Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the following function.
P=f(t)=3t^2+2t+1
Find the rate of population growth at t = 9 min.
To find the rate of population growth at a specific time, we need to find the derivative of the population function with respect to time (t), and then evaluate it at the given time.
Given that the population function is P = f(t) = 3t^2 + 2t + 1, we can find the derivative by using the power rule for differentiation.
The power rule states that if we have a term in the form of x^n, then the derivative is given by:
d/dx (x^n) = n * x^(n-1)
Applying the power rule to each term in the population function, we have:
dP/dt = d(3t^2)/dt + d(2t)/dt + d(1)/dt
Simplifying each term, we get:
dP/dt = 6t + 2 + 0
dP/dt = 6t + 2
Now that we have the derivative of the population function, we can evaluate it at t = 9 min:
dP/dt = 6(9) + 2
= 54 + 2
= 56
Therefore, the rate of population growth at t = 9 min is 56.
f'(t) = 6t+2, so
f'(9) = ?