00:01
All right.
00:01
Hello, everybody.
00:03
So in this example, we're asked to find the difference quotient of the given function 3x squared minus 3x plus 3.
00:10
Excuse me, because of misprint, this should be a 3.
00:14
So 3x squared minus 3x plus 3.
00:17
And then we're going to use that different quotient to find the different quotient difference quotients for the different values, x of an h that are given in that table over here.
00:26
Remember that the difference quotient is f of x plus h.
00:30
So that's all one thing, minus.
00:31
The function at x, all divided by h.
00:36
And so we have our f of x, but we need to find what f of x plus h is too.
00:40
And to do that, basically anywhere we saw an x in the original function, instead we're going to plug in x plus h.
00:46
Then we're going to do some simplifying to simplify it down a little bit.
00:50
So everywhere i see an x, i'm going to do x plus h.
00:53
It's going to be three times x plus h squared minus three times x plus h plus three.
01:01
And then multiplying that out, there's going to be three times x squared plus 2hx plus h squared, minus three times x plus h plus three.
01:15
And even further, i'm going to distribute the threes now.
01:18
So it's going to be 3x squared plus 6hx plus 3h squared minus 3x, minus 3x, because i'm distributing negative 3 now and then plus 3.
01:33
And unfortunately, nothing else combines, no light turns or nothing like that.
01:36
So that's going to be what f of x plus h is right there.
01:40
And let's scroll down a little bit.
01:43
So now if we want to find our different question, i'm just going to call it dq just to abbreviate.
01:48
Our difference quotient, we said was f of x plus h.
01:51
So i'm going to write down what f of x plus h is.
01:54
That really long thing right there.
01:57
3x plus 3h squared, minus 3x, minus 3x, minus 3, plus 3.
02:03
All of that minus f of x, which we said we were given as 3x squared minus 3x plus 3.
02:15
And put all of that over h.
02:19
Nice thing about this is this actually simplifies pretty well because if you notice, i'm subtracting 3x squared from a 3x squared.
02:26
So this and this is going to cancel out when i subtract them.
02:29
I'm subtracting negative 3x from negative 3x.
02:32
So that and that's going to cancel out.
02:34
And then it's subtracting 3 from a 3, so all of that will cancel out.
02:38
And so actually everything that we were subtracting ended up cancelling out pretty nicely.
02:42
And so now what we're left with is up top is 6h x plus 3h squared minus 3h.
02:53
And then all of that is divided by h.
02:57
Well, if we notice, i actually have h in every factor up to every turn up top.
03:03
So i'm going to factor that out.
03:05
So that'll be 6x when i pull out an h...