N(9) = 1000 \times \left(364.5 + 81\right) = 1000 \times 445.5 = 445500
\]

- **\( N'(9) \):**

\[
N'(9) = 1000 \times \left(\frac{3 \times 9^2}{2} + 2 \times 9\right) = 1000 \times \left(\frac{3 \times 81}{2} + 18\right)
\]

\[
N'(9) = 1000 \times \left(\frac{243}{2} + 18\right) = 1000 \times \left(121.5 + 18\right) = 1000 \times 139.5 = 139500
\]

### Step 3: Interpretation
- **\( N(9) = 445,500 \)**: After 9 months, the model predicts that there are 445,500 infectious individuals.
- **\( N'(9) = 139,500 \)**: At 9 months, the rate of change of the number of infectious individuals is 139,500 per month. This suggests that the number of infectious people is rapidly increasing.

Do you want more details or have any questions? Here are some related questions to consider:

1. What does the \( t^3 \) term in \( N(t) \) suggest about the long-term growth of the disease?
2. How does the model behave for smaller values of \( t \)?
3. What is the significance of the large coefficients in this model?
4. How would \( N(t) \) behave as \( t \) approaches infinity?
5. What can we infer about the population if \( N(t) \) continues to grow as predicted by this model?

**Tip:** When interpreting the rate of change in such models, consider both the short-term and long-term implications, especially how fast the disease could potentially spread over time.

Learn how to find the derivative N'(t) for the function N(t) = 1000 * ((t^3/2) + t^2). Evaluate N(9) and N'(9) with a step-by-step solution. Explore the power rule of differentiation and its application to polynomial functions.

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