WEBVTT
Kind: captions
Language: en
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some of you the competent questions are pretty straight forward, be careful you dont psych yourself out
00:00:05.180 --> 00:00:07.320
and turn it into harder questions then they are
00:00:12.300 --> 00:00:14.300
theres no graphing on the exam
00:00:15.580 --> 00:00:17.580
theres no ellipses on the test
00:00:17.580 --> 00:00:19.100
I know that makes you happy
00:00:19.580 --> 00:00:23.500
cant use calculators, what else would people like to know?
00:00:27.260 --> 00:00:32.460
the part one we have the whole take it over in the testing center 2 weeks from now
00:00:34.540 --> 00:00:38.380
at the end on the final there will be a part one and a part two
00:00:38.940 --> 00:00:43.420
part one on the final, thats what you really need to make sure to pass
00:00:45.900 --> 00:00:49.340
thats the real minimum competent part, obbviously
00:00:49.340 --> 00:00:55.460
if you pass that at then end you pass the minimum competent course even if you didnt pass the first few times around
00:00:55.460 --> 00:00:59.460
thats where you demonstrate to us that you can move up to calculus
00:01:02.620 --> 00:01:04.860
theyre not teaching 125 , im sorry
00:01:06.320 --> 00:01:07.200
wasnt up to me
00:01:07.600 --> 00:01:08.720
other questions
00:01:12.540 --> 00:01:16.860
well i think it would probably be the same as last time, 10 out of 16
00:01:19.560 --> 00:01:21.720
im pretty sure, well from memory
00:01:21.720 --> 00:01:24.860
im pretty sure its 16 points and again you need 10 again
00:01:25.580 --> 00:01:29.260
thats about the number, you got to get some of them right
00:01:31.240 --> 00:01:32.360
other questions
00:01:35.040 --> 00:01:37.600
you sure? okay so heres what im gonna do
00:01:37.600 --> 00:01:42.040
im gonna put a bunch of questions on the board that are going to be similar to the minimum competent questions on the test
00:01:42.640 --> 00:01:43.840
if you can do these
00:01:43.840 --> 00:01:45.620
you are minimally competence
00:01:45.620 --> 00:01:46.660
be really happy
00:02:06.920 --> 00:02:10.600
so i will give you these four and youll get the basic idea
00:02:10.800 --> 00:02:12.240
everybody do that one
00:04:37.380 --> 00:04:39.300
okay theres some to start off
00:04:39.300 --> 00:04:43.580
so if you cant read these well because of my beautiful hand writing
00:04:43.580 --> 00:04:47.620
represent log base 8 of 128 as a fraction, interger or radical
00:04:48.680 --> 00:04:53.000
whats the largest domain on which log base 5 to the 5-4x is defined
00:04:55.140 --> 00:04:58.100
solve for x. 16 to the 3x-5 equals 32 to the x+!
00:04:59.720 --> 00:05:02.600
find the equation of a line from 8,pi and 4,1
00:05:02.600 --> 00:05:05.120
dont be scared on pi, pi is just a number
00:05:06.100 --> 00:05:12.740
log base a is 20, log base b is 30, simplify log square root of a/ b cubed, i dont know why i said simplify
00:05:12.740 --> 00:05:16.720
write the equation of a circle with center at 8,-4 and radius 10
00:05:16.920 --> 00:05:21.240
represent log base 10 of 1000 as an interger, radical or fraction
00:05:25.820 --> 00:05:29.980
represent log base 8 of 128, as a interger, fraction or radical
00:05:34.220 --> 00:05:35.980
so you take log base 8 of 128
00:05:38.180 --> 00:05:41.220
you say thats equal to something so we call it x
00:05:43.000 --> 00:05:45.160
you guys can see that over there?
00:05:47.180 --> 00:05:48.780
that means that 8 to the x
00:05:50.700 --> 00:05:51.200
is 128
00:05:55.100 --> 00:05:55.600
so now
00:05:55.600 --> 00:05:59.320
you go how am i suppose to solve 8 to the x, you could solve it
00:05:59.320 --> 00:06:02.120
by doing logs again but thats not what were looking for
00:06:03.340 --> 00:06:04.300
this is 2 cubed
00:06:07.060 --> 00:06:09.140
so 128 must be 2 to the something
00:06:09.520 --> 00:06:11.040
who wants to give a gues
00:06:12.300 --> 00:06:12.800
7
00:06:12.800 --> 00:06:16.040
yea work it out, thats what you have scrap paper for
00:06:19.100 --> 00:06:22.300
so now that we know that 2 to the 3x equals 2 to the 7
00:06:27.660 --> 00:06:28.940
then 3x must equal 7
00:06:30.340 --> 00:06:30.840
x
00:06:33.100 --> 00:06:33.600
is 7/3
00:06:34.520 --> 00:06:36.760
okay, how do we feel about this one
00:06:37.380 --> 00:06:39.940
okay if you can do this one, its 2 points
00:06:41.400 --> 00:06:43.080
if you can practice these
00:06:45.520 --> 00:06:48.080
i posted one like this a couple days ago
00:06:48.420 --> 00:06:50.900
theres some other practice problems
00:06:51.680 --> 00:06:52.960
make sure we do this
00:07:06.640 --> 00:07:08.240
8 to what exponent is 128
00:07:10.220 --> 00:07:10.720
7/3
00:07:14.580 --> 00:07:15.940
cause 8 squared is 64
00:07:16.820 --> 00:07:19.860
and 8 cubed is 512, so somewhere between 2 and 3
00:07:34.220 --> 00:07:37.100
what is the largest domain of this log base 5
00:07:38.680 --> 00:07:39.180
5-4x
00:07:39.720 --> 00:07:40.440
is defined
00:07:42.540 --> 00:07:44.140
now it can be log any base
00:07:44.140 --> 00:07:45.620
it could be natural log
00:07:45.620 --> 00:07:48.560
it can be any base and the rules of the log are the same
00:07:49.640 --> 00:07:53.720
it happens, when you take the log it has to be a positive number
00:07:53.960 --> 00:07:55.480
so whether that says ln
00:07:55.620 --> 00:07:59.540
or that say log, log base whatever it doesnt matter,=. this
00:08:00.680 --> 00:08:02.840
must be greater than 0, 5 minus 4x
00:08:03.780 --> 00:08:04.900
is greater then 0
00:08:09.280 --> 00:08:11.120
so 5 has to be greater then 4x
00:08:12.680 --> 00:08:13.180
or 5/4
00:08:13.180 --> 00:08:15.980
greater then x. or you can flip that around
00:08:16.300 --> 00:08:20.060
x is less then 5/4, you dont need to use interval notation
00:08:21.500 --> 00:08:24.380
you can write it as, any number less then 5/4
00:08:26.120 --> 00:08:28.200
okay you get that? 2 more points
00:08:29.420 --> 00:08:30.220
youre up to 4
00:08:30.240 --> 00:08:33.600
you can call your mom and say maybe i will be a doctor
00:08:34.060 --> 00:08:37.900
well she was hoping, cbecause you have to pay for your time
00:08:37.900 --> 00:08:39.240
got to move to florida
00:08:41.500 --> 00:08:44.700
i told you i want the good doctors from this class
00:08:47.680 --> 00:08:51.120
and if one of you become a billionaire thats fine too
00:08:51.660 --> 00:08:52.380
we can talk
00:09:06.560 --> 00:09:09.120
so this is not very different from that
00:09:11.520 --> 00:09:12.400
16 to the 3x-5
00:09:13.540 --> 00:09:14.500
is 32 to the x+!
00:09:17.860 --> 00:09:21.620
so lets get a common base, what base would you like to use?
00:09:24.040 --> 00:09:26.600
7, i mean 7 would were but thers another
00:09:27.580 --> 00:09:29.580
we can do 2? well 4, does 4 work?
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no, 8, no 2
00:09:33.420 --> 00:09:34.620
so this is 2 to the 4
00:09:36.780 --> 00:09:37.500
to the 3x-5
00:09:39.160 --> 00:09:40.200
equals 2 to the 5
00:09:42.260 --> 00:09:42.900
to the x+1
00:09:45.960 --> 00:09:47.240
so that means that 4
00:09:48.320 --> 00:09:49.040
times 3x-5
00:09:52.920 --> 00:09:53.800
has to equal 5
00:09:55.580 --> 00:09:56.220
times x+!
00:09:57.840 --> 00:09:58.880
hope that works
00:10:03.020 --> 00:10:04.140
so then you get 1x
00:10:04.980 --> 00:10:05.480
-20
00:10:06.680 --> 00:10:07.180
is 5x
00:10:08.540 --> 00:10:09.040
+5
00:10:12.260 --> 00:10:12.900
so 7x is 25
00:10:14.960 --> 00:10:15.520
x is 25/7
00:10:18.020 --> 00:10:18.580
6 points
00:10:21.400 --> 00:10:23.640
all you got to do is stuff like this
00:10:24.140 --> 00:10:25.180
this is minimum
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thats why they are minimum competent
00:10:28.300 --> 00:10:31.500
got to impress us to demostrate your confidence
00:10:32.440 --> 00:10:39.320
whats incompetent, incompetent is for example two outs in the bottom of the 9th and you give up a homerun
00:10:40.920 --> 00:10:41.480
thats it
00:10:57.440 --> 00:10:59.760
so when i find the equation of a line
00:10:59.760 --> 00:11:04.660
see if some one were to watch that video 20 years from now it will make no sense
00:11:07.560 --> 00:11:12.600
so what do we need? we need a slope and a point. remember the equation of a line
00:11:16.360 --> 00:11:16.860
y-y1
00:11:17.820 --> 00:11:19.100
equals m times x-x1
00:11:19.100 --> 00:11:23.040
i know some of you really want to use y=mx+b because you were taught that
00:11:23.040 --> 00:11:25.040
its just so nice and easy to remember
00:11:25.040 --> 00:11:28.220
but its actually not a good way to find the equation of a line
00:11:28.220 --> 00:11:30.440
if you have this kind of information
00:11:32.240 --> 00:11:34.560
in fact i dont know why we teach that
00:11:34.560 --> 00:11:37.500
its not a good formula of a line, its not a good formula at all
00:11:40.800 --> 00:11:43.120
we need m, m is the slope how bout m is
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pi-1
00:11:46.120 --> 00:11:46.680
over 8-4
00:11:48.880 --> 00:11:49.520
or pi-1/4
00:11:50.080 --> 00:11:52.960
so if you have 1-pi/-4, thats the same thing
00:11:53.500 --> 00:11:57.500
and then brace yourself, pick either point it doesnt matter
00:11:58.940 --> 00:11:59.440
y-1
00:12:01.380 --> 00:12:01.880
pi-1/4
00:12:02.780 --> 00:12:05.020
times x-1, and your done, 4 points
00:12:07.320 --> 00:12:09.000
you can pick either point
00:12:12.300 --> 00:12:14.940
you dont have to simplify, pi is not 3.14
00:12:15.680 --> 00:12:16.180
pi is pi
00:12:18.420 --> 00:12:19.300
e is e, pi is pi
00:12:20.100 --> 00:12:21.300
these are numbers
00:12:21.620 --> 00:12:22.580
so far so good?
00:12:23.260 --> 00:12:24.620
you feeling better?
00:12:24.620 --> 00:12:27.720
you think your stomach is getting a little better?
00:12:31.180 --> 00:12:31.680
8-4
00:12:35.780 --> 00:12:36.420
pi-1, 8-4
00:12:38.540 --> 00:12:39.980
gives you 1-pi and 4-8
00:12:40.880 --> 00:12:42.960
then you would have 1-pi over -4
00:12:44.340 --> 00:12:45.060
same thing
00:12:50.760 --> 00:12:55.800
doesnt matter which point you use doesnt matter which way you find the slope
00:12:57.320 --> 00:12:58.440
if you work it out
00:12:58.440 --> 00:13:00.160
youll end up in the same place
00:13:00.180 --> 00:13:03.140
you put it in the mx+b formula its not the same
00:13:04.880 --> 00:13:06.000
wait theres more
00:13:07.140 --> 00:13:07.780
oh my gosh
00:13:10.640 --> 00:13:12.720
we ready? how we doing on these?
00:13:13.620 --> 00:13:14.580
so far so good?
00:13:15.420 --> 00:13:17.420
maybe its not as bad as you thought it would be
00:13:19.200 --> 00:13:22.400
dont worry i havent done the harder problems yet
00:13:22.720 --> 00:13:26.400
but thats okay this is just the minimum competent stuff
00:13:26.460 --> 00:13:29.980
if you can do the minimum competent stuff, youre okay
00:13:31.200 --> 00:13:34.320
if get it right youll get at least a c in the class
00:13:34.420 --> 00:13:36.740
on the exam, not the class, the exam
00:13:39.280 --> 00:13:40.080
log of a is 20
00:13:41.120 --> 00:13:42.080
log base b is 30
00:13:42.620 --> 00:13:45.260
whats the log of square root of a/b cubed
00:13:47.860 --> 00:13:48.740
well lets see
00:13:49.980 --> 00:13:52.620
that is he same as the log square root of a
00:13:53.300 --> 00:13:53.800
minus
00:13:55.100 --> 00:13:56.220
the log of b cubed
00:13:59.220 --> 00:14:00.100
so far so good
00:14:01.220 --> 00:14:03.540
square root of a, is a to what power?
00:14:04.380 --> 00:14:07.580
1/2, is everyone clear it has to be to the 1/2 powe
00:14:08.940 --> 00:14:09.900
you sure? okay
00:14:10.140 --> 00:14:14.620
that means you can now take the powers in front and make this 1/2log a
00:14:16.920 --> 00:14:17.420
-3logb
00:14:25.620 --> 00:14:27.700
thats the same as saying a to 1/2
00:14:33.380 --> 00:14:35.220
is 1/2, cubed root of a is 1/3
00:14:35.220 --> 00:14:38.360
4th root of a is a to the 4th, thats why you use the word 4th
00:14:41.000 --> 00:14:43.720
you put the 1/2 in front and you get a 1/2 log a
00:14:43.720 --> 00:14:45.800
-3logb, now youre gonna substitue
00:14:46.540 --> 00:14:47.180
log a is 20
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so this is 1/2 of 20
00:14:53.300 --> 00:14:54.020
-3 times 30
00:14:55.560 --> 00:14:56.360
comes out -8
00:14:57.240 --> 00:14:58.360
you get -80, yeA?
00:15:01.480 --> 00:15:03.240
whered the half come from?
00:15:04.480 --> 00:15:06.640
square root of a is a to what power
00:15:10.360 --> 00:15:11.640
the square root of a
00:15:12.280 --> 00:15:13.880
is the same as a to the 1/2
00:15:15.660 --> 00:15:17.980
so this is the same as log a to the 1/2
00:15:22.600 --> 00:15:25.240
and our rule is you put the power in front
00:15:26.440 --> 00:15:27.000
1/2 log a
00:15:31.480 --> 00:15:32.520
did you get -80?
00:15:33.800 --> 00:15:37.080
you are doing the minimum competent at this point
00:15:37.600 --> 00:15:40.800
write the equation of a circle with a center 8,-4
00:15:40.800 --> 00:15:44.000
and radius 10, remember the formula for a circle?
00:15:45.620 --> 00:15:46.660
its x-h squared
00:15:50.780 --> 00:15:51.280
+y-k
00:15:52.460 --> 00:15:53.020
squared
00:15:55.500 --> 00:15:57.020
equals r squared where
00:15:58.140 --> 00:15:58.640
hk
00:15:59.640 --> 00:16:00.520
is the center
00:16:01.580 --> 00:16:02.860
and r is the readius
00:16:04.020 --> 00:16:06.420
so this one is very straight forward
00:16:08.740 --> 00:16:10.100
its just gonna be x-8
00:16:10.620 --> 00:16:11.180
squared
00:16:13.540 --> 00:16:14.040
+y+4
00:16:15.040 --> 00:16:15.600
squared
00:16:17.220 --> 00:16:18.420
eqauls 10 squared
00:16:20.920 --> 00:16:23.000
you do not have to simplifythat
00:16:23.000 --> 00:16:26.600
you dont have to multiply it out or anaything, you dont have to make 10 squared into 100
00:16:34.600 --> 00:16:36.680
so another words if you have h-h
00:16:38.880 --> 00:16:40.480
be okay try not to do that
00:16:45.060 --> 00:16:48.100
well put 100 in for 10 squared isnt accidental
00:16:48.380 --> 00:16:50.220
but its perfectly fine, yes
00:16:50.340 --> 00:16:54.020
so you can square root a number out or not, another words
00:16:54.020 --> 00:16:56.380
you dont have to do more work then that
00:16:56.380 --> 00:16:57.740
its the minimum amount of work
00:16:57.860 --> 00:16:59.860
all you have to do is say x minus
00:16:59.860 --> 00:17:00.980
the whole thing squared
00:17:00.980 --> 00:17:02.800
plus y minus the other thing squared
00:17:02.800 --> 00:17:05.580
equals the radius squared, this is just the pythagorean theorem
00:17:05.580 --> 00:17:09.260
so you can turn it to 100 if you want but its not necessary
00:17:11.980 --> 00:17:12.940
so far so good?
00:17:15.760 --> 00:17:16.960
log base 10 of 1000
00:17:20.360 --> 00:17:22.120
log base 10 of 1000 equals x
00:17:30.220 --> 00:17:32.060
that means 10 to the x is 1000
00:17:35.160 --> 00:17:36.520
10 to the what is 1000
00:17:37.120 --> 00:17:39.680
3 if you notice in scientific notation
00:17:40.420 --> 00:17:41.780
just count the zeros
00:17:43.640 --> 00:17:44.600
the log base 10
00:17:44.940 --> 00:17:48.140
1 followed by how many zero is the number of zeros
00:17:50.480 --> 00:17:52.240
you doing good so far? yes?
00:17:55.260 --> 00:17:56.540
you can just write 3
00:17:57.200 --> 00:17:58.320
write a nice big 3
00:17:59.580 --> 00:18:01.740
you ready for some harder stuff?
00:20:10.380 --> 00:20:13.660
there you go, that would keep you busy for a minute
00:20:50.460 --> 00:20:54.380
first when you see this, first you want to do is factor that
00:20:55.680 --> 00:20:57.920
i say wow can i factor that? oh sure
00:21:01.200 --> 00:21:02.560
x suqraed times 4x+3
00:21:04.760 --> 00:21:05.480
pull out a 3
00:21:08.880 --> 00:21:10.720
and thats gonna factor into
00:21:12.380 --> 00:21:12.880
x-3
00:21:14.680 --> 00:21:15.320
x-1 on top
00:21:17.300 --> 00:21:17.800
and x+5
00:21:19.720 --> 00:21:20.220
x+1
00:21:20.820 --> 00:21:21.700
on the bottom
00:21:23.300 --> 00:21:25.060
you do the factoring okay?
00:21:27.940 --> 00:21:30.180
you gonna ask how i factored that?
00:21:34.000 --> 00:21:36.800
what is the largest domain, another words
00:21:36.800 --> 00:21:40.660
domain can be anything you want, you can say its 7 if you felt like it
00:21:40.660 --> 00:21:42.980
so the question is whats the biggest
00:21:42.980 --> 00:21:46.480
section you can do, another words whats the only section you can exclude?
00:21:47.380 --> 00:21:49.860
domain can be x to the anything except
00:21:51.080 --> 00:21:51.580
-5
00:21:51.580 --> 00:21:56.340
or -1, because when you plug those in you make the denominator equal to zero
00:21:57.120 --> 00:22:00.240
the domain is the denominator cant equal t zero
00:22:03.760 --> 00:22:06.000
because you cant have zero on the bottom of a fraction
00:22:06.000 --> 00:22:07.860
have you ever seen a fraction that looks like that?
00:22:08.620 --> 00:22:09.580
i dont think so
00:22:10.620 --> 00:22:13.100
never seen a zero on the bottom before
00:22:13.100 --> 00:22:14.720
you also probably seen better hand writting then that
00:22:15.980 --> 00:22:17.980
okay that was a zero, whatever
00:22:19.780 --> 00:22:23.700
what value, if any does y approach as x approaches infinity
00:22:23.700 --> 00:22:26.560
this is sort of asking for the horizontal asymptotes
00:22:26.560 --> 00:22:28.780
the problem with using the word asymptote is
00:22:30.000 --> 00:22:32.960
its not really an asymptote its end behavior
00:22:32.980 --> 00:22:35.620
so we want to be mathematically correct
00:22:35.620 --> 00:22:37.620
so we just say what is this approach
00:22:38.060 --> 00:22:39.820
as x goes towards infinity
00:22:39.860 --> 00:22:41.300
so remember the rules
00:22:41.380 --> 00:22:43.140
you look at the power on top
00:22:43.280 --> 00:22:45.360
the highest power is 3x squared
00:22:45.360 --> 00:22:48.540
look at the powers on the bottom, the highest power is x squared
00:22:50.420 --> 00:22:52.420
since the powers are the same y
00:22:52.420 --> 00:22:53.700
is going to approach
00:22:54.860 --> 00:22:55.360
3
00:22:55.940 --> 00:22:56.980
because its 3/1
00:23:00.420 --> 00:23:01.380
so far so good?
00:23:06.220 --> 00:23:08.540
now if you have a higher power on top
00:23:08.540 --> 00:23:11.460
then it wouldnt approach anything, youd just say
00:23:11.460 --> 00:23:13.800
you can either write infinity or it does not exist
00:23:14.420 --> 00:23:16.660
okay but the number does not exist
00:23:18.280 --> 00:23:20.040
what dont you understand?
00:23:20.040 --> 00:23:22.280
so you look at the highest power for each
00:23:22.280 --> 00:23:24.360
the numerator and denominator
00:23:24.360 --> 00:23:26.160
the highest power on top is 3x squared
00:23:26.160 --> 00:23:28.800
the highst power on the bottom is x squared
00:23:30.460 --> 00:23:32.220
the power, thats the power
00:23:32.640 --> 00:23:34.160
the powers are the same
00:23:34.160 --> 00:23:35.900
the coefficient of this term
00:23:35.900 --> 00:23:37.700
and the coefficient of this term
00:23:38.500 --> 00:23:39.000
3/1
00:23:40.320 --> 00:23:43.360
if the bottom is a bigger power, then its just 0
00:23:44.200 --> 00:23:47.400
if the top is a bigger power, then it doesnt exist
00:23:47.400 --> 00:23:49.140
and i put this on the instagram the other day
00:23:52.200 --> 00:23:54.120
so agin you look at the powers
00:23:54.820 --> 00:23:55.700
match them up
00:23:55.700 --> 00:24:00.020
if its the same on top and same on bottom its coefficient over coefficient
00:24:06.520 --> 00:24:08.040
x-1 and x+1 dont cancel
00:24:10.360 --> 00:24:12.040
x-1, and x-1 would cancel
00:24:26.240 --> 00:24:27.600
remember your rules
00:24:28.720 --> 00:24:30.560
ill right it down in a second
00:24:32.180 --> 00:24:34.100
what if any are the zeros of fx
00:24:34.700 --> 00:24:37.420
the zeros is where the numerator equals 0
00:24:38.280 --> 00:24:40.600
so where does the numerator equal 0
00:24:42.600 --> 00:24:43.100
at x=3
00:24:43.860 --> 00:24:44.360
x=1
00:24:54.760 --> 00:24:57.480
and what value does the x cross the x axis?
00:24:58.960 --> 00:25:00.880
the y axis sorry, let x equal 0
00:25:01.660 --> 00:25:04.700
when you plug 0 in what are you gonna get on top?
00:25:06.000 --> 00:25:08.640
what do you get on top if you plug in zero?
00:25:10.100 --> 00:25:13.060
what do you get on the bottom when you plug in 0
00:25:15.260 --> 00:25:15.980
so it is 9/5
00:25:24.040 --> 00:25:24.920
one more time
00:25:25.580 --> 00:25:28.140
at what value does f of cross the x axis?
00:25:28.440 --> 00:25:30.360
that means your gonna let x=0
00:25:32.060 --> 00:25:33.580
okay and when x equals 0
00:25:34.220 --> 00:25:34.720
y
00:25:36.540 --> 00:25:37.260
get 9 on top
00:25:38.320 --> 00:25:40.320
and 5 on bottom, we plug 0 on top
00:25:41.420 --> 00:25:43.580
youre gonna get 3 times 0 squared
00:25:45.040 --> 00:25:45.540
+_
00:25:46.140 --> 00:25:46.780
12 times0
00:25:48.060 --> 00:25:49.420
plus 9 so thats just 9
00:25:49.740 --> 00:25:51.340
and on the bottom you get
00:25:52.080 --> 00:25:52.720
0 squared
00:25:55.400 --> 00:25:55.900
plus 5
00:25:57.180 --> 00:25:58.460
look what we get 9/5
00:26:01.920 --> 00:26:03.600
whered i get 3 and 1 again?
00:26:03.860 --> 00:26:05.540
you look at the numerator
00:26:06.040 --> 00:26:08.680
the numerator is equal to 0 when this is 0
00:26:09.400 --> 00:26:10.040
or thats 0
00:26:13.920 --> 00:26:14.800
you set x-3=0
00:26:16.560 --> 00:26:17.060
x=3
00:26:18.600 --> 00:26:19.480
you set x-1=0
00:26:21.360 --> 00:26:21.860
x=1
00:26:29.540 --> 00:26:32.820
you can write 9/5 either 9/5 or you can write 0,9/5
00:26:32.920 --> 00:26:34.120
either one is okay
00:26:34.760 --> 00:26:37.400
this is where end behavioral rule again
00:27:14.480 --> 00:27:16.960
so you have some rational expression
00:27:18.800 --> 00:27:21.360
you have something times x to the m plus
00:27:21.860 --> 00:27:23.220
smaller terms on top
00:27:23.680 --> 00:27:27.360
and something plus x to the n smaller terms on the bottom
00:27:27.520 --> 00:27:29.920
so x to the m is the highest term on top
00:27:29.920 --> 00:27:32.560
x ot the n is the highest term on the bottome
00:27:34.720 --> 00:27:35.760
we go that part?
00:27:39.880 --> 00:27:41.800
now if the top power is bigger
00:27:45.100 --> 00:27:47.180
then it doesnt change anything
00:27:47.960 --> 00:27:49.480
the top power is bigger
00:27:50.400 --> 00:27:52.400
it just goes towards infinity
00:27:59.360 --> 00:28:01.120
the bottom power is bigger
00:28:04.900 --> 00:28:06.820
then the y value approaches 0
00:28:09.300 --> 00:28:11.460
and if the two powers are the same
00:28:14.840 --> 00:28:15.880
then you get a/b
00:28:15.880 --> 00:28:19.420
so here the 2 powers are the same, x squared and x squared
00:28:23.960 --> 00:28:25.880
so you get 3/1 thats equal to 3
00:28:32.860 --> 00:28:35.820
you can write infinity you can write no value
00:28:35.820 --> 00:28:38.040
you can write dne, any of of those things
00:28:39.920 --> 00:28:41.520
just make clear you know
00:28:42.580 --> 00:28:44.340
that it just keeps growing
00:28:44.340 --> 00:28:45.960
not that you have no idea what you are doing
00:28:48.860 --> 00:28:51.900
even if you do have know idea what you are doing
00:28:53.620 --> 00:28:54.900
okay so far so good?
00:28:57.120 --> 00:28:58.560
polynomial division
00:28:58.980 --> 00:29:04.660
im actually gonna put that on the middle board because some sides wont see it very well
00:29:08.360 --> 00:29:09.240
time to erase
00:29:31.400 --> 00:29:33.080
what you want to do is 2x-3
00:29:34.380 --> 00:29:35.420
is gonna go into
00:29:37.820 --> 00:29:38.620
2x to the 4th
00:29:40.580 --> 00:29:41.540
+7x to the 3rd
00:29:44.400 --> 00:29:44.960
plus 16x
00:29:48.260 --> 00:29:48.760
plus 29
00:29:50.140 --> 00:29:50.640
x
00:29:53.080 --> 00:29:53.580
+21
00:29:55.160 --> 00:29:57.320
alright i dont have my glasses on
00:30:07.220 --> 00:30:08.180
so far so good?
00:30:12.600 --> 00:30:14.920
you look at the left term, you get 2x
00:30:16.420 --> 00:30:18.420
what do i have to multiply 2x by
00:30:19.000 --> 00:30:20.200
to get 2x to the 4th
00:30:25.240 --> 00:30:28.440
i take 2x and multiply what to get 2x to the fourth
00:30:29.960 --> 00:30:30.680
x to the 3rd
00:30:33.880 --> 00:30:35.880
x to the 3 times 2x is x to the 4th
00:30:36.740 --> 00:30:37.940
x to the 3rd times 3
00:30:39.460 --> 00:30:40.420
is 3x to the 3rd
00:30:41.240 --> 00:30:42.200
so far so good?
00:30:46.040 --> 00:30:46.920
you subtract
00:30:48.760 --> 00:30:51.320
2x to the 4th times 2x to the 4th answers
00:30:53.200 --> 00:30:54.800
7-3=4 so you get 4x cubed
00:30:57.140 --> 00:30:59.220
and you bring down the next term
00:31:04.000 --> 00:31:04.960
so far so good?
00:31:06.260 --> 00:31:07.380
now we do it again
00:31:08.420 --> 00:31:10.020
2x times what is 4x cubed
00:31:18.940 --> 00:31:21.100
so 2x squared times 2x is 4x cubed
00:31:22.720 --> 00:31:23.760
plus 6x squared
00:31:30.220 --> 00:31:32.780
16x squared -6x squared is 10x squared
00:31:34.060 --> 00:31:35.660
bring down the next term
00:31:41.540 --> 00:31:43.380
2x times what is 10x square
00:31:43.380 --> 00:31:45.380
5x
00:31:50.240 --> 00:31:51.520
you get 10x squared
00:31:53.440 --> 00:31:54.000
plus 15x
00:31:56.760 --> 00:31:57.320
subract
00:32:01.320 --> 00:32:02.760
and bring down the 14x
00:32:04.960 --> 00:32:06.080
bring down the 21
00:32:06.620 --> 00:32:07.740
a little crocked
00:32:09.320 --> 00:32:13.560
and what do i have to multiply 2x to get 14x, i have to mulitply by 7
00:32:19.480 --> 00:32:20.760
and when i subtract
00:32:20.940 --> 00:32:22.380
there is no remainder
00:32:22.380 --> 00:32:26.060
thats very good theres no remainder, what if there is a remainder?
00:32:26.240 --> 00:32:28.400
then i would just add that number
00:32:29.120 --> 00:32:30.240
divide it by 2x+3
00:32:33.040 --> 00:32:35.040
if i had 8, it would be plus 2x+3
00:32:35.040 --> 00:32:37.680
i have a quarter of you paying attention, thats okay
00:32:47.920 --> 00:32:49.360
howd we do on this one?
00:32:56.200 --> 00:32:57.880
so suppose the ramainder
00:32:58.860 --> 00:33:00.780
suppose the remainder was 10
00:33:04.040 --> 00:33:07.080
then you would have, the next term would be 10/
00:33:07.740 --> 00:33:08.240
2x+3
00:33:08.240 --> 00:33:09.880
if you had a remainder of 10
00:33:09.960 --> 00:33:11.560
you can have a remainder
00:33:11.940 --> 00:33:13.700
then you can just ignore it
00:33:23.240 --> 00:33:25.480
you just add so you would have plus
00:33:26.840 --> 00:33:28.200
something over 2x+3
00:33:30.320 --> 00:33:32.240
that would be your remainder
00:33:35.340 --> 00:33:36.700
whatevers left okay
00:33:37.480 --> 00:33:39.320
want to try another problem
00:33:40.440 --> 00:33:42.600
i know how much you want to do this
00:34:22.420 --> 00:34:24.420
how bout that? solve that for x
00:34:24.740 --> 00:34:28.020
i noticed many of you made the same mistake on this
00:34:29.580 --> 00:34:33.340
so first thing youre gonna do is take the log of both sides
00:34:43.220 --> 00:34:45.140
then you put the 3x+2 infront
00:34:46.460 --> 00:34:46.960
so 3x+2
00:34:48.440 --> 00:34:49.160
times log 6
00:34:50.800 --> 00:34:51.360
equals x
00:34:52.460 --> 00:34:52.960
log 9
00:34:54.220 --> 00:34:56.540
many of you forgot the perenthesis
00:34:56.540 --> 00:34:58.200
that wwould get you a wrong answer
00:35:00.740 --> 00:35:04.820
remember the 3x+2 is multiplied by the log 6, not just the log 2
00:35:09.500 --> 00:35:11.820
distribute the log 6, and you get 3x
00:35:13.020 --> 00:35:13.520
log 6
00:35:15.340 --> 00:35:16.060
plus 2 log 6
00:35:17.920 --> 00:35:18.480
equals x
00:35:21.620 --> 00:35:22.120
log 9
00:35:26.220 --> 00:35:28.220
okay remember what you do next
00:35:29.560 --> 00:35:32.600
you put all the terms that containx on one side
00:35:32.600 --> 00:35:35.340
and all the terms that do not contain x on the other side
00:35:37.320 --> 00:35:38.520
so we can get 2 log 6
00:35:41.260 --> 00:35:41.820
is x log 9
00:35:45.080 --> 00:35:45.640
minus 3x
00:35:48.160 --> 00:35:48.660
log 6
00:35:49.640 --> 00:35:53.000
so its always the same thing, first you distribute
00:35:53.260 --> 00:35:56.300
group all the terms that contain x on one sides
00:35:56.300 --> 00:35:59.160
and all the terms that do not contain x on the other side
00:36:03.080 --> 00:36:05.000
factor out the x you get 2 log 6
00:36:05.160 --> 00:36:05.660
is x
00:36:07.960 --> 00:36:08.680
times log 9
00:36:10.460 --> 00:36:11.260
minus 3log6
00:36:15.140 --> 00:36:16.660
and divide you get 2log
00:36:18.400 --> 00:36:18.900
6/
00:36:21.760 --> 00:36:22.560
log9 minus 3
00:36:23.420 --> 00:36:23.920
log 6
00:36:24.440 --> 00:36:26.280
some of you will have -2log6
00:36:26.280 --> 00:36:28.800
and you would have the bottom terms reverse, thats okay
00:36:29.220 --> 00:36:30.660
so go through it again
00:36:31.140 --> 00:36:32.820
take the log of both sides
00:36:34.980 --> 00:36:36.500
take the power in front
00:36:37.360 --> 00:36:39.600
that power goes in perenthesises
00:36:39.820 --> 00:36:40.860
and distribute
00:36:42.060 --> 00:36:43.580
to get 3x, times the log
00:36:43.880 --> 00:36:45.160
plus 2 times the log
00:36:45.580 --> 00:36:46.620
thats this term
00:36:54.980 --> 00:36:57.220
everything with x goes on one side
00:36:57.220 --> 00:36:59.360
everything without and x goes on the other side
00:37:01.760 --> 00:37:03.120
then you factor out x
00:37:03.660 --> 00:37:04.700
then you divide
00:37:08.340 --> 00:37:09.780
how out word problem?
00:37:10.080 --> 00:37:11.520
we love word problems
00:39:07.680 --> 00:39:11.040
you can use yeast in beer, in alochol if your over 21
00:39:11.520 --> 00:39:12.080
or bread
00:39:12.500 --> 00:39:13.940
or all sorts of things
00:39:15.340 --> 00:39:16.700
okay lets see so a(t)
00:39:19.220 --> 00:39:19.940
the amount
00:39:22.920 --> 00:39:23.420
at time
00:39:24.500 --> 00:39:25.000
t
00:39:26.600 --> 00:39:27.100
how
00:39:29.040 --> 00:39:29.540
much
00:39:30.860 --> 00:39:31.740
will you have
00:39:33.420 --> 00:39:34.380
after 24 hours
00:39:39.360 --> 00:39:40.400
2 part question
00:39:49.220 --> 00:39:51.220
were gonna use growth formula
00:39:53.820 --> 00:39:55.020
y=a times b to the x
00:39:55.900 --> 00:39:56.620
this is any
00:39:58.120 --> 00:40:00.200
this is any exppnentail growth
00:40:00.420 --> 00:40:02.020
and we use this equation
00:40:03.340 --> 00:40:05.820
of course were gonna use t instead of x
00:40:05.820 --> 00:40:08.340
and use a of t instead of y, but well fix that at the end
00:40:08.840 --> 00:40:11.560
so what do we know? we know that initially
00:40:12.540 --> 00:40:13.580
theres 20 grams
00:40:15.200 --> 00:40:16.080
5 hours later
00:40:16.360 --> 00:40:17.400
theres 30 grams
00:40:19.060 --> 00:40:21.220
the question is how much is there
00:40:22.980 --> 00:40:23.940
24 hours later
00:40:28.360 --> 00:40:28.860
okay
00:40:31.380 --> 00:40:33.700
so at time 0 were gonna have 20, so 20
00:40:35.840 --> 00:40:36.880
a times b to the 0
00:40:38.580 --> 00:40:40.740
anything to the 0 is 1 except for 0
00:40:41.620 --> 00:40:42.180
so a is 20
00:40:44.600 --> 00:40:46.920
you can fill in one of the variables
00:40:47.860 --> 00:40:52.900
in our equation instead of having a times b to the x, you have 20 times b to the x
00:40:57.420 --> 00:40:58.460
now we also know
00:40:59.580 --> 00:41:00.620
when x is 5 y is 30
00:41:01.360 --> 00:41:02.080
you have 30
00:41:03.340 --> 00:41:03.840
is 20
00:41:05.820 --> 00:41:06.780
times b to the 5
00:41:11.320 --> 00:41:11.820
so 3/2
00:41:14.740 --> 00:41:15.300
b to the 5
00:41:18.280 --> 00:41:18.780
b is 3/2
00:41:21.840 --> 00:41:22.480
to the 1/5
00:41:27.680 --> 00:41:29.440
the equation because y=20
00:41:31.500 --> 00:41:32.140
times 3/2
00:41:34.800 --> 00:41:35.920
1/5 to the x or x/5
00:41:37.660 --> 00:41:39.100
since i asked for a of t
00:41:40.260 --> 00:41:41.460
this would be a of t
00:41:42.660 --> 00:41:43.160
is 20
00:41:44.600 --> 00:41:45.240
times 3/5
00:41:47.480 --> 00:41:47.980
t/5
00:41:49.200 --> 00:41:49.700
3/2
00:41:53.300 --> 00:41:55.140
we just changed the letters
00:41:56.100 --> 00:41:59.700
do we understand that? the first one helps figure out a
00:41:59.700 --> 00:42:02.060
the second one helps us firgure out b
00:42:02.260 --> 00:42:03.700
and now we just plug in
00:42:05.780 --> 00:42:09.700
and now if i want to find out how many there are after 24 hours
00:42:10.900 --> 00:42:11.940
say a of 24 hours
00:42:14.160 --> 00:42:14.660
20
00:42:16.180 --> 00:42:16.820
times 3/2
00:42:20.240 --> 00:42:20.740
24/5
00:42:23.440 --> 00:42:24.720
howd we do on these?
00:42:26.500 --> 00:42:27.300
doing okay?
00:42:33.060 --> 00:42:34.820
you have to write a of 24? no
00:42:35.980 --> 00:42:36.940
you can write a
00:42:39.240 --> 00:42:41.720
we just need to know what you are doing
00:42:42.720 --> 00:42:43.920
one last quick one
00:42:46.900 --> 00:42:48.020
can i cover this?
00:42:50.740 --> 00:42:52.420
ill uncover it in a minute
00:43:42.500 --> 00:43:44.340
if you deposit 2000 dollars
00:43:45.120 --> 00:43:47.520
in the bank or a user fund, something
00:43:48.980 --> 00:43:50.900
at 8% compound continuously
00:43:50.900 --> 00:43:52.820
how much will we have after 10 years?
00:43:52.820 --> 00:43:55.900
remember the formula is just the future amount
00:43:56.460 --> 00:43:57.500
present amount
00:43:59.860 --> 00:44:00.500
e to the rt
00:44:01.980 --> 00:44:03.580
plu in the future amount
00:44:03.900 --> 00:44:04.780
2000 dollars
00:44:06.240 --> 00:44:06.740
e
00:44:08.320 --> 00:44:08.820
.08
00:44:09.140 --> 00:44:09.640
is r
00:44:10.680 --> 00:44:11.180
t is 10
00:44:11.340 --> 00:44:13.580
you can stop that what we need to do
00:44:13.860 --> 00:44:14.500
thats all
00:44:14.500 --> 00:44:17.900
how would you figure that out without a calculator?
00:44:18.520 --> 00:44:19.400
or a log table
00:44:23.700 --> 00:44:24.820
thats the answer