00:01
Okay, we're given the function, f of x is equal to x squared over x squared minus 16, and we were asked to find all of this.
00:11
And all of this information, we're basically collecting information to help us sketch the graph of this function.
00:19
So let's get started.
00:20
We'll find the asymptopes first.
00:25
And so i'm going to focus on the function itself, not do any calculus bits just yet.
00:30
And i'll also number my work, so it's a little easier to follow.
00:34
So the only work i would do here is i would maybe factor the denominator.
00:43
So x squared minus 16, factored out as x plus 4, x minus 4.
00:50
And basically what we're doing is it's helping us to really see where is where is the function undefined.
01:05
And what that's going to tell us is where the vertical asymptotes are.
01:11
So where is it undefined where it's where the denominator is equal to zero.
01:17
And so for that, we have it where x is equal to 4 or negative 4.
01:28
So there's actually two vertical lines.
01:31
And i'll write them out separately.
01:34
Vertical lines, just as a reminder, is usually written as an equation.
01:38
It looks like x equaling a constant number.
01:45
Okay, now with the horizontal asymptope for these rational functions, if the powers, so if the power of the leading term is exactly the same, which it is in this case, we are looking at the ratio of its coefficients.
02:04
You don't see any coefficients here, but we have to think one.
02:08
And so the ratio of 1 over 1 is just 1.
02:13
And we write horizontal lines as y equals a constant number.
02:18
So we'll write it as y equals 1.
02:24
So that's good for now.
02:26
And then we'll move on to the next part.
02:29
Intervals on which f is increasing versus decreasing.
02:33
Also word extreme are max and min.
02:35
We can actually maybe do these kind of in conjunction with each other.
02:40
They're very much related.
02:41
So for this, we will need the first derivative of f.
02:47
So we need f prime.
02:50
Now, f is a rational function, so it would be good if you knew your quotient rule.
03:00
Okay, quotient rule, i have the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator.
03:22
The denominator squared.
03:26
So if you don't have that memorized just yet, the quotient rule would be good to know.
03:34
Let's clean this up a little bit.
03:37
2x cubed minus 32x minus 2x cubed.
03:43
Already i see some cancellation that we can do.
03:49
These go down to 0.
03:50
So we have negative 32x all over x squared minus 16.
03:57
And then all of that squared.
03:58
I would leave it as is.
04:00
Don't multiply it up.
04:06
So what we have here is our first derivative, and we can use that to help us find our critical points.
04:13
So where are the critical points? you want to set your f prime equal to zero.
04:20
So let's do that.
04:24
It's also where your f prime is undefined.
04:27
But if you take a look at the denominator here, it's pretty much the same thing as our original denominator, but squared.
04:36
So it's going to be undefined at those same places, which we have vertical asymptopes for.
04:42
So we don't really need to worry about them because, oh, we'll see it when we sketch it out.
04:55
Let's just finish this first.
04:57
Okay, so solve this equation, we can multiply both sides by this denominator or just realize that if we have fraction equaling zero, if the numerator is equal to zero, then the whole thing is zero.
05:09
So we get that, divide both sides by negative 32.
05:15
X is equal to zero.
05:17
That's our critical number.
05:21
And this is one nice little trick i like to do to kind of visually put in all my information.
05:28
I put in my critical numbers onto a number line.
05:31
And then i look at test values to the left and right of a critical number.
05:39
And then i look at pretty much this form right over here.
05:45
So the test values on the left hand side of your zero here on our number line would be negative.
05:53
And on the right hand side, it would be positive.
05:57
So if i look at, you know, i look at this, i'm going to have a negative 32, essentially timesing a negative number all over something squared.
06:13
That won't be negative.
06:14
It'll be positive.
06:15
So i'll have that.
06:18
And then negative times negative is positive over a positive.
06:22
Overall, we get a positive.
06:27
So that's going to be our f prime in, for any number, less than zero if you stick it in.
06:35
You can double check this, of course, if you're not really sure.
06:39
If i stick in a positive number instead, i'm going to just look at this one more time.
06:44
Negative 32 is always negative times a positive times something squared is always positive.
06:54
Negative times a positive is negative divided by a positive.
06:58
It's still going to be negative.
07:00
So our f prime on this side is going to be negative.
07:04
So what does that tell us? if our first derivative is positive, it means it's increasing.
07:12
So it looks like this.
07:14
And if it is negative, then it's decreasing.
07:21
So it looks like this.
07:23
So if your curve looks like it's going up and then it's going down, that means we have a max in the middle where it changes.
07:34
So that's where the critical number is.
07:37
And we just went ahead and kind of answered a lot of this already.
07:41
I'll write this down in interval notation.
07:44
So we're first increasing from negative infinity all the way to zero, and then we're decreasing on the other side from zero to positive infinity.
07:58
There is a max at x equals zero, and there's no min.
08:11
Oh, and i know this is starting to get a little confusing because we have so many x equals x equals.
08:17
So think of this one as this isn't a value, and these are lines.
08:36
Last but not least, we will find where f is concave up and down and also the inflection points very much related to each other, so we'll do them kind of in conjunction with each other.
08:51
For this, we will need our second derivative.
08:55
So i'm going to need my first derivative to look at.
08:59
We have a rational function, and then we'll go ahead and derive that.
09:06
So let's see.
09:08
We have our denominator times the derivative of our numerator minus our numerator times the derivative of our denominator.
09:28
Okay, so we have to be a little careful here.
09:31
Our denominator is x squared minus 16, all of that squared.
09:36
So be careful about using the chain rule.
09:40
So i'm going to write down two times x squared minus 16.
09:46
The the exponent now goes down to 1, and i need to times that by the derivative of x squared minus 16.
09:55
So i'll write that here, chain.
10:00
Be careful of chain rule.
10:03
And then denominator squared.
10:06
So if we have a square already, square of that brings our exponent to become 4.
10:13
Exponent of 4.
10:16
That's why it's good to keep in this form.
10:20
Okay, now this is going to be a giant mess.
10:23
If you are maybe not trying to cancel or factor along the way, it would be good to try to factor along the way.
10:32
So i'll show you how to do that.
10:34
If you notice, we have two parts of our numerator separated by this minus sign in between.
10:41
Okay, so those are to two parts.
10:45
The parts that are similar, we have this negative 32 for each of those terms...