The average rate of change of
𝑓 as 𝑥 varies from two to 2.6 is negative 1.67. If 𝑓 of two is equal to
negative 13, what is 𝑓 of 2.6?
Remember, the average rate of
change of a function 𝑓 as 𝑥 varies from 𝑎 to 𝑎 plus ℎ is given by 𝑓 of 𝑎
plus ℎ minus 𝑓 of 𝑎 over ℎ. Now, in this question, we don’t
actually know what 𝑓 of 𝑥 is. But we do see that 𝑥 varies
from two to 2.6. So, we let 𝑎 be equal to
two. And then, ℎ is the amount 𝑥
varies by. It’s 2.6 minus two, which is
0.6. We want to find the average
rate of change function, so that’s 𝐴 of ℎ, which is 𝐴 of 0.6. And so, according to our
formula, that’s 𝑓 of two plus 0.6 minus 𝑓 of two over 0.6. This simplifies to 𝑓 of 2.6
minus 𝑓 of two over 0.6.
We’re told, however, that this
is equal to negative 1.67, and also that 𝑓 of two is equal to negative 13. So, we find that negative 1.67
must be equal to 𝑓 of 2.6 minus negative 13 over 0.6. To find 𝑓 of 2.6 as this
question is asking us, we need to solve this equation for 𝑓 of 2.6. We’ll begin by multiplying
through by 0.6. And that gives us negative
1.002 on the left. And then on the right, we’re
left with 𝑓 of 2.6 minus negative 13, which is, of course, 𝑓 of 2.6 plus
13. Next, we subtract 13 from both
sides, and we find that 𝑓 of 2.6 is negative 14.002. Correct to the nearest whole
number, 𝑓 of 2.6 is negative 14.