Discrete vs Continuous Math Example 2

Solution

1. 1
   Discrete ($2^t$): $t=0: 1$, $t=1: 2$, $t=2: 4$, $t=3: 8$.
2. 2
   Continuous ($e^{0.693t}$, noting $\ln 2 \approx 0.693$): $e^0 = 1$, $e^{0.693} \approx 2$, $e^{1.386} \approx 4$, $e^{2.079} \approx 8$.
3. 3
   At integer times the models agree. The continuous model also gives values between integers (e.g., at $t=1.5$: $e^{1.04} \approx 2.83$), whereas bacteria truly exist only in whole numbers.
4. 4
   Discrete is more accurate for small populations; continuous is convenient for calculus-based analysis and large populations.