Video Transcript
If the function 𝑓 is such that 𝑓
of 𝑥 is equal to five 𝑥 minus three, then the variation function 𝑉 of ℎ is equal
to what at 𝑥 is equal to two.
We want to find the variation
function 𝑉 of ℎ for 𝑓 of 𝑥 is five 𝑥 minus three at 𝑥 is equal to two. To do this, we recall that the
variation function for a function 𝑓 of 𝑥 at 𝑥 is equal to 𝑎 is defined as 𝑉 of
ℎ is 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎, where ℎ is the change in 𝑥. In our case, our initial point 𝑥
is equal to two, and that’s 𝑎, so that our variation function 𝑉 of ℎ is 𝑓 of two
plus ℎ minus 𝑓 of two. Now, substituting first 𝑥 is equal
to two plus ℎ into our function 𝑓, we have 𝑓 of two plus ℎ is five times two plus
ℎ minus three, that is, 10 plus five ℎ minus three, which is seven plus five ℎ.
Next, if we substitute 𝑥 is equal
to two into our function 𝑓 of 𝑥, we have 𝑓 of two is five times two minus three,
which is 10 minus three, and that’s seven. With these values in our variation
function, this gives us 𝑉 of ℎ is seven plus five ℎ minus seven. And since seven minus seven is
zero, that’s equal to five ℎ. The variation function 𝑉 of ℎ for
the function 𝑓 of 𝑥 is five 𝑥 minus three at 𝑥 is equal to two is therefore 𝑉
of ℎ is equal to five ℎ.
