If the line 𝑦 equals three 𝑥 plus nine is tangent to the graph of the function 𝑓 at 215, what is 𝑓 prime two?
So if we look at the question here, we look at the information we’ve got, it says that we have a line which is 𝑦 is equal to three 𝑥 plus nine and it’s tangent to a graph. Okay, what does this actually mean? Well I’ve drawn us a little sketch to help us.
This little sketch actually just shows any function and then a line that it’s tangent with at a point. So what I’ve actually done with this sketch of — I’ve just picked a random function and line — is just demonstrate a relationship we have, that at this point, which I’ve marked on, the slope of the function and the line are going to be equal.
And this is gonna be very important cause we’re gonna use this to actually solve the problem that we’re doing now. So thinking about slope, what we’re gonna first do is have a look at our line which is 𝑦 is is equal to three 𝑥 plus nine. But what we can actually see is that our line is in the form 𝑦 is equal to 𝑚𝑥 plus 𝑐 where 𝑚 is our slope and 𝑐 is our 𝑦-intercept.
So therefore, if we take a look at the equation of our line, we can see that our slope or our 𝑚 is gonna be equal to three. And this is because this is our coefficient of 𝑥. Okay, so now we’ve got the slope at this point of our tangent. It’s also worth noting at this point that we said that this is the tangent to the graph of the function at the point two, 15.
So we know that actually when we’re looking at the function itself, this is the point that we’re gonna be using. So now we’re gonna move on to the next part of the question. What is 𝑓 prime two? Well 𝑓 prime 𝑥 actually means the slope function. So again, you could seeing it as 𝑑𝑦 𝑑𝑥.
Well there’s various ways that actually can be represented. But what we’re trying to say that is what is the slope function when 𝑥 is equal to two; so i.e., what is the slope of our function at the point where 𝑥 is equal to two? So it’s at this point we can actually go back to the relationship we looked at earlier.
So what the relationship tells us is that at the point where a tangent actually touches our function, the slope of the function and the tangent of a line at that point is gonna be equal. So therefore, if you bring it into our context, we can say that at the point two, 15, the slope of 𝑦 equals three 𝑥 plus nine and 𝑓 𝑥 are going to be equal.
So therefore, we can say that 𝑓 prime two or the value of the slope function when 𝑥 is equal to two is gonna be equal to the slope of 𝑦 equals three 𝑥 plus nine, because the point when the slope function has a value of 𝑥 equal two is the same as when we have the point where our tangent and function actually meet because it’s two, 15 so the 𝑥-value is two.
So therefore, we can say that the value of 𝑓 prime two must be equal to three because three was actually the slope that we found earlier because we had 𝑚 is equal to three of our tangent three 𝑥 plus nine.