WEBVTT
Kind: captions
Language: en
00:00:14.140 --> 00:00:15.180
okay solve for x
00:00:16.980 --> 00:00:19.140
this is a quadratic in disguises
00:00:19.560 --> 00:00:21.560
so we did ones like this before
00:00:23.180 --> 00:00:25.660
so we see this and multiply by e to the x
00:00:27.300 --> 00:00:28.820
that becomes e to the 2x
00:00:29.400 --> 00:00:31.240
so we get 2x e to the x squared
00:00:32.660 --> 00:00:33.220
minus 12
00:00:34.980 --> 00:00:36.100
equals 4e to the x
00:00:41.520 --> 00:00:44.320
now subtarct 4e to the x so you get e tot he 2x
00:00:45.240 --> 00:00:46.280
mijus 4e to the x
00:00:47.280 --> 00:00:47.840
minus 12
00:00:49.080 --> 00:00:49.640
equals 0
00:00:50.160 --> 00:00:52.880
you cqn do a substitution at this point if
00:00:53.040 --> 00:00:54.400
e to the x bothers you
00:00:55.380 --> 00:00:57.140
youj have y equals e to the x
00:00:57.880 --> 00:00:59.640
which makes this y squared
00:01:00.340 --> 00:01:00.900
minus 4y
00:01:02.020 --> 00:01:02.580
minus 12
00:01:07.500 --> 00:01:09.100
everyone see that okay?
00:01:09.100 --> 00:01:11.080
okay that factors very nicely
00:01:14.860 --> 00:01:15.420
y minus 6
00:01:16.760 --> 00:01:17.260
y plus 2
00:01:23.540 --> 00:01:24.100
an thats
00:01:26.820 --> 00:01:27.620
means that y
00:01:27.620 --> 00:01:28.120
=6
00:01:29.280 --> 00:01:31.120
y=-2 but y is really e to the x
00:01:35.060 --> 00:01:36.340
so e tot he x equals 6
00:01:39.620 --> 00:01:41.620
or e to the x equals -2, e to the x
00:01:42.380 --> 00:01:45.100
always positive so you can throw that out
00:01:46.760 --> 00:01:48.280
and of e to the x equals 6
00:01:49.720 --> 00:01:51.320
x equals natural log of 6
00:01:55.900 --> 00:01:57.660
thats not very hard right?
00:02:00.740 --> 00:02:04.820
well do you think thats a minimum competence or thats a part 2?
00:02:06.300 --> 00:02:09.180
who votes for part 2? there you go its a part 2
00:02:11.460 --> 00:02:13.300
lets try something similar
00:02:13.840 --> 00:02:14.640
what if i had
00:02:30.580 --> 00:02:31.540
what about that
00:02:37.760 --> 00:02:38.640
this factors
00:02:41.600 --> 00:02:42.400
into sinx-3
00:02:43.800 --> 00:02:46.280
sinx-1 again you can do distribution
00:02:46.740 --> 00:02:48.340
if you have y equals sinx
00:02:53.080 --> 00:02:54.280
here sinx equals 3
00:02:55.720 --> 00:02:57.000
the sin will equla 1
00:02:59.180 --> 00:03:00.300
sinx cant equal 3
00:03:00.300 --> 00:03:03.940
x is between 1 and negative 1, including 1 and negative 1
00:03:04.360 --> 00:03:05.800
you can throw that out
00:03:05.800 --> 00:03:09.160
they just need to know where does x equal 1 on this interval
00:03:09.200 --> 00:03:10.720
you are suppose to know
00:03:11.900 --> 00:03:12.780
x equals pi/2
00:03:16.600 --> 00:03:18.440
the review sheet i posted on
00:03:18.440 --> 00:03:21.460
a couple days ago was actually for the 126 class
00:03:21.460 --> 00:03:24.580
so must of that also applies to this final and stuff
00:03:25.360 --> 00:03:26.960
feel free to look at that
00:03:27.220 --> 00:03:28.180
i put that uo on
00:03:29.580 --> 00:03:30.780
saturday, friday
00:03:32.340 --> 00:03:32.900
i forget
00:03:39.760 --> 00:03:41.040
lets do another ond
00:03:41.040 --> 00:03:43.820
lets do another thing you guys should be able to do
00:04:10.060 --> 00:04:14.940
ill think of another one, in the meantime ill give you something else to do
00:05:21.060 --> 00:05:22.900
alright a half life problem
00:05:24.220 --> 00:05:28.860
its a half life problem so what about setting up the half life equation
00:05:46.060 --> 00:05:48.700
anything thats got exponential growth
00:05:50.560 --> 00:05:53.040
you write it a y equals a times b to the x
00:05:56.880 --> 00:06:01.680
so it says initially we have 40 grams so thats when we know that aequlas 40
00:06:01.720 --> 00:06:03.240
or if you want to show it
00:06:08.720 --> 00:06:09.920
that means that 40
00:06:11.000 --> 00:06:12.520
equals a times b to the 0
00:06:16.000 --> 00:06:16.960
so b to the 0 is 1
00:06:18.340 --> 00:06:19.940
so this is just 40 to the a
00:06:19.940 --> 00:06:21.920
so now we can go to our equation
00:06:23.680 --> 00:06:25.360
and you can substitute in
00:06:26.680 --> 00:06:27.180
40 for a
00:06:28.060 --> 00:06:32.620
now we just to find out b, with a full equation we can find the half life
00:06:33.960 --> 00:06:36.680
so i know that 6 hours later i have 30 grams
00:06:38.800 --> 00:06:39.440
30 equals
00:06:41.160 --> 00:06:41.660
40
00:06:43.860 --> 00:06:44.420
b to the 6
00:06:45.200 --> 00:06:46.960
you divide both sides by 40
00:06:48.180 --> 00:06:48.900
you get 3/4
00:06:50.100 --> 00:06:50.660
b to the 6
00:06:51.760 --> 00:06:52.960
you need to solve b
00:06:57.420 --> 00:06:58.220
b equals 3/4
00:07:00.440 --> 00:07:01.080
to the 1/6
00:07:05.800 --> 00:07:09.160
that means i can now plug this in here and y equals 40
00:07:09.940 --> 00:07:10.660
times 3/4\
00:07:14.440 --> 00:07:15.080
to the x/6
00:07:16.580 --> 00:07:18.900
now i just have to find the half life
00:07:20.120 --> 00:07:22.360
whats the half life, divide by 1.6
00:07:23.260 --> 00:07:23.760
times x
00:07:25.880 --> 00:07:27.720
so half life, i want to get 20
00:07:33.020 --> 00:07:35.100
20 equals 40 times 3/4 to the x/6
00:07:35.120 --> 00:07:37.520
divide both sides by 40 you get a half
00:07:39.560 --> 00:07:40.060
3/4
00:07:41.780 --> 00:07:42.420
to the x/6
00:07:42.780 --> 00:07:44.540
i take the log of both sides
00:07:46.000 --> 00:07:49.200
take the log of both sides i can take any log i want
00:07:49.400 --> 00:07:51.640
the natural log i can do common log
00:07:53.100 --> 00:07:54.300
log base anything
00:08:03.640 --> 00:08:05.080
bring the x/6 in front
00:08:07.160 --> 00:08:08.120
you get log 1/2
00:08:10.680 --> 00:08:11.400
equlas x/6
00:08:13.240 --> 00:08:13.880
log of 3/4
00:08:17.800 --> 00:08:19.080
do a little algebra
00:08:20.720 --> 00:08:21.680
and solve for x
00:08:32.520 --> 00:08:33.720
you get 6log of 1/2
00:08:35.260 --> 00:08:36.220
over log of 3/4
00:08:46.840 --> 00:08:48.040
howd we do on that?
00:09:01.240 --> 00:09:03.720
wy is b to the 0, initially the time is 0
00:09:08.600 --> 00:09:09.560
next question
00:09:10.220 --> 00:09:11.900
no lets do something else
00:10:09.500 --> 00:10:12.860
if you has f of x equals 6x squared minus 24x minus 30
00:10:14.000 --> 00:10:15.520
and the bottom you have
00:10:17.960 --> 00:10:19.800
2x squared minus 2x minus 12
00:10:20.060 --> 00:10:25.100
find the vertical asymptote, the horizontal asymptote and the x intercept
00:10:25.460 --> 00:10:26.260
y intercept
00:10:28.140 --> 00:10:30.940
lets find the various pieces of this thing
00:10:32.120 --> 00:10:34.200
we need the verticle asymptote
00:10:34.900 --> 00:10:38.340
the horizontal asymptote, the vertical asymptote
00:10:39.800 --> 00:10:41.080
is also known as the
00:10:41.700 --> 00:10:43.060
the curve and acorss
00:10:44.200 --> 00:10:47.160
the horizontal asymptte is the end behavior
00:10:47.160 --> 00:10:49.040
thats what happens when x goes to infinity
00:10:49.040 --> 00:10:52.440
you can cross the vertical asymptote but you cant cross the horizontal asymptote
00:10:54.460 --> 00:10:56.860
the x intercept which are the values
00:10:57.620 --> 00:10:59.300
where you cross the x axis
00:10:59.540 --> 00:11:00.820
and the y intercept
00:11:01.780 --> 00:11:02.740
is your values
00:11:03.420 --> 00:11:05.020
when you cross the y axis
00:11:07.860 --> 00:11:11.540
well if you want to do this the first thing youy want to do wis demostrate
00:11:12.160 --> 00:11:13.680
something by facoring
00:11:24.740 --> 00:11:26.660
that becomes 6 times x minus 5
00:11:27.720 --> 00:11:28.220
x plus 1
00:11:31.060 --> 00:11:32.260
this is 2 times x-3
00:11:33.960 --> 00:11:34.460
x plus 2
00:11:34.460 --> 00:11:36.420
ots not my job to tell you how to factor
00:11:36.460 --> 00:11:37.820
suppose to know that
00:11:39.700 --> 00:11:40.660
so far so good?
00:11:40.980 --> 00:11:42.340
vertical asymptote
00:11:42.540 --> 00:11:44.780
look for when the denominator is 0
00:11:44.780 --> 00:11:46.460
the denominator is 0 in 2 places
00:11:47.760 --> 00:11:48.260
x=3
00:11:49.800 --> 00:11:50.360
and x=-2
00:11:53.840 --> 00:11:55.600
the horizontal asymptote
00:11:55.600 --> 00:12:00.060
well now we are looking for the end behavior, looks at what happens to thei
00:12:00.360 --> 00:12:01.960
as x goes out to infinity
00:12:01.960 --> 00:12:06.060
so you look at the power on top, the degree of the top is x squared
00:12:06.220 --> 00:12:08.540
the degree of the botto is x squared
00:12:09.540 --> 00:12:12.340
so this is 6x squared and this is 2x squared
00:12:13.600 --> 00:12:14.240
so 6/2 is 3
00:12:14.240 --> 00:12:15.720
so this is going to approach
00:12:16.860 --> 00:12:17.360
y=3
00:12:18.280 --> 00:12:19.400
remember thesE?
00:12:22.580 --> 00:12:23.460
x intercepts
00:12:23.740 --> 00:12:26.460
are where the function is going to equal 0
00:12:27.920 --> 00:12:30.720
or easily where the numerator is equal to 0
00:12:31.620 --> 00:12:34.500
so the numerator will equal to 0 at x equals 5
00:12:35.500 --> 00:12:37.500
or x=-1 or you can write that as
00:12:38.160 --> 00:12:39.040
coordinates
00:12:41.220 --> 00:12:42.900
okay you can do either way
00:12:43.660 --> 00:12:44.620
doesnt matter
00:12:47.320 --> 00:12:50.440
and y intercept is what you get when you plug in 0
00:12:50.440 --> 00:12:53.000
so when you plug in 0 on top what do you get?
00:12:55.160 --> 00:12:55.660
-30
00:12:56.360 --> 00:12:59.080
when you plug in 0 in the bottom you get -12
00:13:02.200 --> 00:13:03.160
so its -30/-12
00:13:04.220 --> 00:13:07.420
you dont need to simplify that if you dont want to
00:13:07.700 --> 00:13:09.220
but if you do you get 5/2
00:13:10.760 --> 00:13:12.920
we would not ask you to graph that
00:13:19.480 --> 00:13:23.320
how do i find the y intercept? y intercept is when x equals 0
00:13:25.240 --> 00:13:27.160
i plug in 0 on top i get 0,0,-30
00:13:27.780 --> 00:13:29.780
and the bottom you get 0,0, -12
00:13:30.480 --> 00:13:31.440
so its -30/-12
00:13:41.280 --> 00:13:43.680
good lets do some other type of stiff
00:13:46.060 --> 00:13:48.460
okay lets do the domains for a minute
00:13:57.600 --> 00:13:59.200
whats the domain of sinx
00:14:07.140 --> 00:14:10.260
all real numbers, you get sin anything you want
00:14:12.620 --> 00:14:13.740
how bout cosine x
00:14:16.460 --> 00:14:17.420
also all reals
00:14:17.900 --> 00:14:19.020
to give a funtion
00:14:19.620 --> 00:14:22.180
to have sinx or cosinx in the numerator
00:14:22.360 --> 00:14:24.120
or theres not denominator
00:14:24.360 --> 00:14:25.000
then that
00:14:25.000 --> 00:14:26.840
part you can plug in anything you want
00:14:27.600 --> 00:14:28.880
what about e to the x
00:14:33.580 --> 00:14:35.580
dont confuse domain and range
00:14:35.640 --> 00:14:37.880
you can take e to any value you want
00:14:39.080 --> 00:14:40.840
you can do e to the anything
00:14:41.100 --> 00:14:42.700
is the domain is all real
00:14:43.640 --> 00:14:46.440
but an important thing to remember about e
00:14:47.560 --> 00:14:50.200
is the range is only positive numbers so
00:14:50.200 --> 00:14:52.520
e to the someting is always positive
00:14:54.960 --> 00:14:56.960
sin and cosin range from -1 to 1
00:14:58.980 --> 00:15:04.020
e to the x always comes out a positive number and you plug in anything you want
00:15:04.540 --> 00:15:08.540
lets see whats the domain of log of x, any log it doesnt matter
00:15:12.040 --> 00:15:14.760
the domain of log of x is positive numbers
00:15:22.340 --> 00:15:25.620
also known as x is greater then 0, remember the log
00:15:25.800 --> 00:15:29.720
is the inverse of e so log always comes out a positive number
00:15:29.980 --> 00:15:34.620
so when you are doing natural log you can only plug in a positive number
00:15:34.920 --> 00:15:39.800
you cannot take the log of 0 and you cannot take the log of a negative number
00:15:42.180 --> 00:15:43.140
what about 1/x
00:15:47.180 --> 00:15:48.860
the domain of x connot be 0
00:15:50.500 --> 00:15:52.180
because you cant have 1/0
00:15:52.540 --> 00:15:55.660
in fact it can be any number on top, any positive
00:15:57.600 --> 00:15:59.920
and last what about square root of x
00:16:02.360 --> 00:16:04.680
x has to be greater then or equal to 0
00:16:05.240 --> 00:16:09.080
because you cannot take a square root of a negative number
00:16:10.640 --> 00:16:13.040
alright now lets put it all together
00:16:13.980 --> 00:16:15.500
and lets say i have f of x
00:16:18.620 --> 00:16:19.120
is
00:16:25.060 --> 00:16:26.180
whats the domain
00:16:31.220 --> 00:16:33.620
so if you want find the domain of this
00:16:33.620 --> 00:16:36.080
this is what you should pay attention to
00:16:36.080 --> 00:16:40.320
when you take the square root you take the square root of a positive number
00:16:41.140 --> 00:16:41.640
so x+5
00:16:43.740 --> 00:16:45.980
has to be greater then or equal to 0
00:16:46.940 --> 00:16:49.420
so x has to be greater then or equal to 5
00:16:52.600 --> 00:16:54.200
and x squared - 3 cant be 0
00:17:00.360 --> 00:17:03.480
x cannot equal plus or minus the square root of 3
00:17:07.420 --> 00:17:08.860
and thats good enough
00:17:13.220 --> 00:17:14.500
lets do another one
00:17:16.960 --> 00:17:17.840
suppose i had
00:17:48.260 --> 00:17:49.780
this is a little tricky
00:17:52.180 --> 00:17:54.020
first look at the numerator
00:17:54.020 --> 00:17:56.300
the numerator can do anything it wants
00:17:57.040 --> 00:18:00.640
im not to concern about the numerator so the key is just
00:18:01.020 --> 00:18:04.700
then you look at the denominator and say well i have a log
00:18:05.340 --> 00:18:07.260
so x-2 has to be greater then 0
00:18:10.080 --> 00:18:11.680
x has to be greater then 2
00:18:13.540 --> 00:18:15.380
but we have another problem
00:18:15.460 --> 00:18:18.580
the problem is you can have 0 in the denominator
00:18:20.020 --> 00:18:20.520
so this
00:18:22.020 --> 00:18:22.980
cant come out 0
00:18:25.200 --> 00:18:26.960
so when is log x-2 equal to 0
00:18:29.280 --> 00:18:31.040
well when you do the log of 1
00:18:35.240 --> 00:18:36.200
the log of 1 is 0
00:18:38.920 --> 00:18:41.080
so that would be when x-2 equals 1
00:18:43.160 --> 00:18:44.040
x cant equal 3
00:18:45.820 --> 00:18:47.980
it can be greater then 2 but ca be 3
00:18:48.640 --> 00:18:51.040
thats probably a little over tricky
00:18:51.040 --> 00:18:54.440
but i thought id throw that in tomake sure you guys know all those things
00:19:00.740 --> 00:19:03.380
alright lets do some other kind of stuff
00:19:25.560 --> 00:19:29.320
im sort of going back and fourth for part 1 and part 2 stuff
00:19:30.360 --> 00:19:32.920
just so youre prepared for everything
00:19:50.380 --> 00:19:52.060
alright thats not to hard
00:19:52.060 --> 00:19:54.520
so if you want to find the equation of a line
00:19:55.060 --> 00:19:58.260
you need a point and the slope but we already have
00:19:58.540 --> 00:19:59.100
2 points
00:19:59.700 --> 00:20:01.060
so we basically want
00:20:02.880 --> 00:20:04.000
use the form y-y1
00:20:04.820 --> 00:20:06.660
equals the slope times x-x1
00:20:07.520 --> 00:20:09.040
we already have a point
00:20:09.040 --> 00:20:12.420
so you can either use 5,10 ad 12,7 either it doesnt matter
00:20:12.780 --> 00:20:14.860
now youre just gonna find slope
00:20:17.040 --> 00:20:18.160
the slope is 10-7
00:20:20.180 --> 00:20:20.820
over 5-12
00:20:26.080 --> 00:20:26.880
so this -3/7
00:20:27.920 --> 00:20:28.420
3/=7
00:20:32.160 --> 00:20:33.360
so the equation is
00:20:35.160 --> 00:20:35.660
y-10
00:20:37.180 --> 00:20:37.680
=3/7
00:20:38.820 --> 00:20:39.460
times x-5
00:20:40.500 --> 00:20:41.780
or you could use y-7
00:20:43.500 --> 00:20:44.780
and -3/7 times x-12
00:20:44.780 --> 00:20:45.680
doesnt matter
00:20:46.700 --> 00:20:50.460
if you simplify you can arrange and end up in the same spot
00:20:50.700 --> 00:20:53.660
so you also have to find the equation of a line
00:21:41.460 --> 00:21:43.700
so we need the equation of a circle
00:21:45.400 --> 00:21:48.040
with the center of h,k and the radius of r
00:21:49.260 --> 00:21:50.060
x-h squared
00:21:51.320 --> 00:21:52.760
plus y minus k squared
00:21:53.880 --> 00:21:55.000
equals r squared
00:21:58.900 --> 00:21:59.860
so for example
00:22:02.280 --> 00:22:03.240
the centers at
00:22:06.240 --> 00:22:06.740
-3,2
00:22:09.140 --> 00:22:10.100
the radius is 5
00:22:12.340 --> 00:22:13.460
and it will be x+3
00:22:14.200 --> 00:22:14.760
squared
00:22:17.180 --> 00:22:17.680
y-2
00:22:18.420 --> 00:22:20.100
squared equals 5 squared
00:22:20.100 --> 00:22:21.360
that wasnt very hard
00:22:21.680 --> 00:22:24.080
how can we make this harder? lets see
00:22:25.160 --> 00:22:26.360
suppose i gave you
00:22:36.100 --> 00:22:37.620
suppose i gave you that
00:22:57.500 --> 00:22:59.900
whats the center? whats the radius?
00:23:00.800 --> 00:23:02.480
i recommend you factor it
00:23:09.220 --> 00:23:13.060
dont need to complete the square you just need to factor it
00:24:27.980 --> 00:24:29.580
x squared plus 8x plus 16
00:24:31.280 --> 00:24:31.920
is x plus 4
00:24:32.540 --> 00:24:33.100
squared
00:24:34.780 --> 00:24:36.940
right thats x plus 4 times x plus 4
00:24:36.940 --> 00:24:40.440
how do you know its the square root of this and the square root of this
00:24:43.260 --> 00:24:44.860
y squared minus 6y plus 9
00:24:46.460 --> 00:24:47.180
is y minus 3
00:24:48.060 --> 00:24:48.620
squared
00:24:49.180 --> 00:24:51.100
cause its the square root of y
00:24:51.540 --> 00:24:52.820
and square root of 9
00:24:53.700 --> 00:24:55.620
so now you know that the cente
00:24:58.920 --> 00:24:59.560
is at -4,3
00:25:01.120 --> 00:25:02.560
and the radius is at 10
00:25:08.460 --> 00:25:10.860
lets do something slightly messier
00:25:17.260 --> 00:25:19.020
well you have the equation
00:25:21.000 --> 00:25:23.160
the equation either this or this
00:25:24.500 --> 00:25:27.700
so if i said whats the center and whats the radius
00:25:28.260 --> 00:25:30.020
when you see it in this form
00:25:30.020 --> 00:25:33.140
you have to be able to transform it into this form
00:25:35.340 --> 00:25:37.180
so if you had something like
00:25:39.900 --> 00:25:40.540
x squared
00:25:41.600 --> 00:25:42.100
plus 3x
00:25:43.960 --> 00:25:44.520
plus 225
00:25:44.680 --> 00:25:46.840
how do you know how to factor that
00:25:47.060 --> 00:25:50.820
well this is a perfect square and this is a perfect square
00:25:53.780 --> 00:25:55.700
and this number in the middle
00:25:56.420 --> 00:25:57.940
is doubled this number
00:25:58.280 --> 00:25:59.960
then it factors into this
00:26:04.180 --> 00:26:06.580
the key is the middle number has to be
00:26:06.580 --> 00:26:07.900
double that square root
00:26:12.700 --> 00:26:13.580
or generally
00:26:21.320 --> 00:26:23.480
x squared plus 2ax plus a squared
00:26:23.900 --> 00:26:25.340
the factors of x plus a
00:26:25.780 --> 00:26:26.340
squared
00:26:27.080 --> 00:26:28.280
so square root of a
00:26:28.280 --> 00:26:29.720
double that number is 2a
00:26:33.520 --> 00:26:35.840
and if you have a minus, so if you had
00:26:41.680 --> 00:26:43.040
square root of 36 is 6
00:26:44.740 --> 00:26:45.700
12 is double 36
00:26:46.560 --> 00:26:48.000
so it owuld be x minus 6
00:26:48.860 --> 00:26:49.420
squared
00:26:50.160 --> 00:26:52.080
if you have that pattern okay
00:26:53.540 --> 00:26:54.180
binomial
00:26:55.140 --> 00:26:56.580
thats the square root
00:27:06.960 --> 00:27:08.880
how about some inverse syuff
00:27:10.420 --> 00:27:11.780
you like the inverse
00:27:49.240 --> 00:27:51.720
alright lets find the inverse of this
00:27:54.420 --> 00:27:57.240
y=8x-4
00:27:58.300 --> 00:27:58.800
over 7
00:27:59.860 --> 00:28:00.980
you switch x and y
00:28:04.200 --> 00:28:05.000
x is 8y cubed
00:28:05.940 --> 00:28:06.580
minus 4/7
00:28:06.580 --> 00:28:09.840
you dont have to switch the x and y you cans witch it at the end
00:28:09.860 --> 00:28:11.300
doesnt really matter
00:28:12.340 --> 00:28:15.220
e is going to isolate the other variable now
00:28:15.220 --> 00:28:17.660
what do we do? you multiply both sides by 7
00:28:19.300 --> 00:28:19.940
you get 7x
00:28:21.720 --> 00:28:22.760
8y cubed minus 4
00:28:26.240 --> 00:28:26.960
and we add 4
00:28:27.700 --> 00:28:28.420
divide by 8
00:28:29.800 --> 00:28:30.680
and so on so 7x
00:28:31.440 --> 00:28:31.940
plus 4
00:28:33.540 --> 00:28:34.260
is 8x cubed
00:28:37.860 --> 00:28:39.140
divide by, 8y sorry
00:28:40.900 --> 00:28:41.620
divide by 8
00:28:46.540 --> 00:28:48.060
and take the cubed root
00:28:50.980 --> 00:28:53.380
and heres something very important
00:28:53.620 --> 00:28:56.100
you get so excited you get to this part
00:28:56.600 --> 00:28:57.640
youd be so happy
00:28:57.640 --> 00:28:59.960
youd be so happy you then forget to write f
00:29:00.280 --> 00:29:01.080
inverse of x
00:29:02.280 --> 00:29:03.560
is the cubed root of
00:29:05.080 --> 00:29:05.640
7x plus 4
00:29:06.080 --> 00:29:08.640
alright dont forget the last step okay
00:29:09.140 --> 00:29:13.700
all you did hear was find y you didnt demonstrate you know the inverse
00:29:15.820 --> 00:29:17.900
were we able to find the inverse
00:29:21.480 --> 00:29:23.720
okay lets move onto the next stuff
00:29:59.480 --> 00:30:00.920
lets see you do those 3
00:30:01.840 --> 00:30:02.340
things
00:30:04.440 --> 00:30:05.160
f of cubed x
00:30:05.800 --> 00:30:06.600
we take g of x
00:30:06.860 --> 00:30:08.460
and we plug it inside of x
00:30:10.100 --> 00:30:12.500
so everywhere i see an x i put in an x/7
00:30:19.820 --> 00:30:22.140
and you can sort of leave that alone
00:30:22.620 --> 00:30:26.460
i dont care if you simplify that but if you really wanted to
00:30:27.360 --> 00:30:28.480
you get x squared
00:30:30.120 --> 00:30:30.620
over 49
00:30:31.580 --> 00:30:32.140
minus 4x
00:30:33.380 --> 00:30:33.880
over 67
00:30:33.880 --> 00:30:36.080
but if you really want to show how great you are
00:30:37.260 --> 00:30:37.900
x squared
00:30:38.920 --> 00:30:39.560
minus 28x
00:30:41.600 --> 00:30:42.100
over 49
00:30:42.700 --> 00:30:45.180
but you know they are all the same to me
00:30:48.020 --> 00:30:51.300
you demonstrate which ones goes inside which one
00:30:53.260 --> 00:30:55.660
what about the last part, f of x plus h
00:30:56.760 --> 00:30:58.280
well what is f of x plus h
00:31:02.300 --> 00:31:03.180
f of x plus h is
00:31:04.340 --> 00:31:04.840
x plus h
00:31:06.920 --> 00:31:08.360
squared minus 4 times
00:31:09.940 --> 00:31:10.440
x plus h
00:31:21.440 --> 00:31:23.040
what is x plus h squared?
00:31:23.840 --> 00:31:25.600
i put that on the instagram
00:31:28.480 --> 00:31:29.520
thats x squared
00:31:30.700 --> 00:31:31.260
plus 2xh
00:31:32.160 --> 00:31:33.120
plus h squared
00:31:33.120 --> 00:31:36.580
i think i put that on that, i recommend you remember that
00:31:37.980 --> 00:31:41.660
if you cant just foil it out, i recommend you remember it
00:31:45.920 --> 00:31:47.040
minus 4x minus 4h
00:31:48.500 --> 00:31:51.300
so if i wanted to find f of x plus h minus f of x
00:31:56.020 --> 00:31:56.740
f of x plus h
00:31:58.000 --> 00:31:58.720
minus f of x
00:32:00.160 --> 00:32:00.660
is
00:32:02.100 --> 00:32:02.600
this
00:32:05.460 --> 00:32:05.960
minus
00:32:07.120 --> 00:32:08.320
f of x which is this
00:32:14.480 --> 00:32:16.080
thats x squared plus 2xh
00:32:17.260 --> 00:32:18.780
plus h squared minus 4x
00:32:19.600 --> 00:32:20.160
minus 4h
00:32:21.640 --> 00:32:23.240
thats f of x plus h, minus
00:32:25.380 --> 00:32:26.580
x squared minus 4x
00:32:28.640 --> 00:32:29.200
the clue
00:32:29.580 --> 00:32:31.660
whenever you have a polynomial
00:32:31.960 --> 00:32:34.280
when you have f of x plus h minus f of x
00:32:34.360 --> 00:32:36.840
all the polynomial terms will cancel
00:32:37.060 --> 00:32:39.780
the terms of the left side, the f of x plus h
00:32:39.780 --> 00:32:42.500
so everything that will be left will have an h in it
00:32:43.940 --> 00:32:45.860
so you would cancel and get 2x
00:32:47.540 --> 00:32:48.580
h plus h squared
00:32:49.380 --> 00:32:51.300
minus 4h, how did we do on that
00:32:51.300 --> 00:32:52.440
able to do that part?
00:32:55.200 --> 00:32:56.320
thats a good sign
00:32:57.080 --> 00:33:00.840
that makes you very happy because thats what part 1 stuff
00:33:06.580 --> 00:33:08.100
lets do something else
00:33:16.600 --> 00:33:17.560
was that hard?
00:34:17.800 --> 00:34:19.560
now we are going to give you
00:34:20.220 --> 00:34:20.940
on the test
00:34:21.980 --> 00:34:23.100
the sin of a plus b
00:34:23.180 --> 00:34:26.460
and the cosin of a plus b ill write that down for you
00:34:28.760 --> 00:34:31.160
pay attention its on the cover sheet
00:34:33.420 --> 00:34:34.540
the sin of a plus b
00:34:37.360 --> 00:34:37.860
is sin a
00:34:39.820 --> 00:34:40.320
cosinb
00:34:42.020 --> 00:34:42.820
plus cosina
00:34:44.540 --> 00:34:45.040
sinb
00:34:48.460 --> 00:34:49.740
and cosin of a plus b
00:34:53.120 --> 00:34:53.760
is cosina
00:34:55.360 --> 00:34:55.860
cosinb
00:34:57.780 --> 00:34:58.500
minus sin a
00:35:02.060 --> 00:35:02.560
sin b
00:35:06.060 --> 00:35:09.340
you dont know this youre not gonna know the others
00:35:09.860 --> 00:35:13.860
if you can figure out the sin plus b you can find out sin minus b
00:35:16.360 --> 00:35:17.960
alright so lets do these
00:35:20.140 --> 00:35:23.100
we have a couple of stages with this so first\
00:35:23.100 --> 00:35:25.480
you just have to figure out missing pieces
00:35:33.380 --> 00:35:34.340
sin of a is 9/10
00:35:56.100 --> 00:35:57.300
the sin of a is 9/10
00:36:00.580 --> 00:36:02.020
and here in quadrant 2
00:36:10.820 --> 00:36:12.260
so that means this is 9
00:36:13.320 --> 00:36:14.200
and this is 10
00:36:14.280 --> 00:36:16.440
and you use pythagorean theorem
00:36:17.880 --> 00:36:20.760
and thats the square root of 19 and negative
00:36:20.760 --> 00:36:25.440
you should just know that in the 2nd quadrant cosin and sin are negative
00:36:29.600 --> 00:36:30.480
the cosin of b
00:36:31.580 --> 00:36:32.620
is negative 7/8
00:36:33.180 --> 00:36:35.180
and were in the third quadrant
00:36:39.680 --> 00:36:41.520
thats negative 7 and thats 8
00:36:41.540 --> 00:36:45.380
thats the square root of 8 square minus negative 7 squared
00:36:46.560 --> 00:36:48.000
which equals minus 49
00:36:48.000 --> 00:36:51.180
thats the square root of 15 and thats gonna be negative
00:36:51.180 --> 00:36:52.580
found in the 3rd quadrant
00:37:11.980 --> 00:37:13.420
so we want to find sina
00:37:15.000 --> 00:37:15.500
cosin b
00:37:15.500 --> 00:37:17.980
and this is minus so its gonna be minus
00:37:19.440 --> 00:37:19.940
cosina
00:37:21.060 --> 00:37:21.560
sinb
00:37:26.880 --> 00:37:28.000
well sina is 9/10
00:37:32.760 --> 00:37:33.560
cosb is -7/8
00:37:37.720 --> 00:37:38.220
cosa
00:37:39.220 --> 00:37:39.720
is
00:37:42.800 --> 00:37:43.840
- radical 19/10
00:37:48.700 --> 00:37:49.260
and sinb
00:37:51.000 --> 00:37:51.500
is -7/8
00:37:52.500 --> 00:37:54.500
i dont really care to simplify
00:38:33.900 --> 00:38:35.180
how do you do cisn2b
00:38:35.180 --> 00:38:38.480
again you dont have to know the double angle formula
00:38:39.820 --> 00:38:41.100
cuase if we tell you
00:38:43.140 --> 00:38:43.940
cosina plu b
00:38:45.480 --> 00:38:47.320
is cosacosb minus sinasinb
00:38:48.820 --> 00:38:50.260
to find the cosin of 2b
00:38:51.180 --> 00:38:53.180
you can write this like b plus b
00:38:54.320 --> 00:38:54.820
so cosb
00:38:57.560 --> 00:38:58.600
cosb minus sinb
00:39:00.580 --> 00:39:01.080
sinb
00:39:05.780 --> 00:39:07.300
thats all you need to do
00:39:09.160 --> 00:39:10.440
so to find the cos2b
00:39:20.200 --> 00:39:21.320
its gonna be -7/8
00:39:24.200 --> 00:39:24.700
-7/8
00:39:27.240 --> 00:39:29.160
minus negative radical 15/8
00:39:31.240 --> 00:39:32.600
negative radical 15
00:39:33.200 --> 00:39:33.700
over
00:39:33.700 --> 00:39:34.200
8
00:40:18.020 --> 00:40:19.380
couple other things
00:40:20.500 --> 00:40:22.260
how do we feel about these?
00:40:23.040 --> 00:40:23.540
good?
00:40:26.700 --> 00:40:28.700
im gonna erase this whole mess
00:41:28.320 --> 00:41:29.040
do this one
00:41:34.540 --> 00:41:37.580
something like this is very straight forward
00:41:40.020 --> 00:41:42.820
basically you do the two parts seperately
00:41:44.220 --> 00:41:47.020
so where does 5x squared minus 2x equal to 0
00:41:51.500 --> 00:41:52.940
well 5x plus 2x squaed
00:41:55.860 --> 00:41:57.220
5/2 equals x squared
00:41:59.180 --> 00:42:01.580
x is plus or minus squared root of 5/2
00:42:04.220 --> 00:42:06.860
but that is the piece for when x has to be 0
00:42:06.860 --> 00:42:09.400
so you can throw out the positive answer
00:42:11.540 --> 00:42:13.940
so you can only use x equals negative
00:42:14.620 --> 00:42:15.820
square root of 5/2
00:42:17.000 --> 00:42:21.000
you look at the other part of the function and set it equal to 0
00:42:23.560 --> 00:42:24.680
3x plus 6 equals 0
00:42:26.080 --> 00:42:26.960
or x equals -2
00:42:28.400 --> 00:42:32.640
but thats in the branch where x has to be greater then or equal to 0
00:42:32.880 --> 00:42:34.240
so you throw that out
00:42:37.540 --> 00:42:40.740
the only solution is negative square root of 5/2