Applications of Exponential & Log Function Derivatives

1. The number of bacteria, N, in a certain culture grows exponentially according to: N = 10,000e^{0.1t} where t is in hours.What is the rate of population increase at a) t = 0?

b) t = 100?

Solution: What is known? What are we looking for? Write this in mathematical terms.

N = 10,000e^{0.1t} and dN/dt = ? when t = 0 and t = 100.

How do we get dN/dt?

We take the derivative of N(t) with respect to time.

dN/dt = (0.1)*10,000e^{0.1t} = 1000e^{0.1t}

Then dN/dt (t = 0) = 1000bacteria/hr

dN/dt (t = 100) = 2.20 x 10^{7} bacteria/hr

Wow! Makes you want to go and wash your hands!