A population’s size after 𝑡 days is given by 𝑓 of 𝑡 equals 11𝑡 squared plus 35923. Find the rate of change in the population when 𝑡 equals 12.
We’re asked for the rate of change in the population. And as we’re not given a period over which to measure this change, we assume that this is the instantaneous rate of change. And we’re asked to find this when 𝑡 equals 12. This means the derivative of the population function 𝑓 of 𝑡 with respect to 𝑡 when 𝑡 is 12. We want to find 𝑓 prime of 12, which by definition is the limit of 𝑓 of 12 plus ℎ minus 𝑓 of 12 all over ℎ as ℎ approaches zero.
Now, we use our expression for 𝑓 of 𝑡. 𝑓 of 𝑡 is 11𝑡 squared plus 35923. So 𝑓 of 12 plus ℎ is 11 times 12 plus ℎ squared plus 35923. And from that, we substitute 𝑓 of 12, which by substituting 12 for 𝑡, we see is 11 times 12 squared plus 35923.
The last thing to do here is to divide by ℎ. Now, we add 35923 and almost immediately subtract it again. So these cancel. We can further simplify by expanding the first term in the numerator. And now, we can notice two terms, which cancel. And both remaining terms in the numerator have a factor of ℎ, which cancels with the ℎ in the denominator to leave the limit of 264. That’s 11 times 24 plus 11ℎ as ℎ approaches zero. And as ℎ approaches zero, 264 plus 11ℎ approaches 264. So that’s our answer.
The rate of change in the population when 𝑡 equals 12 is 264.