Correct equation for exponential growth is :

Options

A. `(dN)/(dt)=rN`
B. `(dN)/(dt)=(b-d)N`
C. `N_t=N_0e^(rt)`
D. Both 'a' and 'c'

Answer

To derive the correct equation for exponential growth, we can follow these steps:

### Step 1: Understand the Concept of Exponential Growth
Exponential growth occurs when resources are unlimited, allowing a population to grow at its maximum potential. This means that the growth rate does not slow down as the population increases.

### Step 2: Define the Variables
- Let \( N \) be the population size at time \( t \).
- Let \( N_0 \) be the initial population size.
- Let \( r \) be the intrinsic rate of increase (also known as biotic potential).
- \( t \) represents time.

### Step 3: Write the Differential Equation
The change in population size over time can be expressed as:
\[ \frac{dN}{dt} = rN \]
This equation indicates that the rate of change of the population (\( \frac{dN}{dt} \)) is proportional to the current population size (\( N \)).

### Step 4: Solve the Differential Equation
To find the solution to this differential equation, we can integrate:
\[ \int \frac{1}{N} dN = \int r \, dt \]
This leads to:
\[ \ln N = rt + C \]
where \( C \) is the constant of integration.

### Step 5: Exponentiate to Solve for \( N \)
By exponentiating both sides, we can solve for \( N \):
\[ N = e^{rt + C} = e^C e^{rt} \]
Let \( e^C = N_0 \), the initial population size, so:
\[ N = N_0 e^{rt} \]

### Step 6: Final Equation for Exponential Growth
The final equation for exponential growth is:
\[ N(t) = N_0 e^{rt} \]
This equation shows how the population size \( N(t) \) changes over time \( t \) given an initial population \( N_0 \) and a constant growth rate \( r \).

### Conclusion
The correct equation for exponential growth is \( N(t) = N_0 e^{rt} \).

---