Video Transcript
If the average rate of change of
the function 𝑓 is 6.67 when 𝑥 varies from two to 2.3, find the change in 𝑓.
Remember that the average rate of
change of the function 𝑓 from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏 is 𝑓 of 𝑏
minus 𝑓 of 𝑎 over 𝑏 minus 𝑎, where 𝑓 of 𝑏 minus 𝑓 of 𝑎 is the change in 𝑓
and 𝑏 minus 𝑎 is the change in 𝑥. This expression is actually another
way of writing the slope function. That is, 𝑦 two minus 𝑦 one over
𝑥 two minus 𝑥 one. 𝑦 two minus 𝑦 one is the change
in 𝑦 and 𝑥 two minus 𝑥 one is the change in 𝑥.
Now we’ve actually been given the
average rate of change of our function 𝑓 which is 6.67 when 𝑥 varies from two to
2.3. So we’ve been given the average
rate of change and the change in 𝑥, and we’re asked to find the change in 𝑓. So let’s begin by substituting what
we know into our average rate of change equation. We know that the average rate of
change of 𝑓 is 6.67. And we know that 𝑥 changes from
two to 2.3 so that in our average rate of change equation, 𝑎 is equal to two and 𝑏
is equal to 2.3.
So if we substitute these values
into our equation, we have 𝑓 of 𝑏, which is 𝑓 of 2.3, minus 𝑓 of 𝑎, which is 𝑓
of two, over 𝑏 minus 𝑎. And that’s equal to our average
rate of change 6.67. Remember, we’re being asked to find
the change in 𝑓. And that’s the numerator. And to find this, all we need to do
is to evaluate the denominator and to cross multiply. In the denominator, 2.3 minus two
is equal to 0.3. And if we cross multiply this with
6.67, we have 0.3 times 6.67 is 𝑓 of 2.3 minus 𝑓 of two, which is the change in
𝑓.
Evaluated to three decimal places,
0.3 times 6.67 is 2.001. So that the change in 𝑓 as 𝑥
varies from two to 2.3 if the average rate of change is 6.67 to one significant
figure is two.
