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 = 10 min.
bacteria per minute
62
Oh, I see we're getting down to the microscopic level here with bacteria population growth! Let's crunch some numbers, shall we?
To find the rate of population growth at t = 10 minutes, we need to find the derivative of the function f(t). So, let's differentiate it:
f'(t) = d/dt (3t^2 + 2t + 1)
= 6t + 2
Now, we can plug in t = 10 into the derivative to find the rate of population growth at that particular time:
f'(10) = 6(10) + 2
= 60 + 2
= 62
Voila! The rate of population growth at t = 10 minutes is 62 bacteria per minute. These little critters sure know how to multiply!
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 and then evaluate it at the given time.
Given the population function: P = 3t^2 + 2t + 1
To find the derivative, we can apply the power rule and the sum rule of differentiation. The power rule states that if we have a term of the form x^n, the derivative is nx^(n-1). And the sum rule states that the derivative of a sum of functions is the sum of their derivatives.
Applying the power rule and the sum rule, we differentiate each term of the function:
dP/dt = d(3t^2)/dt + d(2t)/dt + d(1)/dt
Applying the power rule, we get:
dP/dt = 6t + 2
This is the derivative of the population function.
To find the rate of population growth at t = 10 min, we substitute t = 10 into the derivative:
dP/dt = 6(10) + 2 = 60 + 2 = 62
Therefore, the rate of population growth at t = 10 min is 62 bacteria per minute.
rate of growth at any instant is just dP/dt
P' = 6t + 2
P'(10) = 62 bugs/min