Video Transcript
Suppose a population is 𝑓 of 𝑡
equals 14𝑡 squared plus 33,706 as a function of time 𝑡. What is the average rate of growth
of this population when 𝑡 changes from 𝑡 sub one to 𝑡 sub one plus ℎ?
Were being asked to find the
average rate of growth of the population, in other words the average rate of change,
when 𝑡 changes from 𝑡 sub one to 𝑡 sub one plus ℎ. And so we recall the average rate
of change formula. For a continuous function 𝑓, the
average rate of change over the closed interval 𝑎 to 𝑏 is 𝑓 of 𝑏 minus 𝑓 of 𝑎
over 𝑏 minus 𝑎. Now, our function 𝑓 of 𝑡 is
defined by 14𝑡 squared plus 33,706. And so 𝑓 of 𝑡 is in fact a
polynomial function. Now, thats really useful because we
know that polynomials are continuous over their entire domain. And so were able to use the average
rate of change formula.
Were interested in the rate of
change of the population when 𝑡 changes from 𝑡 sub one to 𝑡 sub one plus ℎ. So those are our values for 𝑎 and
𝑏. So the average rate of change
formula becomes 𝑓 of 𝑡 sub one plus ℎ minus 𝑓 of 𝑡 sub one all over 𝑡 sub one
plus ℎ minus 𝑡 sub one. Now, in fact, lets simplify the
denominator of our fraction by subtracting 𝑡 sub one from 𝑡 sub one. And we’re simply left with ℎ. But what is 𝑓 of 𝑡 sub one plus
ℎ? Well, we need to replace 𝑡 in our
original function with 𝑡 sub one plus ℎ. So we get 14𝑡 sub one plus ℎ
squared plus 33,706.
Now, since it makes a lot of sense
to write 𝑡 sub one plus ℎ squared as 𝑡 sub one plus ℎ times 𝑡 sub one plus ℎ,
lets do that when we replace 𝑓 of 𝑡 sub one plus ℎ in our average rate of change
formula. We also know 𝑓 of 𝑡 sub one will
be 14𝑡 sub one squared plus 33,706. And so before we distribute our
parentheses, we notice something. When we subtract positive 33,706
from the earlier 33,706, we actually get zero. And so lets go ahead and distribute
our parentheses. 𝑡 sub one times 𝑡 sub one is 𝑡
sub one squared. Then, 𝑡 sub one times ℎ and 𝑡 sub
one times ℎ gives us two 𝑡 sub one ℎ. And ℎ times ℎ is ℎ squared. So this all simplifies a little bit
to get 14 times 𝑡 sub one squared plus two 𝑡 sub one ℎ plus ℎ squared minus 14𝑡
sub one squared all over ℎ.
And then we distribute the
parentheses even further. We multiply each term inside by the
14 on the outside. And we get 14𝑡 sub one squared
plus 28𝑡 sub one ℎ plus 14ℎ squared. And next we notice that 14𝑡 sub
one squared minus 14𝑡 sub one squared is zero. And then we divide each remaining
term by the ℎ on the denominator of our fraction. 28𝑡 sub one ℎ divided by ℎ is 28𝑡
sub one, and 14ℎ squared divided by ℎ is simply 14ℎ. And so that leaves us with 28𝑡 sub
one plus 14ℎ, which we might choose to write alphabetically as 14ℎ plus 28𝑡 sub
one. So the average rate of growth of
the population as 𝑡 changes from 𝑡 sub one to 𝑡 sub one plus ℎ is 14ℎ plus 28𝑡
sub one.
