Video Transcript
The slope at the point 𝑥, 𝑦 on
the graph of a function is six 𝑒 to the power of 𝑥 plus two. What is 𝑓 of 𝑥, given that 𝑓 of
the natural log of five equals one?
We’ve been given information about
the slope at a general point 𝑥, 𝑦. And, actually, if we recall, we can
find an expression for the slope of the graph of a function 𝑓 of 𝑥 by finding its
derivative. This means given a function 𝑓 of
𝑥, its derivative 𝑓 prime of 𝑥 at any point 𝑥, 𝑦 here will be defined by six 𝑒
to the power of 𝑥 plus two. Now, what do we know about the
relationship between 𝑓 of 𝑥 and 𝑓 prime of 𝑥?
Well, 𝑓 prime of 𝑥 is the
derivative of our function. But we also know that the opposite
of differentiating is integrating. And so, if we integrate 𝑓 prime of
𝑥 with respect to 𝑥, we will find a general solution for 𝑓 of 𝑥. We’ll then use the fact that 𝑓 of
the natural log of five is equal to one to find a particular solution. So, let’s integrate six 𝑒 to the
power of 𝑥 plus two.
We begin by recalling that the
integral of 𝑒 to the power of 𝑥 is 𝑒 to the power of 𝑥. Then, the integral of two is two
𝑥. Now, of course, this is an
indefinite integral. So, we need that constant of
integration. And we can say that our function is
defined as 𝑓 of 𝑥 equals six 𝑒 to the power of 𝑥 plus two 𝑥 plus some constant
𝑐. Now, we said that 𝑓 of the natural
log of five is equal to one. In other words, when 𝑥 is equal to
the natural log of five, 𝑓 of 𝑥 is equal to one. So, we replace 𝑓 of 𝑥 with
one. And then, each time we see an 𝑥 in
our equation, we replace that with the natural log of five.
So, one is equal to six 𝑒 to the
power of the natural log of five plus two times the natural log of five plus 𝑐. Of course, 𝑒 to the power of the
natural log of five is just five. So, six 𝑒 to the power of the
natural log of five is six times five, which is 30. We now see that we can solve for 𝑐
by subtracting 30 and two times the natural log of five from both sides of our
equation. And when we do, we find that 𝑐 is
equal to negative 29 minus two times the natural log of five.
We then go back to the general
solution for 𝑓 of 𝑥, and we replace 𝑐 with negative 29 minus two times the
natural log of five. And so, we found 𝑓 of 𝑥. It’s six 𝑒 to the power of 𝑥 plus
two 𝑥 minus 29 minus two times the natural log of five.
