A population’s size after 𝑡 days is given by 𝑓 of 𝑡 is equal to 11𝑡 squared plus 35,923. Find the instantaneous rate of change in the population when 𝑡 equals 12.
In this question, we are asked to find the instantaneous rate of change in the population. And we are asked to calculate this when 𝑡 is equal to 12. This means the derivative of the population function 𝑓 of 𝑡 with respect to 𝑡 when 𝑡 is 12.
We know by definition that 𝑓 prime of 𝑎 is equal to the limit as ℎ approaches zero of 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 all divided by ℎ. In this question, we need to calculate 𝑓 prime of 12 as shown. We will begin by using our expression for 𝑓 of 𝑡 to find an expression for 𝑓 of 12 plus ℎ. Replacing 𝑡 with 12 plus ℎ, we have 11 multiplied by 12 plus ℎ squared plus 35,923. Expanding 12 plus ℎ squared gives us 144 plus 24ℎ plus ℎ squared. And multiplying through by 11, we have 𝑓 of 12 plus ℎ is equal to 1,584 plus 264ℎ plus 11ℎ squared plus 35,923.
Next, we can find an expression for 𝑓 of 12. This is equal to 11 multiplied by 12 squared plus 35,923. And 11 multiplied by 12 squared is equal to 1,584. We can now substitute these into our expression for 𝑓 prime of 12. The two constants on the numerator will cancel. And we are left with the limit as ℎ approaches zero of 264ℎ plus 11ℎ squared all divided by ℎ. Since ℎ is approaching zero and will therefore never be equal to zero, we can divide the numerator and denominator by ℎ. This gives us the limit as ℎ approaches zero of 264 plus 11ℎ.
As we now have a polynomial in terms of ℎ, we can use direct substitution such that 𝑓 prime of 12 is equal to 264. And we can therefore conclude that the instantaneous rate of change in the population when 𝑡 is equal to 12 is 264.