During the time interval 0, is less than or equal to, t, is less than or equal to, 21, comma0≤*t*≤21, the rate of change of the population of deer in a forest, in deer per day, can be modeled by the differentiable function C, left parenthesis, t, right parenthesis, comma*C*(*t*), where t*t* is measured in days. At time t, equals, 1*t*=1 there are 250250 deer in the forest. What is the best interpretation of the statement C, prime, left parenthesis, 1, right parenthesis, equals, 80, question mark*C*′(1)=80?

The problem provides a context where \( C(t) \) represents the population of deer in a forest at time \( t \) (in days), and \( C'(t) \) represents the rate of change of the deer population (in deer per day) at time \( t \).

Given that \( C'(1) = 80 \), we need to interpret this statement.

### **Interpretation of \( C'(1) = 80 \):**

At day 1, the population of deer in the forest is increasing at a rate of 80 deer per day.

This means that on the first day, the number of deer in the forest is growing by 80 deer each day.

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If you have any questions or want more details, feel free to ask!

Here are some related questions that might help:

1. What is the significance of a positive rate of change in population?
2. How does \( C(1) = 250 \) relate to the given rate of change?
3. What would it mean if \( C'(1) \) were negative?
4. How can we estimate the total population after a few days using this rate?
5. What does the second derivative \( C''(t) \) tell us about population growth?
6. How does the rate of change vary over time, and what does that imply?
7. What could affect the rate of change \( C'(t) \) in a real-world scenario?
8. How might the population growth trend change as \( t \) increases?

**Tip:** Understanding how a rate of change affects a quantity over time is key in many real-world applications, especially in modeling population dynamics.

Understand how the rate of change of deer population in a forest is interpreted using calculus. Explore the scenario where the rate of change at t = 1 is 80 deer per day, with a starting population of 250 deer. Learn implications of positive rates of change and its real-world applications.

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