WEBVTT
Kind: captions
Language: en
00:00:10.880 --> 00:00:12.640
are we recording this now?
00:00:17.860 --> 00:00:21.380
theres no seats, wait there some seats in the middle.
00:01:17.340 --> 00:01:17.980
alright.
00:01:17.980 --> 00:01:20.160
give you a couple minutes see if you can do that.
00:01:24.580 --> 00:01:25.220
this one.
00:01:26.520 --> 00:01:30.760
we wouldnt give you numbers this hard on the test..thats not me.
00:01:37.280 --> 00:01:39.220
just in case you should know how to set it up
00:01:45.440 --> 00:01:46.960
there you go. alright.
00:01:47.940 --> 00:01:49.100
there were 2 of these.
00:01:49.600 --> 00:01:51.260
and they were both-so now we have 1.
00:01:52.340 --> 00:01:53.920
it works. alright.
00:01:54.920 --> 00:01:55.480
for now.
00:01:59.140 --> 00:02:03.660
we wouldnt actually give you numbers this difficult to work out but the numbers we give you were going to expect you to be able to do.
00:02:04.500 --> 00:02:07.380
so first you want to approximate the area from 2 to 10.
00:02:07.940 --> 00:02:10.720
so 5x^3+1 is just one of these sort of
00:02:11.100 --> 00:02:13.660
upward tilting, polynomial thingys.
00:02:14.720 --> 00:02:15.220
k.
00:02:15.760 --> 00:02:17.200
and you go from 2 to 10.
00:02:19.260 --> 00:02:23.080
i recommend you draw a picture because you might get a point for that i dont know.
00:02:25.740 --> 00:02:29.120
ill put it this way if you draw a picture and you can see where you went wrong
00:02:29.120 --> 00:02:32.520
its better than if you sort of just write stuff down.
00:02:33.160 --> 00:02:36.780
okay the idea of partial credit is that we can see where you made your mistake
00:02:36.780 --> 00:02:38.440
if we think your mistakes not a big one
00:02:38.440 --> 00:02:39.720
you get most of the credit.
00:02:40.440 --> 00:02:44.800
if we cant see what you did if you just scribble stuff you dont get partial credit for scribble.
00:02:45.760 --> 00:02:47.680
so the right hand rectangles
00:02:49.180 --> 00:02:50.460
would look like this.
00:02:51.100 --> 00:02:52.380
theyre going to overestimate.
00:02:53.420 --> 00:02:56.300
and lets find the area of each of those, well
00:02:56.940 --> 00:02:58.580
each of them is 2y.
00:03:00.840 --> 00:03:02.600
and then you would do f(4)
00:03:03.760 --> 00:03:04.620
+ f(6)
00:03:06.300 --> 00:03:07.260
+ f(8)
00:03:08.620 --> 00:03:09.520
+ f(n).
00:03:12.600 --> 00:03:13.100
okay.
00:03:13.660 --> 00:03:16.820
what is f(4)? its 5(4)^3
00:03:18.840 --> 00:03:19.560
+1.
00:03:20.940 --> 00:03:23.500
then 5(6)^3
00:03:24.800 --> 00:03:25.300
+1.
00:03:26.740 --> 00:03:28.340
5(8)^3
00:03:29.820 --> 00:03:30.320
+1.
00:03:31.440 --> 00:03:33.640
and 5(10)^3
00:03:35.700 --> 00:03:36.200
+1.
00:03:37.480 --> 00:03:37.980
that is
00:03:39.280 --> 00:03:42.140
2 times 321
00:03:43.320 --> 00:03:45.400
plus uhh..1,081
00:03:47.320 --> 00:03:50.440
plus uhh 2,561.
00:03:52.260 --> 00:03:53.140
plus 5,0001.
00:03:54.340 --> 00:03:55.460
is that correct?
00:03:56.320 --> 00:03:58.480
alright now i have to add them up.
00:03:59.460 --> 00:04:04.980
uhh lets see 1,402...3,963...8,964 times 2
00:04:07.840 --> 00:04:11.100
alright which is 17,890 nope nope
00:04:11.780 --> 00:04:12.600
17,928.
00:04:15.160 --> 00:04:15.960
that right?
00:04:16.660 --> 00:04:17.160
okay.
00:04:17.740 --> 00:04:18.380
good huh?
00:04:23.940 --> 00:04:25.400
just to show i can actually do it.
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do you want to get that again? ready?
00:04:37.820 --> 00:04:40.640
now how would we write the actual riemann sum?
00:04:40.640 --> 00:04:43.240
he didnt get a chance to do that a lot in the superbowl did he?
00:04:43.240 --> 00:04:45.280
alright im going to erase this.
00:05:20.780 --> 00:05:22.700
so when you do the riemann sum
00:05:22.700 --> 00:05:25.260
you need 2 things you need the width of each of these
00:05:26.040 --> 00:05:26.540
right?
00:05:27.280 --> 00:05:28.700
and then youre going to need
00:05:28.750 --> 00:05:29.500
the f of them.
00:05:29.560 --> 00:05:32.120
so the width while its going from 2 to 10
00:05:32.700 --> 00:05:33.940
so 10-2 is 8.
00:05:36.560 --> 00:05:37.360
this is 8/n.
00:05:40.320 --> 00:05:42.620
and then we have to do f.
00:05:43.160 --> 00:05:43.660
so
00:05:44.880 --> 00:05:47.440
you take the starting point which is 2.
00:05:48.300 --> 00:05:49.340
and then you add
00:05:50.120 --> 00:05:51.980
8/n(i).
00:05:54.840 --> 00:05:57.380
okay thats reimann sum thats all thats involved
00:05:57.740 --> 00:06:00.340
and then youre going to take that and plug it into the equation.
00:06:01.140 --> 00:06:02.340
its going to be 8/n
00:06:03.460 --> 00:06:05.320
sigma i=1 to n
00:06:06.020 --> 00:06:06.520
of
00:06:07.580 --> 00:06:11.140
[2+(8i/n)]^3
00:06:12.140 --> 00:06:12.640
+1.
00:06:13.620 --> 00:06:14.120
and 5.
00:06:14.720 --> 00:06:15.220
ok?
00:06:15.940 --> 00:06:17.240
thats the reimann sum, yes?
00:06:21.600 --> 00:06:23.620
you always start at 1.
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i always goes from 1 to n.
00:06:26.760 --> 00:06:28.920
this is always the left end point
00:06:31.240 --> 00:06:34.800
this is width times i and so this
00:06:35.120 --> 00:06:36.400
is the same as this.
00:06:37.700 --> 00:06:38.200
ok?
00:06:46.040 --> 00:06:48.600
will i get points off if you take 8 out?
00:06:52.260 --> 00:06:54.280
if i leave the 8/n inside the sigma thats fine.
00:06:54.300 --> 00:06:55.580
you could put it either way.
00:06:55.580 --> 00:06:57.840
okay because since theres no i in the term
00:06:57.840 --> 00:07:00.360
you can take it out of the sigma or put it in the sigma.
00:07:04.020 --> 00:07:05.260
umm now
00:07:05.260 --> 00:07:07.660
if you wanted to compute the riemann sum
00:07:07.660 --> 00:07:11.220
you could either do it the hard way or the easy way i recommend the easy way.
00:07:24.020 --> 00:07:26.260
okay so you would find
00:07:26.960 --> 00:07:28.740
the integral from 2 to 10
00:07:29.600 --> 00:07:31.360
5x^3+1
00:07:33.120 --> 00:07:34.440
dx.
00:07:36.380 --> 00:07:38.780
so what is the integral of 5x^3+1 dx?
00:07:39.380 --> 00:07:45.480
well lets see..integrate you get 5x^4/4 +x.
00:07:47.360 --> 00:07:48.160
from 2 to 10.
00:07:49.680 --> 00:07:51.820
plus in 10, plug in 2 and subtract.
00:07:52.800 --> 00:07:54.660
so lets see thats uhh
00:07:55.440 --> 00:07:59.340
12,510
00:08:00.700 --> 00:08:01.260
minus..
00:08:03.480 --> 00:08:03.980
22.
00:08:05.940 --> 00:08:08.420
okay which is 12,488.
00:08:11.380 --> 00:08:12.340
so far so good?
00:08:13.540 --> 00:08:15.760
its not perfect, sorry i'll choose a better one.
00:08:19.000 --> 00:08:19.880
we like that?
00:08:19.880 --> 00:08:21.840
of course as i said you can work it out long hand
00:08:21.840 --> 00:08:24.400
but i cant imagine any of you actually want to do that to yourselves.
00:08:27.200 --> 00:08:27.760
alright
00:08:28.520 --> 00:08:30.580
we all understand riemann sum stuff?
00:08:31.780 --> 00:08:32.280
yes?
00:08:33.380 --> 00:08:34.400
hello yes?
00:08:46.360 --> 00:08:47.800
2 is the left end point
00:08:47.800 --> 00:08:49.400
youre starting from x=2.
00:08:52.700 --> 00:08:56.020
youre taking 2 and first youre moving 8/n to the right of it.
00:08:57.160 --> 00:08:59.640
okay so thats why you start at 2+(8/n)
00:09:00.600 --> 00:09:01.480
because i=1.
00:09:01.860 --> 00:09:04.660
if i was 0 youd be doing the left end points.
00:09:08.120 --> 00:09:10.520
you do not have to if you dont want to.
00:09:22.560 --> 00:09:23.280
why is 8/n?
00:09:24.140 --> 00:09:24.640
well
00:09:24.860 --> 00:09:26.780
how far is it for 2 10s 8
00:09:26.820 --> 00:09:28.640
meaning you cut it into n rectangles
00:09:28.640 --> 00:09:30.700
so the width of each rectangle is 8/n.
00:09:30.700 --> 00:09:33.020
now why is the 8/n in the outside than in the inside?
00:09:33.020 --> 00:09:34.660
it doesnt matter it could be either place.
00:09:35.600 --> 00:09:36.100
ok?
00:09:36.960 --> 00:09:40.320
you can move it out of the sigma and put it in the sum.
00:09:43.620 --> 00:09:45.980
do you need to write the limit? i dont think you need to.
00:09:46.420 --> 00:09:49.060
but if you want to, just in case, write..
00:09:49.660 --> 00:09:50.300
the limit
00:09:51.380 --> 00:09:53.640
n goes to infinity.
00:09:54.460 --> 00:09:54.960
ok?
00:09:57.680 --> 00:09:59.700
the important thing is the sigma not the limit.
00:10:16.980 --> 00:10:18.180
other questions?
00:10:18.600 --> 00:10:21.260
i could do another one of these if you want but theyre time consuming.
00:10:23.420 --> 00:10:25.440
lets do some other stuff we can always come back.
00:10:26.420 --> 00:10:30.500
alright we can do some stuff on the review sheet or i can make up stuff similar to the review sheet?
00:10:32.460 --> 00:10:34.760
new stuff? okay.
00:10:34.760 --> 00:10:36.620
sorry about the minus signs, you know
00:10:36.620 --> 00:10:39.900
thats why i dont punish you guys if you occasionally miss a minus sign.
00:10:40.040 --> 00:10:42.040
i was doing this late at night.
00:10:42.120 --> 00:10:44.520
you know..sitting in the arm chair.
00:10:45.000 --> 00:10:46.040
kind of mellow.
00:10:46.040 --> 00:10:48.700
and i said maybe i should work out some of these.
00:10:49.420 --> 00:10:51.880
and instead of falling asleep i decided to do them.
00:10:52.340 --> 00:10:53.620
this is the result.
00:10:54.700 --> 00:10:55.260
let see.
00:11:07.380 --> 00:11:09.780
b says f(x)=
00:11:10.180 --> 00:11:13.000
e^x+ (4/x)
00:11:14.060 --> 00:11:18.280
and x at 1 =0 and i should say f(1)=0 sorry about that.
00:11:22.600 --> 00:11:23.240
that says
00:11:24.220 --> 00:11:24.720
e^x.
00:11:25.820 --> 00:11:26.320
ok?
00:11:26.740 --> 00:11:28.420
e^x+(4/x)
00:11:30.020 --> 00:11:31.340
f(1)=0
00:11:38.520 --> 00:11:39.720
repeating 1 more.
00:12:00.660 --> 00:12:02.100
alright next one now.
00:12:08.300 --> 00:12:11.520
f'(x) is x^2- radical x+4
00:12:12.380 --> 00:12:13.900
if you want to find f(x)
00:12:14.060 --> 00:12:15.340
we integrate that.
00:12:15.920 --> 00:12:16.960
the integral of
00:12:17.440 --> 00:12:22.640
x^2-radical x +4 dx
00:12:23.560 --> 00:12:25.640
is x^3/3
00:12:28.240 --> 00:12:30.960
what is square root of x? well thats x^1/2
00:12:31.540 --> 00:12:34.100
so this becomes x^3/2
00:12:35.080 --> 00:12:35.640
over 3/2
00:12:37.300 --> 00:12:37.800
+4x
00:12:38.700 --> 00:12:39.820
plus a constant.
00:12:40.000 --> 00:12:43.120
in dividing by 3/2 its like multiplying by 2/3.
00:12:43.580 --> 00:12:44.380
so its x^3/3
00:12:45.360 --> 00:12:49.000
minus (2x^3/2)/3
00:12:50.420 --> 00:12:50.920
+4x
00:12:51.860 --> 00:12:52.360
+c.
00:12:52.960 --> 00:12:54.540
okay thats f(x).
00:12:56.740 --> 00:12:58.740
okay? again, were just integrating.
00:12:59.740 --> 00:13:00.300
and then
00:13:00.980 --> 00:13:02.460
we know that f(1)=8.
00:13:03.340 --> 00:13:05.980
so who knows how that comes out, 8 is 1/3
00:13:06.420 --> 00:13:07.640
minus 2/3
00:13:08.680 --> 00:13:09.180
+4
00:13:10.160 --> 00:13:10.660
+c.
00:13:14.180 --> 00:13:14.900
so lets see
00:13:15.320 --> 00:13:20.300
that is uh, 11/3...8..take it away, 13/3
00:13:21.780 --> 00:13:22.420
equals c.
00:13:25.780 --> 00:13:27.800
soo f(x)
00:13:28.760 --> 00:13:34.140
is (x^3)/3-[(2x^3/2/)3]+4x+13/3.
00:13:35.240 --> 00:13:36.200
so far so good?
00:13:37.020 --> 00:13:38.220
you like that one?
00:13:38.940 --> 00:13:39.580
alright.
00:13:49.100 --> 00:13:49.740
how about
00:13:50.840 --> 00:13:52.840
the second one. so were going to integrate
00:13:53.720 --> 00:13:54.220
e^x..
00:13:55.800 --> 00:13:57.700
+4/x dx
00:13:59.620 --> 00:14:01.380
the integral of that is e^x
00:14:03.920 --> 00:14:04.600
+4...
00:14:06.800 --> 00:14:08.800
ln of the absolute value of x+c
00:14:09.880 --> 00:14:11.560
and we know that f(1) is 0.
00:14:14.360 --> 00:14:15.560
thats f(x) right?
00:14:17.500 --> 00:14:19.320
so 0 is e^1
00:14:20.120 --> 00:14:23.540
+ 4ln(1), ln(1) is 0.
00:14:25.800 --> 00:14:26.840
c is negative e.
00:14:30.340 --> 00:14:31.620
so this would be e^x
00:14:32.100 --> 00:14:34.260
+4lnx-e.
00:14:37.020 --> 00:14:38.300
did i do that wrong?
00:14:40.120 --> 00:14:41.600
e^1 is e, right?
00:14:43.460 --> 00:14:46.200
e^1 is e because anything to the 1 is itself right?
00:14:46.480 --> 00:14:48.820
pi^1 is pi..good.
00:14:48.900 --> 00:14:50.820
We like to put that stuff in.
00:14:52.600 --> 00:14:55.380
You put in pi, immediately 20% won't get the question.
00:14:57.360 --> 00:14:58.000
alright.
00:14:58.880 --> 00:15:01.420
now ive got sinx +2cosx.
00:15:02.620 --> 00:15:03.120
sinx
00:15:04.420 --> 00:15:05.600
+2cosx
00:15:06.960 --> 00:15:07.460
dx
00:15:08.880 --> 00:15:10.080
so f(x) would be..
00:15:11.740 --> 00:15:13.920
the integral of sin is -cos
00:15:15.980 --> 00:15:17.980
the integral of cosine is sine
00:15:20.500 --> 00:15:21.000
+c.
00:15:22.760 --> 00:15:23.900
and f(0) is 1.
00:15:25.940 --> 00:15:28.960
so 1 =....the cos of 0 is 1.
00:15:29.100 --> 00:15:30.360
so this is -1
00:15:31.360 --> 00:15:32.780
+0+c
00:15:32.880 --> 00:15:33.900
so c is 2.
00:15:35.120 --> 00:15:38.320
on the exam dont just write +c rewrite the answer
00:15:39.940 --> 00:15:41.300
so here you would have f(x)
00:15:42.780 --> 00:15:44.040
is -cosx
00:15:45.520 --> 00:15:47.020
+2sinx
00:15:49.260 --> 00:15:49.760
+2.
00:15:50.740 --> 00:15:52.360
dont forget that last step okay?
00:15:52.800 --> 00:15:54.720
here you have to write out e^x
00:15:55.200 --> 00:15:56.860
+4lnx-e.
00:15:58.100 --> 00:15:58.980
and so on, ok?
00:16:01.720 --> 00:16:02.520
questions?
00:16:05.120 --> 00:16:06.720
so far so good, alright.
00:16:07.060 --> 00:16:08.220
lets give you some more stuff.
00:16:08.420 --> 00:16:10.260
let me know when i can erase.
00:16:12.180 --> 00:16:12.820
its good?
00:16:26.220 --> 00:16:27.020
how about..
00:17:04.860 --> 00:17:06.620
and lets make up one more..
00:17:18.800 --> 00:17:20.240
there ya go, theres 3.
00:17:27.100 --> 00:17:28.540
lets do the first one.
00:17:29.120 --> 00:17:36.040
the integral of x^3 square root of 5+x^4 dx
00:17:37.360 --> 00:17:39.080
you look for function and its derivative.
00:17:39.760 --> 00:17:42.960
well the derivative of x^4 is 4x^3 you have an x^3
00:17:43.080 --> 00:17:44.440
thats pretty close.
00:17:45.480 --> 00:17:46.280
so lets let u
00:17:49.960 --> 00:17:53.320
equal 5+x^4.
00:17:56.480 --> 00:17:56.980
then du
00:17:57.580 --> 00:17:58.820
is 4x^3.
00:18:04.240 --> 00:18:05.920
so i look at the integrand
00:18:06.340 --> 00:18:07.640
i dont see 4x^3
00:18:08.540 --> 00:18:09.040
i see..
00:18:10.520 --> 00:18:11.020
x^3.
00:18:11.780 --> 00:18:12.280
so i
00:18:12.740 --> 00:18:14.260
divide both sides by 4.
00:18:15.040 --> 00:18:16.160
and you get 1/4du
00:18:18.380 --> 00:18:19.920
is x^3dx.
00:18:21.300 --> 00:18:23.540
so now i can rewrite that integral
00:18:25.340 --> 00:18:31.140
1/4 the integral of square root of u du.
00:18:33.400 --> 00:18:35.720
now square root is to the 1/2 power.
00:18:36.040 --> 00:18:37.560
so the integral of that
00:18:39.360 --> 00:18:44.680
is (1/4u^3/2)/(3/2) plus a constant.
00:18:46.540 --> 00:18:47.820
flip the fraction
00:18:48.760 --> 00:18:49.960
and you get 1/6...
00:18:52.020 --> 00:18:54.600
u^3/2 is 1/6 of...
00:18:55.400 --> 00:18:57.240
5+x^4
00:18:58.700 --> 00:18:59.200
3/2
00:19:00.200 --> 00:19:00.700
+c.
00:19:00.980 --> 00:19:02.420
howd we do on that one?
00:19:03.580 --> 00:19:04.700
that one is okay?
00:19:05.760 --> 00:19:06.560
not too bad?
00:19:09.960 --> 00:19:11.960
difficult..medium..happy?
00:19:13.700 --> 00:19:15.060
lets do another one.
00:19:16.900 --> 00:19:18.180
the integral from 0
00:19:18.920 --> 00:19:20.240
to pi/2
00:19:21.440 --> 00:19:23.960
xcosx^2
00:19:25.560 --> 00:19:26.060
of dx.
00:19:26.060 --> 00:19:29.220
two ways you can do when you have limits of integration.
00:19:29.220 --> 00:19:32.640
one is after you do substitution you can change the limits
00:19:32.640 --> 00:19:35.240
as well then you have to substitute back.
00:19:35.340 --> 00:19:37.740
or you can sort of ignore the limits.
00:19:37.740 --> 00:19:41.920
when youre done with the integration you substitute back and then you put the limits back.
00:19:41.940 --> 00:19:44.660
so for now thats the only way ive taught it
00:19:44.660 --> 00:19:46.360
so well do that way for the moment.
00:19:47.320 --> 00:19:48.280
i look and i see
00:19:48.720 --> 00:19:51.920
x^2 and an x so im going to let u=
00:19:52.560 --> 00:19:53.320
x^2
00:19:54.280 --> 00:19:56.840
the du would be 2x
00:19:58.080 --> 00:19:58.580
dx.
00:20:01.700 --> 00:20:03.400
so 1/2du
00:20:06.480 --> 00:20:07.500
is xdx.
00:20:10.040 --> 00:20:12.180
so ignoring the limits
00:20:12.340 --> 00:20:14.980
i would just have 1/2
00:20:16.040 --> 00:20:17.440
cosu
00:20:18.300 --> 00:20:18.800
du
00:20:20.080 --> 00:20:20.880
which is 1/2
00:20:22.760 --> 00:20:23.260
sinu
00:20:25.320 --> 00:20:26.120
which is 1/2
00:20:27.800 --> 00:20:29.320
sin(x^2)
00:20:30.280 --> 00:20:32.820
now we go from 0 to pi/2.
00:20:36.300 --> 00:20:36.800
ok?
00:20:40.960 --> 00:20:42.640
that equals 1/2
00:20:45.120 --> 00:20:46.300
sin of..
00:20:47.300 --> 00:20:48.700
pi^2/4
00:20:49.020 --> 00:20:52.780
which i have no idea what that is minus 1/2sin0 which is 0.
00:20:58.560 --> 00:20:59.520
so far so good?
00:20:59.520 --> 00:21:01.380
how did we do on the second one?
00:21:03.420 --> 00:21:04.780
eh? good?
00:21:06.380 --> 00:21:09.740
alright my goal is to send some of you to med school.
00:21:22.560 --> 00:21:24.640
no its not the sin thats squared
00:21:24.680 --> 00:21:26.280
its the x thats squared.
00:21:27.040 --> 00:21:29.040
the pi/2 is squared.
00:21:31.400 --> 00:21:34.240
its not the sin thats squared its the pi/2 thats squared.
00:21:35.160 --> 00:21:36.040
you got that?
00:21:42.560 --> 00:21:44.600
youre supposed to know the unit circle.
00:21:46.320 --> 00:21:49.760
you got a little more than 24 hours to rememorize it.
00:21:50.660 --> 00:21:51.940
its up on the insta.
00:21:53.800 --> 00:21:56.340
you can find it i put it up there about a week ago.
00:21:57.480 --> 00:21:59.240
got a lot of likes for that.
00:21:59.240 --> 00:22:02.100
i mean who puts a unit circle on their instagram and gets likes?
00:22:06.200 --> 00:22:08.220
alright lets do this one.
00:22:09.000 --> 00:22:10.280
i could let u
00:22:11.100 --> 00:22:13.320
equal e^x+5
00:22:13.320 --> 00:22:15.520
why would i let u equal the denominator?
00:22:15.940 --> 00:22:17.860
well if i let u= the numerator
00:22:18.000 --> 00:22:21.600
du is going to end up down here and have like 5 plus du's.
00:22:21.780 --> 00:22:23.380
have no idea how to do it.
00:22:23.820 --> 00:22:26.300
but if i let u=e^x+5
00:22:27.380 --> 00:22:27.880
then du
00:22:29.180 --> 00:22:30.540
is e^xdx
00:22:31.380 --> 00:22:32.820
and this just becomes
00:22:34.100 --> 00:22:34.920
du/u.
00:22:42.380 --> 00:22:45.260
the integral of du/u is just the natural log
00:22:46.980 --> 00:22:48.500
the absolute value of u
00:22:49.180 --> 00:22:50.300
plus a constant.
00:22:51.580 --> 00:22:53.340
so this would be the ln of..
00:22:54.440 --> 00:22:56.260
e^x+5
00:22:57.740 --> 00:22:58.240
+c.
00:22:58.960 --> 00:23:02.820
you dont actually need the absolute value bars because e^x+5 is always positive.
00:23:03.940 --> 00:23:05.860
okay e^x is always positive.
00:23:07.200 --> 00:23:08.880
never 0, never negative.
00:23:10.200 --> 00:23:13.320
alright i had a request for a specific problem.
00:23:23.680 --> 00:23:27.080
this one i was kind of proud of its slightly nasty but thats okay.
00:23:45.860 --> 00:23:46.360
i have
00:23:48.000 --> 00:23:49.140
f(x) is...
00:23:50.600 --> 00:23:53.080
this is number 2c on the review sheet.
00:23:53.560 --> 00:23:54.060
cosx
00:23:55.580 --> 00:23:56.080
sinx
00:23:57.840 --> 00:24:00.260
square root of 1-t^2
00:24:01.940 --> 00:24:02.440
dt
00:24:04.620 --> 00:24:08.300
we have to find f'(3) and f'(5pi/4).
00:24:08.940 --> 00:24:09.580
but first
00:24:10.280 --> 00:24:11.720
lets just find f'(x).
00:24:12.660 --> 00:24:15.360
this again, this is on the review sheet number 2c.
00:24:22.000 --> 00:24:24.000
okay so the derivative of this
00:24:25.720 --> 00:24:28.840
youre going to get the square root of 1-sin^2x.
00:24:30.040 --> 00:24:32.480
you plug in the top function.
00:24:32.480 --> 00:24:34.040
times the derivative of sinx
00:24:34.040 --> 00:24:34.920
which is cosx.
00:24:37.680 --> 00:24:38.240
minus..
00:24:39.260 --> 00:24:42.140
the bottom which is 1-cos^2x
00:24:43.420 --> 00:24:45.680
times the derivative of cos which is -sinx.
00:24:48.280 --> 00:24:52.760
so then you look at this and you panic because you say what am i going to do i dont know how to find sin of 3.
00:24:52.980 --> 00:24:55.620
but remember in integral trigonometry
00:24:55.620 --> 00:24:58.080
you learned all those trig identities for a reason.
00:24:58.080 --> 00:25:00.760
trig identities make your life much much happier.
00:25:02.180 --> 00:25:04.500
so 1-sin^2 is cos^2.
00:25:11.640 --> 00:25:13.360
and 1-cos^2
00:25:14.260 --> 00:25:15.300
is sin^2.
00:25:20.380 --> 00:25:22.340
and whats the square root of cos^2?
00:25:22.860 --> 00:25:24.880
cos. and cos times cos is?
00:25:25.440 --> 00:25:25.940
cos^2.
00:25:29.120 --> 00:25:30.620
square root of sin square root of sin?
00:25:30.840 --> 00:25:32.360
sin times sin is sin^2.
00:25:32.980 --> 00:25:34.660
this becomes plus sin^2
00:25:37.920 --> 00:25:38.940
equals 1.
00:25:38.940 --> 00:25:42.160
it doesnt matter what you plug in youre going to get 1 no matter what you do.
00:25:42.920 --> 00:25:43.980
isnt that annoying?
00:25:46.460 --> 00:25:49.340
i mean you hate problems like that. remember when i told you when i doubt put 0?
00:25:49.340 --> 00:25:50.540
you could also put 1.
00:25:52.920 --> 00:25:56.200
just look at this and write..its obvious..its 1.
00:25:56.200 --> 00:25:56.980
lets move on.
00:25:56.980 --> 00:25:59.420
make a little contempt face on the test.
00:25:59.420 --> 00:26:02.700
how dare you insult me with such a simple problem?
00:26:05.700 --> 00:26:08.020
even a fool could see that this is 1.
00:26:08.960 --> 00:26:10.720
alright lets try something else.
00:26:12.880 --> 00:26:16.720
you guys were able to do everything on the review sheet with my explanations and all that?
00:26:16.800 --> 00:26:19.760
thats pretty good. lets try something then.
00:26:24.660 --> 00:26:26.340
how about that integral?
00:26:26.340 --> 00:26:29.220
this chalk is sort of skipping on the board.
00:26:29.940 --> 00:26:32.020
sin of the cube root of x
00:26:32.180 --> 00:26:33.860
over the cube root of x^2.
00:26:35.360 --> 00:26:35.860
dx.
00:26:37.480 --> 00:26:40.040
dont be intimidated its not that hard.
00:26:40.300 --> 00:26:41.180
you can do it.
00:26:42.160 --> 00:26:43.600
howd we do on this one?
00:26:43.820 --> 00:26:44.860
we ace this one?
00:26:46.040 --> 00:26:47.800
its not hard come on folks.
00:26:49.200 --> 00:26:50.560
actually it is hard.
00:26:50.680 --> 00:26:51.560
take it back.
00:26:51.960 --> 00:26:53.720
thats alright we can do it.
00:26:55.640 --> 00:26:57.400
the cube root of x^2 come on
00:26:57.400 --> 00:27:00.120
thats gotta be somehow related to the cube root of x.
00:27:01.580 --> 00:27:02.080
so
00:27:02.360 --> 00:27:03.880
if u is the cube root of x
00:27:05.120 --> 00:27:07.360
also known as x^1/3
00:27:08.240 --> 00:27:08.920
then du
00:27:09.540 --> 00:27:13.340
would be 1/3x^-2/3.
00:27:14.520 --> 00:27:21.160
which is the same thing as 1/3 times the cube root of x^2.
00:27:23.700 --> 00:27:25.460
gotta have a little faith.
00:27:26.580 --> 00:27:27.940
gotta figure this..
00:27:27.940 --> 00:27:29.540
how do i integrate this?
00:27:30.180 --> 00:27:31.140
so 3du
00:27:31.960 --> 00:27:32.840
oh this is dx.
00:27:34.160 --> 00:27:35.120
would equal dx
00:27:36.940 --> 00:27:39.480
over the cube root of x^2.
00:27:40.640 --> 00:27:41.680
so this would be
00:27:43.440 --> 00:27:45.160
3sinu
00:27:47.060 --> 00:27:47.560
du.
00:27:57.680 --> 00:28:00.440
and the integral of sinu is -cosu
00:28:01.740 --> 00:28:04.740
this is -3cosu + a constant.
00:28:05.240 --> 00:28:08.820
=-3cos(x^3)
00:28:10.860 --> 00:28:11.360
+c.
00:28:15.440 --> 00:28:16.400
so far so good?
00:28:21.280 --> 00:28:23.280
why is the dx the numerator?
00:28:24.500 --> 00:28:25.000
here?
00:28:26.700 --> 00:28:28.540
put the dx in the 1 okay?
00:28:28.900 --> 00:28:30.980
dont be afraid, ask questions.
00:28:38.000 --> 00:28:40.000
oh yeah im sorry you know that.
00:28:43.380 --> 00:28:46.100
i need you guys to proofread my thoughts.
00:28:46.180 --> 00:28:47.300
be very helpful.
00:28:52.640 --> 00:28:53.820
lets give you another question.
00:29:42.160 --> 00:29:43.740
okay so g(x)
00:30:26.320 --> 00:30:26.820
ok.
00:30:27.080 --> 00:30:28.600
heres 3 fun questions.
00:30:33.460 --> 00:30:34.900
thats a minus..
00:30:37.080 --> 00:30:37.580
minus 4.
00:30:54.140 --> 00:30:56.200
i dont know if any of you are watching the videos.
00:30:57.440 --> 00:30:59.520
are you watching these at home?
00:31:24.640 --> 00:31:27.320
its hard to make this stuff up in class-its gotta be 2 right?
00:31:27.320 --> 00:31:29.300
otherwise its not a semicircle.
00:31:30.360 --> 00:31:32.360
did that change your answers?
00:31:33.120 --> 00:31:34.240
want to make it 4?
00:31:36.220 --> 00:31:38.780
it screws up the other part..alright.
00:31:39.620 --> 00:31:40.960
so g(0) well
00:31:41.360 --> 00:31:43.920
the integral from 0 to 0 is just 0 right?
00:31:44.720 --> 00:31:46.380
cause youre not going anywhere.
00:31:46.380 --> 00:31:49.380
any time these 2 numbers are the same you get 0.
00:31:49.380 --> 00:31:50.940
cause you havent integrated anything.
00:31:50.980 --> 00:31:53.460
you start and finish at the same spot.
00:31:53.460 --> 00:31:57.040
when you get the top and when you get the bottom you subtract and get the same answer.
00:31:59.380 --> 00:32:01.140
alright we're going from 0
00:32:01.740 --> 00:32:02.560
to 8.
00:32:03.500 --> 00:32:04.780
thats what g(8) is.
00:32:06.940 --> 00:32:07.440
from 0
00:32:08.500 --> 00:32:09.000
to 8
00:32:09.580 --> 00:32:10.920
of f(t)
00:32:11.400 --> 00:32:11.900
dt
00:32:12.220 --> 00:32:13.580
t is a dummy variable
00:32:13.580 --> 00:32:14.780
doesnt really matter what it is.
00:32:14.780 --> 00:32:17.440
well that would be the area of the semi circle.
00:32:18.020 --> 00:32:18.980
which would be
00:32:19.500 --> 00:32:21.520
pi(2^2)/2
00:32:21.800 --> 00:32:23.240
feel free to plug in 4.
00:32:23.920 --> 00:32:24.420
ok.
00:32:25.080 --> 00:32:25.640
minus..
00:32:26.180 --> 00:32:27.940
the area of this triangle.
00:32:29.300 --> 00:32:31.940
well that has a base of 4 and a height of 4.
00:32:32.920 --> 00:32:34.520
so thats got an area of 8.
00:32:35.540 --> 00:32:37.620
so that would be the answer so it would be 2pi-8.
00:32:38.120 --> 00:32:40.840
but for those who used 4 it would be 8pi-8.
00:32:44.340 --> 00:32:46.860
now where is g(x) a maximum?
00:32:46.860 --> 00:32:50.360
well you could do it 2 ways 1 is you could do it intuitively.
00:32:50.860 --> 00:32:52.780
by just looking at the areas.
00:32:52.780 --> 00:32:55.860
the other thing is say well look i get some area here
00:32:56.220 --> 00:32:56.720
right?
00:32:56.720 --> 00:32:58.460
then i subtract a bunch of stuff
00:32:58.460 --> 00:33:00.020
and then i add some stuff back
00:33:00.020 --> 00:33:04.100
so the question is have i added back more here than i subtracted here?
00:33:04.520 --> 00:33:06.440
no so it has to be bigger here.
00:33:07.480 --> 00:33:08.600
that make sense?
00:33:08.740 --> 00:33:10.820
i had to get my maximum area here
00:33:10.820 --> 00:33:12.400
because then i subtracted
00:33:13.180 --> 00:33:15.380
8 and i only added back
00:33:16.820 --> 00:33:17.320
uhh 2.
00:33:19.500 --> 00:33:21.200
the other is you take the derivative.
00:33:21.980 --> 00:33:22.540
because
00:33:22.540 --> 00:33:26.380
g is a maximum well were going to want to find where is g prime
00:33:28.120 --> 00:33:28.840
equal to 0?
00:33:28.920 --> 00:33:30.280
well what is g prime?
00:33:30.800 --> 00:33:31.640
g'(x)
00:33:32.200 --> 00:33:33.120
is f(x).
00:33:33.860 --> 00:33:34.360
right?
00:33:35.040 --> 00:33:37.980
if g(x) is the integral from 0 to x of f(t)
00:33:38.480 --> 00:33:40.340
then g'(x) is just f(x).
00:33:42.180 --> 00:33:44.460
so f(x) is 0 in 2 places..here
00:33:46.580 --> 00:33:49.460
here. and here it goes from + to -
00:33:50.240 --> 00:33:52.700
its going from up to down so thats going to be a maximum.
00:33:52.700 --> 00:33:55.500
here it goes from minus to plus so it has a maximum
00:33:56.020 --> 00:33:56.520
at
00:33:57.540 --> 00:33:59.200
x=4
00:34:00.040 --> 00:34:03.260
and itd be a minimum if we cared at x=8.
00:34:04.900 --> 00:34:05.400
ok?
00:34:06.640 --> 00:34:09.440
if i ask about ???? you take the derivative again.
00:34:13.320 --> 00:34:14.120
did i skip b?
00:34:14.500 --> 00:34:15.920
oh i forgot g'(4).
00:34:16.360 --> 00:34:17.960
well sure, whats g'(x)?
00:34:20.200 --> 00:34:20.700
g'(x)
00:34:21.860 --> 00:34:22.420
is f(x).
00:34:24.340 --> 00:34:25.420
so g'(4)
00:34:26.220 --> 00:34:26.720
is 0.
00:34:26.720 --> 00:34:29.060
remember what i said when in doubt what do you guess?
00:34:29.680 --> 00:34:30.480
there ya go.
00:34:33.920 --> 00:34:35.840
not really a great strategy.
00:34:37.040 --> 00:34:40.320
right its like baseball trivia when in doubt guess satchel page
00:34:45.220 --> 00:34:46.420
how did i get g(8)?
00:34:47.280 --> 00:34:49.040
its the integral from 0 to 8
00:34:49.040 --> 00:34:50.900
so its going to be the area of the semicircle
00:34:52.760 --> 00:34:55.960
which is 1/2 of (pi)r^2 so itd be 2pi.
00:34:58.300 --> 00:34:58.800
minus
00:34:58.800 --> 00:35:00.360
the area of this triangle.
00:35:00.360 --> 00:35:03.380
the area of the triangle is 1/2 base times height is 8.
00:35:03.920 --> 00:35:05.360
so itd be this minus 8.
00:35:12.480 --> 00:35:13.280
thats good?
00:35:17.880 --> 00:35:19.960
alright everybody study hard.
00:35:19.960 --> 00:35:22.440
see you tomorrow night in javits at 8:45.