If the function 𝑓 of 𝑥 equals
five 𝑥 plus seven over four 𝑥 plus two, determine its rate of change when 𝑥 is
equal to two.
We recall the definition for the
rate of change of the function or its derivative. It’s the limit as ℎ approaches zero
of 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 over ℎ assuming that limit exists. In this question, 𝑓 of 𝑥 is equal
to five 𝑥 plus seven over four 𝑥 plus two. And we want to find the rate of
change when 𝑥 is equal to two. So we’re going to let 𝑎 be equal
to two. So we need to evaluate 𝑓 prime of
two, the rate of change or the derivative of our function, when 𝑥 equals two.
By our definition, it’s the limit
as ℎ approaches zero of 𝑓 of two plus ℎ minus 𝑓 of two over ℎ. Let’s work out 𝑓 of two plus ℎ and
𝑓 of two. To find 𝑓 of two plus ℎ, we
replace each instance of 𝑥 in our original function with two plus ℎ. And we get five times two plus ℎ
plus seven over four times two plus ℎ plus two. And when we distribute our
parentheses and simplify, we get 17 plus five ℎ over 10 plus four ℎ. Similarly, 𝑓 of two is five times
two plus seven over four times two plus two which is 17 tenths. And we now see that 𝑓 prime of two
is the limit as ℎ approaches zero of the difference between these all over ℎ.
There are two fractions in our
numerator. So we’re going to simplify by
creating a common denominator there. We’ll multiply the numerator and
denominator of the first fraction on the numerator by 10 and the second fraction on
our numerator by 10 plus four ℎ. And when we do, we achieve the
numerator shown. Well, this doesn’t make a lot of
sense. But actually, dividing this entire
fraction by ℎ is the same as timesing it by one over ℎ. So we rewrite our denominator as a
ℎ times 100 plus 40ℎ. And then, we simplify a numerator
to negative 18ℎ. And you might now see that we can
simplify further by dividing through by ℎ.
And we’re now ready to perform
direct substitution. By replacing ℎ with zero, we find
that 𝑓 prime of two is negative 18 over 100, which simplifies to negative nine over
50. The rate of change of our function
𝑓 of 𝑥 when 𝑥 is equal to two is negative nine over 50.