Which of the following equations corretly represents the exponential population growth curve ?

Options

A. `dN//dt = rN`
B. `dN//dt = rN((K-N)/(K))`
C. `N_(t) = N_(0)e^(rt)`
D. Both (a) and (c)

Explanation

To determine which equations correctly represent the exponential population growth curve, we need to understand the characteristics of exponential growth and the equations that describe it.

### Step-by-Step Solution:

1. **Understanding Exponential Growth**:
- Exponential growth occurs when a population has unlimited resources and can reproduce at its maximum potential. This leads to rapid population increase over time.

2. **Identifying the Equation**:
- The mathematical representation of exponential growth is given by the differential equation:
\[ \frac{dn}{dt} = Rn \]
Here, \( \frac{dn}{dt} \) represents the rate of change of the population size (n) over time (t), and \( R \) is the intrinsic rate of natural increase.

3. **Integral Form of the Equation**:
- The integral form of the exponential growth equation can be derived from the differential equation. When you integrate, you get:
\[ n(t) = n_0 e^{Rt} \]
where \( n_0 \) is the initial population size and \( e \) is the base of the natural logarithm.

4. **Analyzing Given Options**:
- From the options provided, we need to identify which equations match the forms described above.
- **Option A**: If it matches \( \frac{dn}{dt} = Rn \) or \( n(t) = n_0 e^{Rt} \), it is correct.
- **Option C**: If it is in the integral form \( n(t) = n_0 e^{Rt} \), it is also correct.
- **Option B**: If it includes a carrying capacity (K), it represents logistic growth, not exponential growth.

5. **Conclusion**:
- After analyzing the options, both **Option A** and **Option C** correctly represent the exponential growth model. **Option B** represents logistic growth due to the presence of carrying capacity (K).
- Therefore, the correct answer is that both **Option A and C are correct**.

### Final Answer:
Both Option A and Option C correctly represent the exponential population growth curve.