WEBVTT
Kind: captions
Language: en
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what will be on the exam
00:00:02.520 --> 00:00:04.560
you need to be able to take the derivative of stuf
00:00:04.680 --> 00:00:06.840
you need to know how to power rule
00:00:08.680 --> 00:00:10.200
we'll right these down
00:00:16.760 --> 00:00:20.920
the power rule is the derivative of something like x to the 10th
00:00:21.460 --> 00:00:22.500
of 5x to the 10th
00:00:22.500 --> 00:00:25.260
so when you have a polynomial you do the derivative of all those terms
00:00:25.260 --> 00:00:28.820
so thats really easy people dont struggle wih power rule to much
00:00:30.180 --> 00:00:31.060
product rule
00:00:35.760 --> 00:00:36.720
quotient rule
00:00:42.460 --> 00:00:43.180
chain rule
00:00:44.800 --> 00:00:48.240
chain rule is where it starts to get really annoying
00:00:48.800 --> 00:00:51.040
we just did the webassign on these
00:00:52.760 --> 00:00:54.360
implicite diffraction
00:01:06.880 --> 00:01:09.920
be able to find the derivative of exponentials
00:01:19.800 --> 00:01:20.520
functions
00:01:24.720 --> 00:01:25.760
all logarithms
00:01:30.340 --> 00:01:31.940
inverse trig functions
00:01:31.960 --> 00:01:35.480
guess what were doing today, inverse trig functions
00:01:35.480 --> 00:01:39.880
you also should be able to do something called logarithmic diffraction
00:01:39.880 --> 00:01:42.400
which sounds very fancy but its actually really easy
00:01:44.380 --> 00:01:45.340
its easy for me
00:01:47.280 --> 00:01:48.880
so thats kind of the list
00:01:48.880 --> 00:01:51.200
you need to be able to take the derivative of anything
00:01:51.200 --> 00:01:55.940
and at the end of next week in theory you should be able to take the derivative of anything'
00:01:56.320 --> 00:02:00.400
you might get in wrong but at least in theory you know what to do
00:02:01.980 --> 00:02:05.180
and the rest of the course is going to be applying
00:02:05.740 --> 00:02:08.380
how calculus tools problems that i know youve been dying to do
00:02:09.120 --> 00:02:11.840
you need to know all those types of things
00:02:12.420 --> 00:02:15.540
we will be review that, we have a review session
00:02:15.540 --> 00:02:18.360
we have a review monday, we have stuff i have to put up
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lets figure out how to do the derivative of inverse
00:02:33.680 --> 00:02:35.840
sin, why because theres a reason
00:02:45.580 --> 00:02:47.500
if you have y= inverse sin of x
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that can also be written as arc sin of x
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feel free to use either notation
00:02:57.320 --> 00:02:59.960
so we either write arc sin or inverse sin
00:03:01.900 --> 00:03:03.900
that means that x is the sin of y
00:03:06.260 --> 00:03:10.500
what does it mean when we say x is the sin of y, so we draw a triangle
00:03:13.840 --> 00:03:15.440
and you have some angle y
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remember soh, cah toa
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opposite over hypothenuse
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so that means that the sin of y
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is x or x/1
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and you can use the pythagorean theorem
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defined this side and get
00:03:40.080 --> 00:03:41.840
square root of 1-x squared
00:03:42.260 --> 00:03:43.780
so thats your triangle
00:03:44.900 --> 00:03:45.780
so far so good
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now were going to figure out how to find
00:03:48.400 --> 00:03:50.080
the derivative of arc sin
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so y is the src sin of x
00:03:53.360 --> 00:03:55.600
inverse sin of x, which means arc sin of y
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i take the derivative
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you have one, because x is 1
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and this is cosin of y
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dy/dx
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because we have to do it implicitly
00:04:06.040 --> 00:04:07.720
to get the derivative of y
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now lets isolate dy/dx
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derivative of 1 over the cos of y
00:04:17.980 --> 00:04:18.540
is dy/dx
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why did i draw this triangle because
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what is the cos of y
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remember soh, cah, toa
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cos of y
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is the adjacent
00:04:31.460 --> 00:04:32.740
over hypothenuses
00:04:33.300 --> 00:04:36.180
adjacent is the squared root of 1-x squared
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the hypothenuse is just 1
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so this says that dy/dx
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1/square root of 1-x squared
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that is the derivative of inverse sin
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y= derivative of sinx
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dy/dx
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1 over
00:05:04.940 --> 00:05:06.700
square root of 10x squared
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that wasnt so bad
00:05:12.920 --> 00:05:15.880
you dont have to prove it you just have to do it
00:05:15.880 --> 00:05:17.660
so this is what you memorize
00:05:17.980 --> 00:05:24.980
my job is to explain where it comes from, your job is to know how to do it, you should also know where it comes from of course
00:05:28.760 --> 00:05:30.440
so what if i wanted to find
00:05:37.500 --> 00:05:39.180
the derivative of sin pi x
00:05:55.900 --> 00:05:59.180
we got over formula 1/ squared root of 1-x squared
00:06:03.680 --> 00:06:06.720
so here it would be 1/ the square root of 1 minus
00:06:07.580 --> 00:06:08.080
pix
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squared, dont forget to squared both of them
00:06:12.700 --> 00:06:14.860
why pi x because its this squared
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times the derivative of pi x
00:06:19.780 --> 00:06:22.020
because of the chain rule, this is
00:06:24.760 --> 00:06:26.440
pi squared root of 1 minus
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pi x squared
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or pix squared x squared
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you do not have to simplify it
00:06:36.820 --> 00:06:41.060
by the way that reminds me, on the exam youll have so messy problems
00:06:41.060 --> 00:06:45.540
you dont have to simplify if you're doing product rule and quotient rule
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sometimes its handy to simplify but if we just say whats the derivative
00:06:49.840 --> 00:06:51.340
you dont have to simplify
00:06:52.220 --> 00:06:54.540
but if we say the second derivative
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youre better off simplifying the first and second
00:07:00.280 --> 00:07:03.880
so the question is can you just leave it in this form yea
00:07:05.460 --> 00:07:06.260
absolutely
00:07:07.240 --> 00:07:09.320
we make that annoying by asking
00:07:09.320 --> 00:07:13.320
for the second derivative but of course whats the derivative of pi
00:07:16.360 --> 00:07:17.640
is it 0 are you sure?
00:07:18.040 --> 00:07:19.960
really what about pi squared
00:07:22.140 --> 00:07:24.780
you sure, its a number pi is just a number
00:07:24.980 --> 00:07:27.460
whats the derivative of pi to the 10th
00:07:28.700 --> 00:07:30.300
sill 0, how out pi to the x
00:07:32.020 --> 00:07:33.380
we did that on monday
00:07:33.380 --> 00:07:35.820
since everyone was here on monday i dont need to do it again
00:08:11.980 --> 00:08:13.260
lets do inverse tan
00:08:24.820 --> 00:08:27.540
so we said x is inverse tan of y, also known
00:08:28.080 --> 00:08:28.800
arctan or y
00:08:36.640 --> 00:08:38.640
so if x is tan inverse of y thats
00:08:38.720 --> 00:08:40.000
the same as saying y
00:08:42.620 --> 00:08:43.260
to tan of x
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did i write that backward, yes actually sorry
00:08:55.820 --> 00:08:56.860
y is tangent of x
00:08:58.220 --> 00:09:00.380
x is the tan of y, sorry about that
00:09:03.960 --> 00:09:06.600
if x is the tangent of y we draw a triangle
00:09:08.160 --> 00:09:10.640
i should make you do this yourself but
00:09:11.480 --> 00:09:12.920
so we have some angle y
00:09:14.900 --> 00:09:16.580
the tangent is x so its x/1
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you use your favorite theorem, the one you only memorized pythagorean theorem
00:09:40.220 --> 00:09:40.780
x is tan y
00:09:40.780 --> 00:09:43.940
that means when we have a triangle and thats y that means that tangent is x/1
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so lets take the derivative, derivative of x is 1
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the derivative of tangent is secant squared y
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dy/dx, just like the last thing
00:10:02.160 --> 00:10:03.680
divide and we get 1 over
00:10:38.520 --> 00:10:39.880
oh i just forgot that
00:10:44.760 --> 00:10:47.080
its the squared root of 1+x squared
00:10:48.340 --> 00:10:49.300
dy/dx is 1 over
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secant squared of y
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and the secant of y
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is the square root of 1+x squared
00:10:56.740 --> 00:10:57.240
over 1
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because its the reciprocal of cosine
00:11:01.220 --> 00:11:02.660
so this would be 1 over
00:11:03.980 --> 00:11:05.740
square root of 1+x squared
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squared
00:11:07.200 --> 00:11:08.720
which is 1/1+x squared
00:11:16.080 --> 00:11:16.580
if y
00:11:19.900 --> 00:11:21.100
is tan inverse of x
00:11:23.240 --> 00:11:23.740
dy/dx
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1/1+x squared
00:11:32.220 --> 00:11:35.340
well the square and square root and they go away
00:11:35.400 --> 00:11:39.320
everythings positive so you dont need absolute value bars
00:11:40.540 --> 00:11:41.260
so if i said
00:11:51.280 --> 00:11:52.080
if i had that
00:11:53.600 --> 00:11:54.240
lets find
00:11:55.940 --> 00:11:58.420
the derivative of tan inverse of 4x+3
00:12:27.980 --> 00:12:28.620
its 1 over
00:12:30.420 --> 00:12:31.540
1 plus this thing
00:12:33.020 --> 00:12:33.580
squared
00:12:36.200 --> 00:12:38.120
times the derivative of 4x+3
00:12:39.680 --> 00:12:40.320
which is 4
00:12:40.760 --> 00:12:43.160
and of course you can move the 4 on top
00:12:46.500 --> 00:12:47.940
and write it like that
00:12:51.520 --> 00:12:53.440
you guys still confused here
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thats 1 thats x right
00:12:56.360 --> 00:12:58.300
pythagorean theorem square root of 1 plus x squared
00:13:00.680 --> 00:13:02.120
so 1/secant squared y
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is the same thing as saying cosine squared of y
00:13:07.980 --> 00:13:08.480
cos of y
00:13:08.480 --> 00:13:11.060
is 1/ suqared root of 1+x squared, so you are squaring it
00:13:11.460 --> 00:13:12.180
you get 1/1
00:13:19.020 --> 00:13:20.780
because theres no squared
00:13:20.780 --> 00:13:22.400
the inverse sin theres no squaring
00:13:26.060 --> 00:13:27.180
memorize those 2
00:13:27.180 --> 00:13:28.560
they will be very handy
00:13:28.700 --> 00:13:31.020
were not going to do inverse cosine
00:13:31.380 --> 00:13:32.420
inverse secant
00:13:32.540 --> 00:13:34.620
inverse cot you dont need those
00:13:36.020 --> 00:13:38.420
if anybody asks you in another class
00:13:38.420 --> 00:13:40.640
whats the derivative of the inverse cosine of x
00:13:40.640 --> 00:13:42.880
its exactly the same as inverse cosx
00:13:43.360 --> 00:13:44.080
which is -1
00:13:44.280 --> 00:13:46.840
okay so theres nothing really to learn
00:13:47.400 --> 00:13:49.160
if they ask for inverse cot
00:13:49.160 --> 00:13:50.940
its the same as inverse tangent
00:13:51.140 --> 00:13:52.180
except thats -1
00:13:52.180 --> 00:13:53.540
you can figure it out on your own
00:13:54.700 --> 00:13:56.780
were not going to do inverse sec
00:13:58.260 --> 00:14:02.420
you can always figure them out if you need to, you have the trick
00:14:08.840 --> 00:14:11.160
lets try another practice problem
00:14:32.580 --> 00:14:35.940
lets see if you guys can take the derivative of that
00:14:35.940 --> 00:14:38.740
notice youre going to need the product rule
00:14:41.940 --> 00:14:42.440
dy/dx
00:14:43.300 --> 00:14:46.260
alright we have 2 functions you either do the
00:14:46.260 --> 00:14:47.720
first times the derivative of the second
00:14:47.720 --> 00:14:49.600
plus the second times the derivative of the first
00:14:49.600 --> 00:14:51.560
or the other one, doesnt really matter
00:14:51.560 --> 00:14:53.540
so we leave the first one alone
00:14:56.960 --> 00:14:59.680
and now we do the derivative of the second
00:14:59.680 --> 00:15:01.620
the derivative of inverse sin
00:15:01.880 --> 00:15:04.200
is 1/ the square root of 1-x squared
00:15:05.240 --> 00:15:07.720
so this is going to be 1/ square root of
00:15:08.640 --> 00:15:09.140
1-2x
00:15:10.900 --> 00:15:11.460
squared
00:15:12.140 --> 00:15:13.900
times the derivative of 2x
00:15:15.520 --> 00:15:16.020
2
00:15:17.500 --> 00:15:18.000
plus
00:15:18.000 --> 00:15:20.820
pkay so this is productrule so this is just the first half
00:15:24.300 --> 00:15:25.260
no you reverse
00:15:25.580 --> 00:15:27.180
we do the inverse sin of 1
00:15:32.520 --> 00:15:35.000
and we have to do the derivative of sin
00:15:35.400 --> 00:15:35.900
3x
00:15:35.900 --> 00:15:37.180
the derivative of sin of 3x
00:15:38.400 --> 00:15:39.120
is cos of 3x
00:15:41.860 --> 00:15:42.360
times 3
00:15:45.280 --> 00:15:47.280
you do not need to simplify tht
00:15:47.280 --> 00:15:48.620
you do not want to simplify that
00:15:49.780 --> 00:15:51.860
i wouldnt want to simplify that
00:15:53.520 --> 00:15:56.720
we would never ask you to do anything beyond that
00:15:57.020 --> 00:15:58.540
of plugging in a number
00:16:00.060 --> 00:16:01.740
we do something fun at x=0
00:16:05.020 --> 00:16:06.220
yea thats about it
00:16:06.380 --> 00:16:08.380
and you say what happens at x=0
00:16:09.440 --> 00:16:10.640
solving that for 0
00:16:10.640 --> 00:16:12.560
you cant do it you have to use a calculator
00:16:13.780 --> 00:16:14.660
or a computer
00:16:16.420 --> 00:16:17.780
howd we do on that one
00:16:19.540 --> 00:16:21.620
should we practice another one
00:16:22.440 --> 00:16:22.940
yes
00:16:46.800 --> 00:16:48.800
okay do the derivative of that
00:16:51.360 --> 00:16:55.840
how did we do the derivative of e to the tan inverse of square root of x
00:16:59.600 --> 00:17:01.360
well we just leave it alone
00:17:01.360 --> 00:17:03.700
its e to the tan inverse square root of x
00:17:05.220 --> 00:17:08.900
because when you do the derivative of e to the something
00:17:09.280 --> 00:17:11.200
that term always stays there
00:17:12.260 --> 00:17:13.860
then we have to multiply
00:17:13.860 --> 00:17:16.980
by the derivative of the power because of the chain rule
00:17:17.440 --> 00:17:20.240
so whats the derivative of tan inverse of x
00:17:21.000 --> 00:17:21.500
its 1
00:17:22.780 --> 00:17:23.500
over 1 plus
00:17:24.160 --> 00:17:25.760
square root of x squared
00:17:26.580 --> 00:17:27.540
which is just x
00:17:28.420 --> 00:17:31.300
times the derivative of the square root of x
00:17:32.500 --> 00:17:33.700
which we memorize
00:17:34.860 --> 00:17:36.700
is 1/2 to the square root of x
00:17:38.540 --> 00:17:40.460
okay do you remmeber that one
00:17:40.880 --> 00:17:45.680
ill put that down, thats one of the ones i keep telling you guys o memorize
00:17:47.500 --> 00:17:48.220
very handy
00:17:48.560 --> 00:17:51.520
you should know where it comes from of course
00:17:52.220 --> 00:17:53.900
if y is the square root of x
00:17:57.140 --> 00:17:59.620
the derivative is 1/ 2 square root of x
00:18:00.500 --> 00:18:03.140
that comes from writing it y=x to the 1/2
00:18:03.140 --> 00:18:04.720
we are taking the derivative
00:18:06.100 --> 00:18:09.060
if you dont have it memorized i suggest you do
00:18:10.020 --> 00:18:11.300
same with y= 1 over x
00:18:13.260 --> 00:18:14.300
the derivative
00:18:15.700 --> 00:18:16.820
-1 over x squared
00:18:16.820 --> 00:18:18.760
why do i tell you to remmeber these
00:18:18.760 --> 00:18:20.160
youre going to see them a lot
00:18:20.160 --> 00:18:22.620
and these are the kinds of things we ask you
00:18:23.100 --> 00:18:25.420
to simplify or set things equal to 0
00:18:27.520 --> 00:18:31.280
the square root cancels so if you wanted to simplify this
00:18:32.600 --> 00:18:33.800
not a terrible idea
00:18:36.900 --> 00:18:38.180
this would be 1/1+x
00:18:40.240 --> 00:18:41.360
times 1/2 square root of x
00:18:41.360 --> 00:18:46.780
and if you wanted to show off you could put these to terms together but you dont need to
00:18:58.880 --> 00:19:00.000
the question was
00:19:00.120 --> 00:19:02.280
you are doing the chain rule okay
00:19:02.280 --> 00:19:04.500
this is the chain rule, the first step
00:19:04.680 --> 00:19:07.240
doing e to a function the derivative is
00:19:07.240 --> 00:19:09.900
e to the function, thats the derivative to the e part
00:19:09.900 --> 00:19:12.200
thats the first outter most function
00:19:12.200 --> 00:19:15.320
the second function you have is tan inverse of x
00:19:15.940 --> 00:19:18.740
that 1 over, 1+ the square root of x squared
00:19:18.740 --> 00:19:20.660
then you have to do the derivative of the square root of x
00:19:21.000 --> 00:19:22.280
which is 1/2 to the x
00:19:22.560 --> 00:19:24.240
so thats your three parts
00:19:25.620 --> 00:19:27.700
does that answer that question
00:19:28.180 --> 00:19:30.020
lets do something similar
00:19:58.780 --> 00:20:00.940
lets take the derivative of that
00:20:04.680 --> 00:20:07.160
how do we do derivative of natural log
00:20:07.160 --> 00:20:09.880
you guys know the derivative of natural log
00:20:10.480 --> 00:20:12.160
thats all we did on monday
00:20:13.500 --> 00:20:15.020
and everybody was here
00:20:16.400 --> 00:20:17.120
so lets see
00:20:20.540 --> 00:20:21.820
y is the natural log
00:20:24.540 --> 00:20:26.220
and the derivative is 1/x
00:20:27.400 --> 00:20:28.200
and you get y
00:20:29.080 --> 00:20:30.840
is the log of a function of x
00:20:35.140 --> 00:20:36.580
and the derivative is
00:20:37.740 --> 00:20:42.780
you make a fraction with the function on the bottoma nd the derivative on top
00:20:46.400 --> 00:20:47.840
i hope you can see that
00:20:47.940 --> 00:20:50.340
thats the derivative of natural log
00:20:50.380 --> 00:20:51.740
we did that on monday
00:20:51.740 --> 00:20:53.280
and i know how much you pay attention
00:20:56.260 --> 00:20:58.740
here we put the function on the bottom
00:21:04.300 --> 00:21:06.940
and whats the derivative of inverse sin
00:21:08.780 --> 00:21:09.280
1 over
00:21:09.720 --> 00:21:11.720
the square root of this thingy
00:21:14.080 --> 00:21:14.640
squared
00:21:16.380 --> 00:21:18.140
times the derivative of 4x
00:21:18.700 --> 00:21:21.500
which is 4, you put the 4 where ever you want
00:21:22.860 --> 00:21:24.940
thats it, thats all you got to do
00:21:24.940 --> 00:21:26.740
now of course you can simplify that
00:21:26.920 --> 00:21:28.200
but you dont need to
00:21:32.580 --> 00:21:34.260
that simplifies to 4 over
00:21:35.160 --> 00:21:36.120
sin inverse 4x
00:21:37.520 --> 00:21:38.800
square root of 1-4x
00:21:40.380 --> 00:21:40.940
squared
00:21:42.380 --> 00:21:44.860
but you certainly dont need to do that
00:21:45.700 --> 00:21:47.460
you can just leave it there
00:21:49.260 --> 00:21:51.260
that is chain rule so once agin
00:21:51.260 --> 00:21:54.200
the derivative of natural log, the function on the bottom
00:21:54.200 --> 00:21:57.320
derivative on top, the derivative of inverse sin 4x
00:21:59.120 --> 00:22:00.560
1/ square root of 1-4x
00:22:00.860 --> 00:22:02.060
quantity squared
00:22:02.060 --> 00:22:03.740
times the derivatie of 4x
00:22:03.740 --> 00:22:05.820
which is 4, you also put the 4 up here
00:22:07.140 --> 00:22:08.580
you put the 4 above the
00:22:08.580 --> 00:22:10.460
fraction right there it doesnt matter
00:22:12.340 --> 00:22:13.620
did you get that one
00:22:16.260 --> 00:22:17.780
one more thing to learn
00:22:18.280 --> 00:22:20.040
well do some more practice
00:22:45.700 --> 00:22:47.220
why is the radical here
00:22:50.200 --> 00:22:54.200
remember you said, function on the bottom derivative on top
00:22:54.200 --> 00:22:56.120
that should say the derivative of the bottom okay
00:22:56.120 --> 00:22:57.500
its one over the function
00:22:57.500 --> 00:22:59.600
ties the derivative of the function
00:22:59.600 --> 00:23:00.900
this goes in the numerator
00:23:02.760 --> 00:23:04.920
however since that is a fraction
00:23:04.920 --> 00:23:08.780
if you want to do the algebra you can make it like this which is much easier
00:23:12.340 --> 00:23:12.840
the 4
00:23:13.820 --> 00:23:15.820
would also be up here like that
00:23:16.560 --> 00:23:17.840
or it could be there
00:23:17.840 --> 00:23:19.420
the reason that this is true
00:23:19.960 --> 00:23:20.840
this is over 1
00:23:21.320 --> 00:23:23.000
so if you flip it multiply
00:23:23.060 --> 00:23:25.460
this is in the denominator with this
00:23:25.460 --> 00:23:26.820
this is up in the neminator
00:23:53.160 --> 00:23:55.320
its confusing because remember
00:23:55.320 --> 00:23:57.720
the inverse sin has a radical for the whole thing
00:23:57.720 --> 00:24:01.300
inverse tan doesnt have a radical at all, you take the radical out
00:24:11.620 --> 00:24:12.500
okay suppose
00:24:12.880 --> 00:24:15.360
instead of asking something fun like
00:24:19.460 --> 00:24:19.960
y=x
00:24:19.960 --> 00:24:23.120
squared what if i asked you to find the derivative y=x
00:24:24.300 --> 00:24:25.020
to the sinx
00:24:25.020 --> 00:24:28.520
you know you go to be kidding me who would want to do that
00:24:29.580 --> 00:24:30.220
you would
00:24:31.300 --> 00:24:34.500
it is not bring the sinx in front and reduce it by 1
00:24:34.980 --> 00:24:36.420
it would be way to easy
00:24:37.680 --> 00:24:39.120
so how would we do that
00:24:41.980 --> 00:24:42.620
any ideas
00:24:42.620 --> 00:24:44.760
people who took calculus before
00:24:45.500 --> 00:24:48.860
here is what you could do, take the log of both sides
00:24:57.260 --> 00:24:58.940
natural log of both sides
00:25:02.040 --> 00:25:04.200
then we could use the rules of log
00:25:04.720 --> 00:25:06.320
bring the power in front
00:25:07.400 --> 00:25:09.640
and you get natural log of y is sinx
00:25:12.080 --> 00:25:12.580
log x
00:25:18.640 --> 00:25:21.360
this is called logarithmic diffraction
00:25:21.360 --> 00:25:22.800
take the log of both sides
00:25:22.880 --> 00:25:24.560
then use the rules of logs
00:25:25.140 --> 00:25:26.260
put sinx in front
00:25:26.800 --> 00:25:27.760
remember that
00:25:29.340 --> 00:25:30.060
log of a of b
00:25:30.760 --> 00:25:32.120
is b times the log of a
00:25:32.780 --> 00:25:34.780
remember that rule, now you do
00:25:35.540 --> 00:25:39.460
so now you can take the derivative because this is calculus
00:25:39.680 --> 00:25:43.280
so whats the derivative of natural log of y, this is 1/y
00:25:44.680 --> 00:25:45.180
dy/dx
00:25:45.920 --> 00:25:46.640
implicite
00:25:47.580 --> 00:25:49.660
right the natural log of y is 1/y
00:25:49.920 --> 00:25:53.600
and now we have to use the product rule for the other side
00:25:56.520 --> 00:25:57.020
sinx
00:25:59.040 --> 00:26:01.120
derivative of log x which is 1/x
00:26:02.900 --> 00:26:03.400
plus
00:26:04.540 --> 00:26:05.040
log x
00:26:07.340 --> 00:26:07.840
cosx
00:26:13.660 --> 00:26:14.620
not quite done
00:26:15.940 --> 00:26:17.620
now we need to find, dy/dx
00:26:19.440 --> 00:26:22.000
we have 1/y times dy/dx so multiply 1/y
00:26:23.040 --> 00:26:23.920
you get dy/dx
00:26:26.460 --> 00:26:27.420
equals y times
00:26:29.820 --> 00:26:30.940
six/1x or sinx/x
00:26:33.400 --> 00:26:34.360
plus logxcosx
00:26:42.040 --> 00:26:43.960
and what is y, y is x to the sin x
00:27:01.120 --> 00:27:03.440
so it repeats as i see puzzled faces
00:27:05.840 --> 00:27:08.640
is y=x to the sinx how do i know how to do this
00:27:08.640 --> 00:27:11.320
because i have a function raised to a function
00:27:12.500 --> 00:27:15.060
okay so i cant use the rules i had before
00:27:15.380 --> 00:27:17.620
take the natural log of both sides
00:27:18.740 --> 00:27:21.860
and use the power rule, the rule of logarithmic
00:27:22.340 --> 00:27:22.840
powers
00:27:22.840 --> 00:27:24.420
to bring the sinx in front
00:27:24.700 --> 00:27:25.820
so now i have this
00:27:26.980 --> 00:27:28.020
natural log of y
00:27:28.820 --> 00:27:29.860
is sinx times log
00:27:29.860 --> 00:27:30.360
x
00:27:31.460 --> 00:27:32.480
take the derivative of both sides
00:27:32.920 --> 00:27:35.160
the derivative of lny is 1/y dy/dx
00:27:36.360 --> 00:27:38.280
and the derivative of (sinx)(lnx) is
00:27:38.480 --> 00:27:41.440
sinx times the derivative of log which is 1/x
00:27:43.060 --> 00:27:43.700
plus log x
00:27:43.700 --> 00:27:45.700
times the derivative of sinx which is cosx
00:27:46.280 --> 00:27:50.040
and now we do some rearranging, put the y on the other side
00:27:50.960 --> 00:27:52.960
and replace y with x to the sinx
00:27:58.700 --> 00:28:00.540
lets do another one of those
00:28:09.380 --> 00:28:10.020
how about
00:28:11.920 --> 00:28:12.480
y equals
00:28:14.740 --> 00:28:15.780
x to the e to the x
00:28:19.280 --> 00:28:21.280
so we have e, y=x to the e to the x
00:28:21.280 --> 00:28:26.620
so if you see that you know you need to use logarithmic diff. take the log of both sides
00:28:27.500 --> 00:28:28.540
natural log of y
00:28:29.080 --> 00:28:30.120
natural log of x
00:28:31.560 --> 00:28:32.520
to the e to the x
00:28:33.400 --> 00:28:34.680
things dont cancel
00:28:35.120 --> 00:28:36.240
its not that easy
00:28:37.880 --> 00:28:39.960
you can put the e to the x in front
00:28:40.240 --> 00:28:41.440
and you get log of y
00:28:42.240 --> 00:28:42.960
is e to the x
00:28:44.020 --> 00:28:44.520
lnx
00:28:45.640 --> 00:28:47.640
those e's and logs dont cancel
00:28:47.640 --> 00:28:51.080
i know you want them to because they would make life so great but they dont
00:28:52.780 --> 00:28:56.700
alright so i take the derivative of the left side so i get 1/y
00:28:59.040 --> 00:28:59.540
dy/dx
00:29:01.400 --> 00:29:04.520
and now lets do the derivative of the other side
00:29:05.160 --> 00:29:06.280
you have e to the x
00:29:08.660 --> 00:29:10.340
natural log of log x is 1/x
00:29:14.760 --> 00:29:16.120
plus natural log of x
00:29:16.600 --> 00:29:19.000
times e to the x which is just e to the x
00:29:21.300 --> 00:29:22.180
so far so good
00:29:22.180 --> 00:29:24.060
alright now you just do rearranging
00:29:24.520 --> 00:29:26.440
technically this is correct
00:29:26.700 --> 00:29:29.740
if you want to put the y on the other side, dy/dx
00:29:33.640 --> 00:29:34.200
y times e
00:29:35.140 --> 00:29:36.020
to the x over x
00:29:37.020 --> 00:29:37.660
plus log x
00:29:39.500 --> 00:29:40.060
e to the x
00:29:41.760 --> 00:29:43.120
notice no canceling
00:29:43.400 --> 00:29:44.520
thats not the log
00:29:44.980 --> 00:29:45.700
of e to the x
00:29:45.960 --> 00:29:47.800
thats log of x times e to the x
00:29:49.100 --> 00:29:51.580
now you replace y with x to the e to the x
00:30:06.740 --> 00:30:07.780
lets do one more
00:30:07.940 --> 00:30:09.620
make sure everyone has it
00:30:15.880 --> 00:30:19.400
oh if you forget to replace y, if you leave it like this
00:30:20.260 --> 00:30:21.540
i would not take off
00:30:21.960 --> 00:30:23.480
ill check with the TA's
00:30:23.480 --> 00:30:25.300
i think thats a correct answer
00:30:25.300 --> 00:30:27.680
but you should know thats nearly the correct answer
00:30:27.680 --> 00:30:30.740
but thats okay i dont think theres a big difference
00:30:30.740 --> 00:30:33.440
but ill talk to professor sutherland to see
00:30:34.940 --> 00:30:39.340
alright lets give you one more of these to make sure you get the idea
00:30:45.220 --> 00:30:46.500
how bout y=x to the x
00:30:47.900 --> 00:30:49.820
you take the log of both sides
00:30:51.400 --> 00:30:51.900
log of y
00:30:54.460 --> 00:30:55.660
the log of x to the x
00:30:58.400 --> 00:30:59.680
bring the x in front
00:31:05.840 --> 00:31:07.040
and you get y= xlnx
00:31:10.160 --> 00:31:10.660
1/y
00:31:11.880 --> 00:31:14.360
dy/dx that left side is kind of boring
00:31:16.240 --> 00:31:18.160
the right side, product rule
00:31:19.060 --> 00:31:21.860
x times the derivative of log x which is 1/x
00:31:23.620 --> 00:31:25.940
plus 1 times the derivative of log x
00:31:28.600 --> 00:31:30.440
now we are going to simplify
00:31:34.800 --> 00:31:36.880
so put the y on the other side and
00:31:39.040 --> 00:31:40.000
x times 1/x is 1
00:31:44.040 --> 00:31:45.160
why would i have 1
00:31:46.220 --> 00:31:47.740
because x times 1/x is 1
00:31:51.840 --> 00:31:52.960
and replace the y
00:31:54.680 --> 00:31:55.320
dx to the x
00:31:55.780 --> 00:31:56.820
you get x to the x
00:31:57.400 --> 00:31:58.520
times 1 plus log x
00:32:04.360 --> 00:32:08.440
i would put something like that on the exam, maybe i wouldnt do
00:32:09.860 --> 00:32:10.900
x to the x to the x
00:32:16.020 --> 00:32:19.220
im not writing the exam professor sutherland is
00:32:21.680 --> 00:32:24.000
other stuff you should be able to do
00:33:30.320 --> 00:33:31.520
alright i give you
00:33:34.800 --> 00:33:37.040
f(x)=2x cubed +9x squared+12x-4
00:33:37.120 --> 00:33:40.240
where does f of x have a horizontal tangent line
00:33:41.000 --> 00:33:43.960
what does that mean horizontal tangent line
00:33:43.960 --> 00:33:47.780
that means the tangent line is horizontal what does the slope mean when its horizontal
00:33:48.740 --> 00:33:51.620
0, how do you find the slope of a tangent line
00:33:51.620 --> 00:33:54.000
take the derivative because what class is this
00:33:54.040 --> 00:33:56.360
calculus do you know anything else
00:33:56.360 --> 00:33:59.420
you know only one thing, you only got one thing in your tool box
00:33:59.420 --> 00:34:02.140
take the derivative, if you dont know what to do
00:34:02.140 --> 00:34:04.220
take the derivative thats what youre here for
00:34:04.220 --> 00:34:06.380
lets do this, lets take the derivative
00:34:06.380 --> 00:34:09.040
find the horizontal tnagent line and set it to 0
00:34:13.940 --> 00:34:16.580
thats an easy derivative its 6x squared
00:34:19.920 --> 00:34:20.420
+18x
00:34:22.260 --> 00:34:22.760
+12
00:34:24.640 --> 00:34:26.400
so we can set that equal to 0
00:34:28.700 --> 00:34:30.060
and you get x squared
00:34:32.600 --> 00:34:33.560
+3x+2 equals 0
00:34:34.200 --> 00:34:35.320
if you divide by 6
00:34:38.180 --> 00:34:40.820
you couldve used the quadratic formula
00:34:40.820 --> 00:34:44.980
or if you were clear you would of divided everything by 6 first and throw the 6 out
00:34:45.600 --> 00:34:47.600
and this factors quite nicely
00:34:48.700 --> 00:34:49.200
x plus 1
00:34:50.540 --> 00:34:51.420
times x plus 2
00:34:52.780 --> 00:34:53.340
equals 0
00:34:56.620 --> 00:34:57.500
so thats x=-1
00:34:59.300 --> 00:34:59.800
x=-2
00:35:01.600 --> 00:35:04.000
we could also ask you something like
00:35:04.000 --> 00:35:07.380
where is the function increasing and where is the function decreasing
00:35:08.440 --> 00:35:09.720
so what you would do
00:35:10.700 --> 00:35:11.900
make a number line
00:35:13.220 --> 00:35:14.340
and put on the 0's
00:35:16.380 --> 00:35:18.300
this would help us figure out
00:35:18.700 --> 00:35:20.940
where the derivative is positive
00:35:20.940 --> 00:35:22.780
positive means the function is increasing
00:35:22.780 --> 00:35:25.140
where the derivative is negative decreasing
00:35:26.180 --> 00:35:28.660
so you make a number line, put on the 0s
00:35:30.120 --> 00:35:31.400
and now pick a value
00:35:31.640 --> 00:35:33.640
in each part of the number line
00:35:33.680 --> 00:35:35.280
pick a value less than -2
00:35:35.880 --> 00:35:37.560
a value between -1 and -12
00:35:37.560 --> 00:35:39.120
and a value greater than -1
00:35:39.400 --> 00:35:41.240
plug it into the derivative
00:35:41.240 --> 00:35:43.600
and see if you got a positive of negative value
00:35:43.780 --> 00:35:46.740
remember we did this in precalc for graphing
00:35:48.180 --> 00:35:49.140
say we tried -3
00:35:50.660 --> 00:35:51.220
-3+1=_2
00:35:53.060 --> 00:35:53.620
-3+2=-1
00:35:53.620 --> 00:35:56.120
a negative times a negative is a positive
00:35:56.200 --> 00:36:00.920
the function is positive there, that means its increasing and going up
00:36:02.800 --> 00:36:03.600
increasing
00:36:04.240 --> 00:36:05.840
from negative infinity
00:36:07.320 --> 00:36:07.820
to -3
00:36:10.280 --> 00:36:13.480
now you pick a number between -2 and -1, like -1/2
00:36:15.220 --> 00:36:17.140
-1/2 +1=-1/2 thats negative
00:36:19.680 --> 00:36:20.560
-1/2+2=+1/2
00:36:20.560 --> 00:36:21.400
so thats positive
00:36:21.400 --> 00:36:23.480
negative times a positive is a negative
00:36:23.480 --> 00:36:24.760
so its negative in this region
00:36:26.480 --> 00:36:29.520
so that means the function is going down there
00:36:31.540 --> 00:36:32.340
decreasing
00:36:36.740 --> 00:36:37.240
from -2
00:36:38.580 --> 00:36:39.080
to -1
00:36:42.360 --> 00:36:44.120
so if you were graphing thi
00:36:44.480 --> 00:36:45.760
the grap is going up
00:36:45.760 --> 00:36:47.600
then its coming down so it has a maximum
00:36:48.360 --> 00:36:52.040
now its going down, what about the right of -1, how bout 0
00:36:52.680 --> 00:36:54.920
0 you get 1 times which is positive
00:36:54.940 --> 00:36:58.060
positive is the function is going back up again
00:36:58.760 --> 00:37:00.280
so its also increasing
00:37:00.780 --> 00:37:04.060
when the value is greater than -1, so we can ask you
00:37:04.060 --> 00:37:06.440
where is the horizontal tangent line
00:37:06.440 --> 00:37:10.680
where is the function increasing or decreasing, lets do another one
00:37:12.580 --> 00:37:14.660
one is never enough to practice
00:37:43.920 --> 00:37:45.200
suppose we had that
00:37:47.000 --> 00:37:50.120
once again wheres the horizontal tangent line
00:37:50.120 --> 00:37:52.120
wheres the function increasing
00:37:52.120 --> 00:37:54.000
wheres the function decreasing
00:37:55.380 --> 00:37:56.740
take the derivative
00:37:59.700 --> 00:38:01.140
and you get 3x squared
00:38:02.320 --> 00:38:02.880
times 6x
00:38:04.300 --> 00:38:04.800
-24
00:38:06.120 --> 00:38:06.840
if this is a
00:38:07.460 --> 00:38:08.500
6 point problem
00:38:08.500 --> 00:38:10.400
yoju earned yourself a couple of points
00:38:11.860 --> 00:38:13.300
now we set it equal to 0
00:38:19.080 --> 00:38:20.440
we could factor that
00:38:20.440 --> 00:38:23.120
but if youre clever youll first divide it by 3
00:38:27.700 --> 00:38:29.460
and then youll factor that
00:38:30.280 --> 00:38:31.480
that factors into
00:38:34.100 --> 00:38:34.600
x-4
00:38:36.740 --> 00:38:37.240
x+2=0
00:38:38.180 --> 00:38:40.980
so you would get horizontal tangent lines
00:38:47.520 --> 00:38:48.020
at x=4
00:38:49.900 --> 00:38:50.400
x=-2
00:38:51.680 --> 00:38:56.880
so if the question says where does the function have a horizontal tangent line
00:38:59.520 --> 00:39:00.800
you stop at =4, x=-2
00:39:00.800 --> 00:39:03.520
what if we said where is the function increasing and decreasing
00:39:04.540 --> 00:39:05.040
now
00:39:06.820 --> 00:39:07.940
you would use our
00:39:09.120 --> 00:39:12.080
number line test, this is called sign testing
00:39:14.560 --> 00:39:15.440
sign not sine
00:39:19.140 --> 00:39:20.820
pick a number less than -2
00:39:21.540 --> 00:39:24.100
and test it in the derivative so like -3
00:39:24.260 --> 00:39:26.180
pick a number between -2 and 4
00:39:26.600 --> 00:39:27.100
like 0
00:39:27.180 --> 00:39:29.180
pick a number greater than 4, 5
00:39:29.180 --> 00:39:32.980
okay you are going to plug those into the derivative and see if you get a positive
00:39:33.180 --> 00:39:34.460
or negative answer
00:39:34.700 --> 00:39:35.980
you take negative 3
00:39:37.000 --> 00:39:39.960
you can plug it in anywhere in the derivative
00:39:40.620 --> 00:39:42.700
but its easiest when youre here
00:39:44.820 --> 00:39:45.380
-3-4=-7
00:39:47.600 --> 00:39:48.160
-3+2=-1
00:39:48.160 --> 00:39:51.160
a negative times a negative will give you a positive
00:39:51.540 --> 00:39:53.780
the function is increasing there
00:39:55.740 --> 00:39:58.140
pick a number between -2 and 4 i have 0
00:39:59.640 --> 00:40:03.160
plug in the derivative -2 times 2 is a negative number
00:40:03.220 --> 00:40:05.540
so your function is negative there
00:40:07.640 --> 00:40:09.720
then pick a value greater than 4
00:40:10.360 --> 00:40:10.860
like 5
00:40:10.860 --> 00:40:11.880
pick any number you want
00:40:12.260 --> 00:40:15.300
plug it in, you get a positive times a positive
00:40:16.140 --> 00:40:17.500
do its positive here
00:40:19.160 --> 00:40:21.640
so the function is increasing either
00:40:22.660 --> 00:40:24.020
when x is less than -2
00:40:26.320 --> 00:40:28.000
or when x is greater than 4
00:40:30.740 --> 00:40:32.820
and the function is decreasing
00:40:34.840 --> 00:40:36.440
between negative 2 and 4
00:40:36.920 --> 00:40:39.320
notice gonna use interval notation
00:40:39.940 --> 00:40:41.940
you do not use square brackets
00:40:42.960 --> 00:40:46.080
it is not dreasing or increasing at an end point
00:40:46.080 --> 00:40:48.620
its increasing or decreasing in the middle
00:40:48.620 --> 00:40:52.140
lets do one more of these, lets make one slightly more annoying
00:41:17.460 --> 00:41:19.300
suppose you had f of x=xsinx
00:41:27.520 --> 00:41:30.640
can you do that one, hang on that might be to hard
00:41:31.640 --> 00:41:33.560
to hard without a calculator
00:41:36.600 --> 00:41:37.560
no you can do it
00:41:47.340 --> 00:41:48.220
thats to ahrd
00:41:50.820 --> 00:41:53.220
you cant do that without a caculator
00:42:05.660 --> 00:42:07.340
lets take the derivative
00:42:08.120 --> 00:42:09.320
the derivative is
00:42:12.480 --> 00:42:14.880
e to the x times the derivative of sin
00:42:16.480 --> 00:42:17.280
which is cos
00:42:19.580 --> 00:42:22.700
plus the derivative e to the x, which is e to the x
00:42:23.980 --> 00:42:24.480
sinx
00:42:26.640 --> 00:42:30.080
thats a product rule right, so its e to the x times sin
00:42:30.080 --> 00:42:32.620
plus sinx times derivative of e to the x
00:42:32.620 --> 00:42:33.740
which is e to the x
00:42:34.040 --> 00:42:35.240
set that equal to 0
00:42:35.960 --> 00:42:38.920
and notice you can subtract your e to the x out
00:42:39.200 --> 00:42:40.480
and you get e to the x
00:42:41.500 --> 00:42:42.220
times cos x
00:42:43.500 --> 00:42:44.140
plus sinx
00:42:47.760 --> 00:42:49.760
so theres two possibiulities
00:42:51.240 --> 00:42:52.440
either e to the x =0
00:42:55.760 --> 00:42:57.200
or cosx +sinx equals 0
00:43:00.540 --> 00:43:02.300
where is e to the x equal to 0
00:43:05.240 --> 00:43:06.440
e to the x is never 0
00:43:07.900 --> 00:43:09.260
its always positive
00:43:10.880 --> 00:43:11.600
its never 0
00:43:12.040 --> 00:43:13.480
just pay no attention
00:43:15.640 --> 00:43:16.680
how often is it 0
00:43:17.800 --> 00:43:19.080
never good alright
00:43:20.800 --> 00:43:22.000
firgure the graph
00:43:23.720 --> 00:43:24.600
cosx+sinx=0
00:43:25.920 --> 00:43:26.640
where sinx
00:43:28.640 --> 00:43:29.920
is negative of cosx
00:43:29.920 --> 00:43:32.160
of course you know this from the unit circle
00:43:32.300 --> 00:43:34.620
or if you were to divided 2 from cosx
00:43:35.200 --> 00:43:36.880
you would get tangent of x
00:43:39.220 --> 00:43:41.380
is -1, you can solve it either way
00:43:43.060 --> 00:43:44.820
where is sinx equal to cosx
00:43:45.660 --> 00:43:46.860
45 degrees or pi/4
00:43:47.160 --> 00:43:49.080
its equal to negative cosinx
00:43:50.000 --> 00:43:50.960
at 135 degrees
00:43:51.700 --> 00:43:52.820
fourth quadrant
00:43:54.820 --> 00:43:57.700
ASTC, we all know what that stands for right
00:44:00.680 --> 00:44:02.200
all students take crap
00:44:06.360 --> 00:44:10.200
its going to be in the second quadrant and the 4th quadrant
00:44:10.240 --> 00:44:12.000
its either going to be here
00:44:13.280 --> 00:44:13.780
or here
00:44:17.580 --> 00:44:19.580
so pi/4 in the second quadrant
00:44:21.740 --> 00:44:22.300
is 3pi/4
00:44:23.760 --> 00:44:25.520
and in the fourth quadrant
00:44:27.000 --> 00:44:27.560
is 7pi/4
00:44:30.520 --> 00:44:33.320
you should not write the answer in degrees
00:44:34.100 --> 00:44:36.340
i think we maybe would not take off
00:44:36.500 --> 00:44:38.900
but in calc everything is in radians
00:44:39.420 --> 00:44:40.860
nothing is in degrees
00:44:41.460 --> 00:44:46.020
alright ill see some of you on sunday, ill see the rest of you on monday