WEBVTT
1
00:00:01.100 --> 00:00:04.320 A:middle L:90%
we want to sketch the curve of Why is it
2
00:00:04.330 --> 00:00:08.109 A:middle L:90%
to execute over X minus one? Now, in
3
00:00:08.109 --> 00:00:10.300 A:middle L:90%
this tractor, they give us this laundry list of
4
00:00:10.310 --> 00:00:13.839 A:middle L:90%
steps which Paula, Let's just go ahead and follow
5
00:00:13.839 --> 00:00:15.449 A:middle L:90%
it. So the first thing they tell us we
6
00:00:15.449 --> 00:00:18.289 A:middle L:90%
should do is to determine what our domain is going
7
00:00:18.289 --> 00:00:20.719 A:middle L:90%
to be. And we know for rational functions,
8
00:00:21.149 --> 00:00:23.230 A:middle L:90%
we just need to check to see where our denominator
9
00:00:23.239 --> 00:00:25.820 A:middle L:90%
is equal to zero. So we don't want our
10
00:00:25.820 --> 00:00:27.780 A:middle L:90%
done. I'm there to be zero in this case
11
00:00:27.780 --> 00:00:29.739 A:middle L:90%
, and that would be one exit one. It's
12
00:00:29.739 --> 00:00:32.850 A:middle L:90%
our domain is gonna be negative. Entity to one
13
00:00:33.009 --> 00:00:37.369 A:middle L:90%
union, one the impending. The next thing they
14
00:00:37.369 --> 00:00:38.979 A:middle L:90%
tell us to find it is to look for our
15
00:00:38.990 --> 00:00:47.490 A:middle L:90%
interests. So intercepts. So let's look for X
16
00:00:47.490 --> 00:00:49.740 A:middle L:90%
intercept first. That's when we set. Why equals
17
00:00:49.740 --> 00:00:53.000 A:middle L:90%
zero you get there was he would execute over X
18
00:00:53.000 --> 00:00:55.020 A:middle L:90%
minus one. So nothing the denominator. You never
19
00:00:55.020 --> 00:00:56.689 A:middle L:90%
make this good to zero. So I just had
20
00:00:56.700 --> 00:00:59.869 A:middle L:90%
the numerical zero, which tells us zero is equal
21
00:00:59.869 --> 00:01:03.469 A:middle L:90%
X. And since zero is equal to X,
22
00:01:03.590 --> 00:01:07.760 A:middle L:90%
that also tells us what are why intercept is going
23
00:01:07.760 --> 00:01:10.510 A:middle L:90%
to be. We'll also just B Y 00 So
24
00:01:10.510 --> 00:01:11.599 A:middle L:90%
the only intercept we have 00 so we don't need
25
00:01:11.599 --> 00:01:15.890 A:middle L:90%
assault for that. The next thing they tell us
26
00:01:15.890 --> 00:01:22.359 A:middle L:90%
to look for is symmetry. So rational functions aren't
27
00:01:22.359 --> 00:01:23.409 A:middle L:90%
really known for being periodic. But we can check
28
00:01:23.420 --> 00:01:26.450 A:middle L:90%
to see if this is even or odd. So
29
00:01:27.010 --> 00:01:32.109 A:middle L:90%
would get negative X Q times negative X minus one
30
00:01:33.719 --> 00:01:37.890 A:middle L:90%
. Now, negative excuse would just be negative times
31
00:01:37.900 --> 00:01:41.069 A:middle L:90%
X cube, so we can factor that out and
32
00:01:41.069 --> 00:01:42.230 A:middle L:90%
then distribute it to the denominator. And we get
33
00:01:42.310 --> 00:01:47.719 A:middle L:90%
X cubed over X plus one. Now, this
34
00:01:47.719 --> 00:01:49.730 A:middle L:90%
here does not equal to FX, nor is it
35
00:01:49.730 --> 00:01:53.010 A:middle L:90%
negative FX. So there is no symmetry, at
36
00:01:53.010 --> 00:01:57.700 A:middle L:90%
least no symmetry about the why access wore about the
37
00:01:57.900 --> 00:02:01.099 A:middle L:90%
origin. The next thing they tell us to look
38
00:02:01.109 --> 00:02:08.650 A:middle L:90%
for is Assam tips. So let's go ahead and
39
00:02:09.300 --> 00:02:12.139 A:middle L:90%
see. So we're no, we're going to have
40
00:02:12.719 --> 00:02:15.800 A:middle L:90%
a vertical ascent over at Exit one. So let's
41
00:02:15.810 --> 00:02:20.490 A:middle L:90%
go ahead on and see the behavior as we approach
42
00:02:20.490 --> 00:02:25.460 A:middle L:90%
one from the right of at the books. So
43
00:02:25.469 --> 00:02:30.650 A:middle L:90%
this is gonna be one from the right, huge
44
00:02:30.879 --> 00:02:35.659 A:middle L:90%
over one from the right, minus one now,
45
00:02:35.889 --> 00:02:37.500 A:middle L:90%
one from the right is going to be a positive
46
00:02:37.500 --> 00:02:38.669 A:middle L:90%
number. So if I cube still going to be
47
00:02:38.680 --> 00:02:43.840 A:middle L:90%
possible and if I take one from the right Well
48
00:02:43.840 --> 00:02:46.189 A:middle L:90%
, that's something slightly larger than one, and I
49
00:02:46.189 --> 00:02:47.530 A:middle L:90%
subtract it from one that's still gonna be positive.
50
00:02:47.849 --> 00:02:51.879 A:middle L:90%
So from the right of one, which of the
51
00:02:51.879 --> 00:02:59.020 A:middle L:90%
approaching positive infinity And since explain, this one has
52
00:02:59.360 --> 00:03:00.539 A:middle L:90%
a odd degree or a degree of one. We
53
00:03:00.539 --> 00:03:05.270 A:middle L:90%
know that this vertical Assam tote should have opposite in
54
00:03:05.270 --> 00:03:07.580 A:middle L:90%
behavior. So from the left of one that should
55
00:03:07.580 --> 00:03:12.509 A:middle L:90%
go too negative. Infinity. All right, now
56
00:03:12.509 --> 00:03:15.530 A:middle L:90%
something else we can go ahead and check. Is
57
00:03:15.560 --> 00:03:23.860 A:middle L:90%
our in behavior for this as well? So let's
58
00:03:23.860 --> 00:03:27.310 A:middle L:90%
see. So the limit it puts us in green
59
00:03:29.240 --> 00:03:32.560 A:middle L:90%
, so the limit has X approaches. Infinity of
60
00:03:32.569 --> 00:03:36.439 A:middle L:90%
F of X is going to be well X cubed
61
00:03:36.439 --> 00:03:39.400 A:middle L:90%
approaches positive energy and X minus one Purchase positivity.
62
00:03:39.599 --> 00:03:45.490 A:middle L:90%
So this is going to go to possibility and the
63
00:03:45.490 --> 00:03:47.729 A:middle L:90%
limit as ex purchase negative fnd f of X is
64
00:03:47.729 --> 00:03:51.840 A:middle L:90%
going to go too well. Negative. Vex goes
65
00:03:51.840 --> 00:03:53.159 A:middle L:90%
to negative infinity and negative x my swamp. It's
66
00:03:53.159 --> 00:03:57.409 A:middle L:90%
negative infinity. So overall would go to positive.
67
00:04:00.240 --> 00:04:03.159 A:middle L:90%
And we know that our horizontal asado is going to
68
00:04:03.169 --> 00:04:08.349 A:middle L:90%
go deposit or negativity just due to the fact of
69
00:04:08.939 --> 00:04:11.379 A:middle L:90%
O. R. Degree is larger in the new
70
00:04:11.379 --> 00:04:15.889 A:middle L:90%
mayor than in the denominator. Now, the next
71
00:04:15.889 --> 00:04:20.060 A:middle L:90%
thing they want us to find is our intervals were
72
00:04:20.060 --> 00:04:27.810 A:middle L:90%
the function is increasing and decreasing as well as any
73
00:04:27.819 --> 00:04:35.750 A:middle L:90%
local. Max is four minutes. So we're gonna
74
00:04:35.750 --> 00:04:38.850 A:middle L:90%
need throughout. What? Why Prime is equal to
75
00:04:38.860 --> 00:04:43.290 A:middle L:90%
so see that another page? Why is he going
76
00:04:43.290 --> 00:04:46.670 A:middle L:90%
to execute over X minus one? So take the
77
00:04:46.670 --> 00:04:47.529 A:middle L:90%
story that we're gonna need to apply. Questionable,
78
00:04:47.829 --> 00:04:59.860 A:middle L:90%
Questionable says hello. Hi. Minus high d low
79
00:05:04.240 --> 00:05:11.360 A:middle L:90%
all over the square of what is below. So
80
00:05:11.360 --> 00:05:13.899 A:middle L:90%
we know the derivative of X Cube is going to
81
00:05:13.910 --> 00:05:15.759 A:middle L:90%
be three x squared using power rule and the drill
82
00:05:15.759 --> 00:05:18.290 A:middle L:90%
Bit of experience. One role during the Texas one
83
00:05:18.290 --> 00:05:24.470 A:middle L:90%
derivative negative 10 But that's just one day. Now
84
00:05:24.470 --> 00:05:27.139 A:middle L:90%
if you go ahead and do this, algebra here
85
00:05:27.149 --> 00:05:33.879 A:middle L:90%
should be left with X squared two X minus three
86
00:05:34.689 --> 00:05:49.540 A:middle L:90%
Products Street all over X minus one squared. So
87
00:05:49.540 --> 00:05:53.160 A:middle L:90%
let's go ahead. And I should probably do that
88
00:05:53.160 --> 00:05:55.920 A:middle L:90%
. Let's find our possible critical value for our critical
89
00:05:55.920 --> 00:05:58.920 A:middle L:90%
values so we can find her possible Max's Airmen's.
90
00:06:00.180 --> 00:06:02.279 A:middle L:90%
So this is gonna tell us either X is equal
91
00:06:02.279 --> 00:06:05.949 A:middle L:90%
to zero or two. X minus three is zero
92
00:06:06.069 --> 00:06:09.829 A:middle L:90%
. So that gives us specs. Easy to perhaps
93
00:06:09.939 --> 00:06:12.810 A:middle L:90%
so possible. Backs is a men's or a zero
94
00:06:12.810 --> 00:06:18.149 A:middle L:90%
ed three house, but so we have X squared
95
00:06:19.509 --> 00:06:25.949 A:middle L:90%
U X minus three all over X minus one squared
96
00:06:27.810 --> 00:06:30.660 A:middle L:90%
. Now this function will be increasing. Where,
97
00:06:30.660 --> 00:06:33.870 A:middle L:90%
Why? Print is strictly larger than zero and already
98
00:06:33.879 --> 00:06:38.029 A:middle L:90%
went ahead and solved this beforehand. And this is
99
00:06:38.040 --> 00:06:43.100 A:middle L:90%
negative. Infinity to zero union, 0 to 1
100
00:06:43.209 --> 00:06:48.980 A:middle L:90%
union, three halves to a bed and this function
101
00:06:48.980 --> 00:06:51.959 A:middle L:90%
will be decreasing. Or were why promise? Strictly
102
00:06:51.959 --> 00:06:58.009 A:middle L:90%
less than zero on the last piece of this,
103
00:06:58.009 --> 00:07:02.089 A:middle L:90%
which is 123 house. Now let's go ahead and
104
00:07:02.160 --> 00:07:05.810 A:middle L:90%
put those values we come before, So he had
105
00:07:05.970 --> 00:07:12.600 A:middle L:90%
X x zero X is equal to three hubs.
106
00:07:14.430 --> 00:07:19.649 A:middle L:90%
Well, we know to the left of zero,
107
00:07:20.939 --> 00:07:30.750 A:middle L:90%
the function is increasing and to the right of zero
108
00:07:31.069 --> 00:07:38.339 A:middle L:90%
, it will be increasing until one and then from
109
00:07:38.339 --> 00:07:43.180 A:middle L:90%
1 to 3/2 the function is going to be decreasing
110
00:07:45.839 --> 00:07:50.459 A:middle L:90%
and then to the right of three halves, the
111
00:07:50.459 --> 00:07:57.970 A:middle L:90%
function is going to be increasing. So this tells
112
00:07:57.970 --> 00:08:00.490 A:middle L:90%
us that at X equal dessert we will have a
113
00:08:00.490 --> 00:08:05.649 A:middle L:90%
salad point and excessive with three House will be a
114
00:08:07.240 --> 00:08:16.209 A:middle L:90%
mogul minimum. Now the last thing they suggest we
115
00:08:16.209 --> 00:08:22.439 A:middle L:90%
find is where are function has any inflection points.
116
00:08:22.709 --> 00:08:28.769 A:middle L:90%
So we need to find Khan Cappie con on inflection
117
00:08:28.769 --> 00:08:31.580 A:middle L:90%
points. So we need to know if I double
118
00:08:31.580 --> 00:08:35.230 A:middle L:90%
primates, let's go ahead and find that alibi.
119
00:08:37.940 --> 00:08:39.250 A:middle L:90%
So why Double Prime is going to equal to,
120
00:08:41.519 --> 00:08:46.480 A:middle L:90%
so we're going to need to use quotient, rule
121
00:08:46.649 --> 00:08:52.179 A:middle L:90%
and product cool for this one. So first,
122
00:08:52.179 --> 00:08:54.259 A:middle L:90%
let's go ahead and apply the product. Cool.
123
00:08:56.730 --> 00:09:00.769 A:middle L:90%
I mean, the questionable and I might be a
124
00:09:00.769 --> 00:09:07.659 A:middle L:90%
little bit more space on this, so it's still
125
00:09:07.659 --> 00:09:16.049 A:middle L:90%
going to be low. Hi X squared times two
126
00:09:16.049 --> 00:09:20.279 A:middle L:90%
X minus three. I actually don't even need to
127
00:09:20.279 --> 00:09:22.610 A:middle L:90%
do product because we could distribute that. It's let's
128
00:09:22.610 --> 00:09:31.509 A:middle L:90%
do that first, Actually, sufficiency two two x
129
00:09:31.509 --> 00:09:39.549 A:middle L:90%
cubed minus three X quick and then minus then in
130
00:09:39.549 --> 00:09:46.039 A:middle L:90%
the opposite order to execute minus squared times the derivative
131
00:09:48.340 --> 00:09:50.860 A:middle L:90%
. Oh, but we have ended in, um
132
00:09:50.860 --> 00:09:56.279 A:middle L:90%
, there's two nice ones and then all over what
133
00:09:56.279 --> 00:10:00.539 A:middle L:90%
we have in our denominator squared X minus one,
134
00:10:00.549 --> 00:10:01.110 A:middle L:90%
and it was squared before. So now it should
135
00:10:01.120 --> 00:10:07.539 A:middle L:90%
be to be power. Now to take the derivative
136
00:10:07.139 --> 00:10:09.090 A:middle L:90%
, uh, here, we're gonna need to use
137
00:10:09.090 --> 00:10:11.370 A:middle L:90%
power will preach. So it's gonna be six x
138
00:10:11.370 --> 00:10:16.450 A:middle L:90%
squared, minus six x and the derivative of exploits
139
00:10:16.450 --> 00:10:20.059 A:middle L:90%
. One swear they need to power and changeable.
140
00:10:20.149 --> 00:10:24.340 A:middle L:90%
So it's gonna be two times X minus one times
141
00:10:24.350 --> 00:10:26.570 A:middle L:90%
the derivative of X minus one. Did it change
142
00:10:26.570 --> 00:10:30.639 A:middle L:90%
Will, Which would just be one? And if
143
00:10:30.639 --> 00:10:33.019 A:middle L:90%
we go through and simplify all this algebra wound up
144
00:10:33.019 --> 00:10:39.049 A:middle L:90%
with two ex over X squared minus three x mostly
145
00:10:41.940 --> 00:10:48.159 A:middle L:90%
all over X minus one. Cute. Now we
146
00:10:48.159 --> 00:10:50.059 A:middle L:90%
want to set the secret zero so we can find
147
00:10:50.059 --> 00:10:52.450 A:middle L:90%
our possible points of inflection that we're going to get
148
00:10:52.870 --> 00:10:58.320 A:middle L:90%
ex busy with zero or X squared minus three X
149
00:10:58.389 --> 00:11:01.710 A:middle L:90%
plus three zero. So it turns out that this
150
00:11:01.710 --> 00:11:09.250 A:middle L:90%
year has no riel solutions. So the only possible
151
00:11:09.250 --> 00:11:11.159 A:middle L:90%
point of reflection will have is that X is equal
152
00:11:11.159 --> 00:11:15.529 A:middle L:90%
to zero. So let's go ahead and write down
153
00:11:15.529 --> 00:11:18.309 A:middle L:90%
our second derivative here, which is going to be
154
00:11:20.799 --> 00:11:31.149 A:middle L:90%
two x x squared minus reacts plus three all over
155
00:11:31.840 --> 00:11:35.580 A:middle L:90%
X minus one. Cute and again, I just
156
00:11:35.580 --> 00:11:39.440 A:middle L:90%
went ahead and it's all for it's gonna become came
157
00:11:39.440 --> 00:11:41.379 A:middle L:90%
up. Calm down, look for hand. So
158
00:11:41.940 --> 00:11:43.070 A:middle L:90%
conch a boat is going to be where this is
159
00:11:43.539 --> 00:11:48.970 A:middle L:90%
strictly larger than there And this happens to be from
160
00:11:50.000 --> 00:12:00.769 A:middle L:90%
negative and infinity 20 Union one to infinity and the
161
00:12:00.779 --> 00:12:05.509 A:middle L:90%
function is conch aid down when? Why Double prime
162
00:12:05.519 --> 00:12:13.700 A:middle L:90%
strictly less than zero all over zero now are possible
163
00:12:13.700 --> 00:12:18.000 A:middle L:90%
point of inflection Was that exit so to the left
164
00:12:18.009 --> 00:12:20.919 A:middle L:90%
of zero the functions calm keep up and to the
165
00:12:20.929 --> 00:12:24.210 A:middle L:90%
right of zero functions concrete down. So this year
166
00:12:24.210 --> 00:12:30.809 A:middle L:90%
will be a inflection. And once we did this
167
00:12:31.230 --> 00:12:33.639 A:middle L:90%
, it said we can go ahead and actually started
168
00:12:33.639 --> 00:12:37.549 A:middle L:90%
graphing. So let's put our intercept first. We
169
00:12:37.549 --> 00:12:39.700 A:middle L:90%
have an intercept at only the organ. We have
170
00:12:39.700 --> 00:12:43.399 A:middle L:90%
no symmetry. We know are absent toes. So
171
00:12:43.399 --> 00:12:46.779 A:middle L:90%
the in behavior is going to be to infinity on
172
00:12:46.779 --> 00:12:50.259 A:middle L:90%
each side. And we have vertical awesome totes at
173
00:12:50.269 --> 00:12:56.399 A:middle L:90%
Exit one, so X is equal to one.
174
00:13:00.340 --> 00:13:05.679 A:middle L:90%
So to be right of this, we should go
175
00:13:05.679 --> 00:13:07.860 A:middle L:90%
into infinity and to the left. We should be
176
00:13:07.860 --> 00:13:16.240 A:middle L:90%
going too negative Infinity. So at X is equal
177
00:13:16.240 --> 00:13:18.970 A:middle L:90%
to three house we know we're gonna have a local
178
00:13:18.970 --> 00:13:22.240 A:middle L:90%
men. Let's go ahead. So say this is
179
00:13:22.269 --> 00:13:26.590 A:middle L:90%
what happens here. And since we're coming from positive
180
00:13:26.590 --> 00:13:31.830 A:middle L:90%
infinity and we have no other Exeter sets or anything
181
00:13:31.830 --> 00:13:33.029 A:middle L:90%
like that, we know that our minimum is going
182
00:13:33.039 --> 00:13:37.159 A:middle L:90%
to be like that. And we know we have
183
00:13:37.159 --> 00:13:41.350 A:middle L:90%
a point of contact bitty at X. So we
184
00:13:41.500 --> 00:13:43.629 A:middle L:90%
wanted all of our important pieces. Now we could
185
00:13:43.629 --> 00:13:45.659 A:middle L:90%
just go ahead and start connecting lines. So let's
186
00:13:45.659 --> 00:13:48.360 A:middle L:90%
go ahead and start to the right of our words
187
00:13:48.360 --> 00:13:52.759 A:middle L:90%
on plastic, our bird a classic. So we're
188
00:13:52.759 --> 00:13:54.519 A:middle L:90%
starting from positive, Benny, and we're gonna go
189
00:13:54.519 --> 00:14:00.830 A:middle L:90%
until we hit our local men. And then it's
190
00:14:00.830 --> 00:14:03.850 A:middle L:90%
going to just go up and come next like that
191
00:14:05.620 --> 00:14:09.340 A:middle L:90%
and then on the other side. Well, we
192
00:14:09.340 --> 00:14:15.419 A:middle L:90%
know that at exit with zero is intercept and we
193
00:14:15.419 --> 00:14:18.450 A:middle L:90%
also share the changing cavity. So it should look
194
00:14:18.019 --> 00:14:22.250 A:middle L:90%
something like this here, and I'm just gonna reset
195
00:14:22.250 --> 00:14:26.309 A:middle L:90%
and hate over they're connected. But this here should
196
00:14:26.320 --> 00:14:33.029 A:middle L:90%
be a nice little sketch of our graph. You
197
00:14:33.029 --> 00:14:35.529 A:middle L:90%
could possibly go back in and actually say what this
198
00:14:35.529 --> 00:14:39.090 A:middle L:90%
minimum is here. But since we're just trying to
199
00:14:39.100 --> 00:14:41.549 A:middle L:90%
sketch graph, I think this is sufficient