Video Transcript
Determine the average rate of
change function 𝐴 of ℎ for 𝑓 of 𝑥 equals four 𝑥 squared plus three 𝑥 plus
two at 𝑥 equals one.
Remember, the average rate of
change of a function 𝑓 of 𝑥 between two points defined by 𝑎, 𝑓 of 𝑎 and 𝑎
plus ℎ, 𝑓 of 𝑎 plus ℎ is 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 all over ℎ. We can see in this question
that 𝑓 of 𝑥 has been defined for us. It’s four 𝑥 squared plus three
𝑥 plus two. We want to find the average
rate of change function for 𝑓 of 𝑥 at 𝑥 equals one. So, we let 𝑎 be equal to
one. We don’t actually know what ℎ
is, but that’s fine. This question is essentially
asking us to derive a function that will allow us to find the average rate of
change for any value of ℎ with this function. Let’s break this down and begin
by working out what 𝑓 of 𝑎 plus ℎ is.
We said 𝑎 is equal to one, so
we’re actually looking to find 𝑓 of one plus ℎ. We go back to our function 𝑓
of 𝑥, and each time we see an 𝑥, we replace it with one plus ℎ. So, 𝑓 of one plus ℎ is four
times one plus ℎ squared plus three times one plus ℎ plus two. Let’s distribute our
parentheses. One plus ℎ all squared is one
plus two ℎ plus ℎ squared, and three times one plus ℎ is three plus three ℎ. We can then distribute these
parentheses and we get four plus eight ℎ plus four ℎ squared. Finally, we collect like terms
and we get four ℎ squared plus 11ℎ plus nine.
Next, we work out 𝑓 of 𝑎. Well, of course, we know that
that’s 𝑓 of one. This one is slightly more
straightforward than 𝑓 of one plus ℎ. We simply replace 𝑥 with
one. And we get four times one
squared plus three times one plus two. And that’s equal to nine. We’re now ready to substitute
everything into the average rate of change formula. We have 𝑓 of one plus ℎ minus
𝑓 of one and that’s all over ℎ. Well, nine take away nine is
zero, and then we can divide through by ℎ. And it simplifies really nicely
to four ℎ plus 11. And so, the average rate of
change function 𝐴 of ℎ for 𝑓 of 𝑥 equals four 𝑥 squared plus three 𝑥 plus
two at 𝑥 equals one is four ℎ plus 11.
