Stony Brook MAT 125 Spring 2015
Lecture 17: Related rates
April 6, 2015

Start   I didnt get a chance to cover on wednesday because we just didnt get that far lets review a little bit before we do stuff what does it mean when we say find the linear equation, remember i told you we have the local linear approximation erase that immediately, oh that right
0:46theres two different ways you can think of whats going on they are the same thing but we write them in two different forms this ones linear regressions should've given myself a little more room, but why would i
1:01thats the first form but they both say the same thing, when you're trying to find f of x what you can do is use the tangent line and get close and thats by taking the function and just adding it tangent line times the rule different so if we said something like find the linear regression of something im not gonna do that one obviously that would be to easy, lets do something totally different
1:41that could be more different i hate to do something to similar so what do we mean by the linear regression, well first, what class is this? its calculus so take the derivative you guys know there answer to that by now
2:05so we want to find what we would use tangent line to approximate this we can just create the equation we used the first one of these so f of a is what you get when you plug in two here which is 44 and you get prime of a plugging in 2 here
2:32equals 76 you say f of x is about 44 plus 76 times x minus 2 i picked nice big numbers plus 76x minus 108 thats the linear regression
3:02and then if i said, okay so firgure it out at 2.1 you would plug in 2.1 here okay, notice on the exam we cant ask you to do these crazy numbers cause you cant use calculators its more just like we ask you to set it up thats what linear regression looks like
3:35so a reminder again, lets say you have an approximation like 2.001 cubed, say you wanted to do 4.002 cubed is about what?
well first off you know its around 64 because 4 cubed is 64 so if we wanted to approximate now youre going to use the second function you can use the first, theyre really the same thing
4:01you say, well lets say f of x is x cubed then f prime of x is 3x squared now what value am i doing this at, well i want to find it at 4.002 cubed so if x is four
4:30delta x would be .002 thats how much were off by so this equation, second one says that if i want to find f of 4.002 thats about f of four plus f prime of 4
5:02times .002 so that would be well lets see four cubed is 64 f prime of 4 is 48
5:31times .002 so you do that and you would get .096 got the idea so thats how you use linear regression to approximate function. we can ask you something like that on the exam, make the numbers come out relatively easy
6:00so what were we talking about when we said differentials how are we doing so far on this?
got the concept?
by the way, having real trouble accessing webassign before were many of you having equal difficulties yes, okay sorry its not us, its web assigns fault of course we can blame congress very important suppose we wanted to do differentials so remember differential means a little bit
6:44so this for example, find the differential of y=e to the x over 2 take something totally different, y equals e to the 2x what do we mean by the differential, we want to know when you change y by a little bit what happens so take the derivative
7:03remember that dy means a little change because its the slope, dx means a little change dy/dx would be e to the 2x times 2, chain rule so you just multiply across by dx that is the differential so if you were looking at 7 that would be the first box something like that
7:32and then to evaluate we just plug in thats not so hard what we mean by differential, we mean basically what happens to y when you change it by a little bit it will change by whatever you get when you get in here.
so if you have the area of a cicle alright so what is the differential?
8:10for, that might not be right, the area of a circle when its radius is increased
8:40from 4 to 4.01 centimeters remember how much does the area change?
okay what is the differential of the area of a circle when the radius increase from 4 to 4.01 cm? Well
9:08the area of a circle is pi r squared.
by the way if you need to know, if you have an exam with this qestion and you use the area of a cirlce were not going to tell you the formula for area of a circle you need to know its pi r squared.
we expect you to know your geographic formulas
9:32your in college we expect you to know the area of a circle, prime of a sphere stuff like that.
what would the differential be? well lets take the derivative dr over dr is 2pir isnt that fun you take the derivative of the area and you get the circumference?
you think thats a coincidence? its not Definitely not
10:02nothing with this math stuff is a coincidence you cross multiply and you get da is 2pir dr. alright, so the r is 4 and the dr is .01 thats how much we changed the radius.
so da would be 2pi times 4 times .01
10:31which would be .08 pi, so thats how much it would change.
thats .08 pi cm squared thats not to tough right theres one other thing i didnt notice we dint get to.
i think that would be relative error stuff
11:04how we doing so far> so relative error means the error prepared for the whole thing and error of one centimeter is a lot when youre looking at something 3 centimeters but not a lot when youre looking at something 3 meters the radius of the disc is 19 cm
11:30with maximum error of .2 centimeters its very similar to this we have the area of the disc which is pi r squared and the differential of the area 2 pi r dr
12:07how would you calculate the error? Well okay so if i tell you the radius is 19 centimeters got and error of .2 you do 2 pi times 19 times .2 thats not very hard
12:30thats .76 pi squared cm so what do we mean by relative error? Well the relative error would be da over a the differential and the area divided by the area thats the relative error thats 2pi r
13:00dr, over pi r squared see you cancel pi and you cancel r and its 2 dr over r so here it would be 2 times .2 over 19.
and if you are asked percentage, you just multiply by 100.
13:32thats the kind of stuff we didnt get a chance to do on wednesday so we did it today.
now i feel like i left you for air also paper homework we havent done number 2 yet, okay?
we have done number one but we have not done number two
14:03take the homework rr cause were no longer numbering them is not due for a bit, okay?
its due next week or another week, so you still have plenty of time to do the homework alright, funs over so far so good? questions?
no, nobody has any questions we all excited for the basketball game tonight?
15:03talking about basketball game, (Not important.) you sure theres no questions, im moving on to a new topic so this is the topic that bothers people a lot
15:32im not sure why, i really don think its that bad but can have a little trouble setting it up this is where were gonna enter the word problem part of the course yay. okay?
were now going to imply our tools to something new say i have a pond, a nice smooth pond and i throw a rock into the center of the pond the ripples come out and make a shape of a circle in general
16:03till the end of the pond im going to try and frigure out how fast those circles are moving and we wanna know what happens to the area of that circle so you drop your rock in a pond hey not bad. So your rock hit right here.
and after a second the shock waves traveled here and then to here, and to here
16:31so we want to keep track how fast that area is changing if we know how fast that shock wave is moving so in movies by the way, they have the glass steven sugolis running and he out runs the shock wave, that doesnt actually happene you usually get about this far sometimes youll get this far, the waves are just moving at the speed of sound.
so its past you you dont actually out run it its fun when it happens in a movie but in real life its a really bad thing to be near one of these things
17:04this is a little pond that you dropped a little rock and you get these pretty little ripples how are we going figure out its changing? well you know the area of the circle is pi r sqaured if the radius is increasing
17:33at 4 meters per second thats very slow how fast is the area increasing when the radius is
18:005 meters by the way the increase of area from here to here is different from the area from here to here different from area from here to here because the radius expands its a bigger and bigger area each time but its not linear were gonna get to the bottom of this is a second ignore the phrase its not linear it changes a different amount depending on the radius
18:31you were at calculus is all about finding derivatives. derivatives are the rate of change lets turn this into numbers we know that the area of a circle is pi r squared and we know how fast the radius is changing that would be the change in radius with respect to time, how fas the time how fast the change is over time dr, dt so were going to be doing everything in related radius
19:02in d in variable dt, in respect to time thats why we use rate this is increasing at positive 4 We want to find ADT how fast the area changes when the radius is pi
19:35the area of a circle is pir squared we know the dr/dt is 4 m/s what is the da/dt at the moment when the radius is 5 meters well, lets take the derivative of a=pi r sqaured with respect to time that would be da/dt is, the derivative of pi r squared is 2 pi r
20:04dr/dt so notice this is like Implicit depreciation in fact this is implicit depreciation.
because everything is in respect to time so everything gets a variable of dt and now you just plug in.
da/dt is 2 pi times 5 times 4
20:33which is 40 pi meters squared per second. How do I know what the units are?
well area is in square meters as we respect to seconds, respect the time in seconds 40 pi square meters per second do we undertsand that?
not so bad sure. lets do something very similar
21:05ill leave that, not ill erase that. you guys are taking notes.
Always watch this again
22:09excellent grammar the sides of a cube are growing at 2 cm per second the sides of a cube are growing at 2 cm/s how fast is the volume of the cube growing when the sides are 10 cm
22:31this is very similar to what we just did lets see if we can firgure it out and im not gonna tell you the formula of the bottom of a cube youre suppose to know.
alright thats long enough alright so we have a cube
23:01or something like a cube if you can draw a little pretty picture that might help on the final might not and we know the volume of a cube. Well lets say each side is x theyre all the same because its a cube the volume is x cubed in fact thats why we say cubed.
rather than to the 3rd, okay?
23:34we know that the sides are growing at 2 cm/s or each side is growing at 2 cm/s dx/dt, the change in the side is 2 how fast is the volume growing so what if dv/dt when x is 10 of course you couldve used a different letter
24:07well lets look at this we got v is x cubed so dv is 3x squared dx/dt right? now we just plug in.
so this says. oh this is dv/dt sorry
24:30the change in volume over time.
the rate the bottom is changing is 3x squared times the rate of the side changing so dv/dt so 3 times 10 squared, because its the time at 10 cm times 2 which is 600 cm cubed per second. i dont know if we'll take off for units they do on the av, but this is not the av of course
25:02but you should know what the units are the units are volume so thatd be cubic centimeters per second go the idea?
should we do another one of these?
oh yea, lets make sure you guys have the idea down
25:37a spherical balloon
26:31alright, a spherical balloon is shrinking at 4 pi cm/s how fast is the radius shrinking when the volume of the sphere is 36 pi cubic centimeters this time only ill tell ya the volume of a sphere 4/3 pi r cubed. While im at it the surface area of a sphere is 4 pi r squared notice the surface area formula is the derivative of the bottom area of the formula
27:03is that a coinencedensce?
spherical balloon is shrinking at 4pi cubic centimeers per secnd, that means youre going to use a negative number for that, -4 pi how fast is the radius shrinking when the volume when the sphere is 36 pi taken the derivative okay we got a sort of spherical balloon part two circles on the board
27:31and we know the volume of a sphere is 4/3 pi r cubed and i tell you dv/dt is negative 4 pi cubic centimeters per second how fast is the radius shrinking? What is dr/dt?
when v is 36 pi
28:05so this is called related rates i want to relate the rate the volume is chaning to the rate the radius is changin i got v is 4/3 pi r cubed dv/dt 4 pi r squared thats what you get when you take the derivative.
28:31times dr/dt so now i just plug in i know dv/dt is 4 pi im looking for dr/dt, i just have to plug in for r but wait i dont have r i have v so in have to figure out what r is that happens we do that to you guys so 36 pi is 4/3 pi r cubed, cancel both pi's
29:03we get 27 is r cubed r is 3 Questions from student: how did i get the 4 here?
that formula is for volume i wrote it over there.
and of course you have to memorize these for the final, just like unit circle
29:34so much fun if yure not taking calculus 126, this is your last calculus class you got one more month of math, thats it its over, forever, youll be very sad okay now that we know r is 3, we plug in and you get 4 pi
30:05is 4 pi times 3 squared dr/dt and if you do a tiny bit of algebra, youd get dr/dt 1/9 of a centimeter per second cause the 4 pi's cancel and leave you with one
30:31i know that for some people geometry was a long time ago so lets get some formulas down
31:23thats enough, okay?
31:30the volume of a sphere is 4 pi r cubed and the surface area is 4pi r squared and the volume of a cube is x cubed, and the surface area is 6x squared the volume of a cylinder is pi r squared h and the surface area, that doesnt show up much is 2 pi r h + 2 pi r squared. A cylinder
32:03the tube is expanding and the top and bottom are expanding thats why you have two components in the surface area we just gave you whats called the lateral area thats just the tube part then it would just be 2 pi r h the volume of a cone 1/3 pi r sqaured h we wont ask you what the surface area of a cone. I can tell you, you want to know the volume of a cone formula for the proble coming up
32:31soemtime between now and wednesday make sure you know that i dont know if ill get to it today yes, oh that would be a negative sign dr/dt is -1/9 sorry about that folks its shrinking not decreasing you get no partial credit i you miss that sign, its 0
33:02alright lets do another one question first?no, you should want to memorize these
34:51a 15 foot ladder is leaning against a wall.
the bottom starts to slide out at 2 feet per second. you hope youre not on the ladder when this happens
35:02how fast is the top sliding down?
when the top is 12 feet above the ground heres what happening, initially the ladder is leaning like this and then it starts to slide down the ladder is initially like this and then it starts to slide out like that, okay?
35:30the one side goes out when the other side comes down now 2 things are changing at once not just ona you what to try this on your own first?
or should we do this as a team we should do the team. the team approach? can i be on the team
36:34so the 15 foot ladder is leaning against the wall this is 15 feet the length of the ladder does not change now lets call this distance
37:01lets call the distance from the ladder, the foot of the ladder to the wall a.
and the distance from then top of the ladder tot he ground, B.
now i need to relate how fast the bottom is sliding out which would be da/dt store all my information over here i know that da/dt is sliding out at 2
37:31feet per second i want to find how fast db/dt is chanign when b is 12 do i have a way to relate a and b can anyone think of a good way to relate a and b everyones favorite theorem the only one anyone really memorized
38:00starts with pi Pythagorean theorem a squared plus b squared equals 1 squared thats not a hard theorem to memorize, we all memorized that one just fine of course if i rearrange a and b it will give us the answers but i want to find bd/dt, so lets take the derivative so you get 2a da/dt
38:33plus 2b db/bt equals 0. Why does it equal 0?
because 15 squared is a constent the derivative of constant is 0 thats why thats 0 who wants to write for a minute and look up for a minute and say "hey why is that 0?" so 2a da/dt plus 2b db/dt
39:00is 0 now i ccan plug in because i know b i kno db/dt, oh wait i dont know a.
so how do i figure out a?
well i know the Pythagorean theorem i know that when b is 12 i can find a a squared plus 12 squared
39:30is 225 15 squared sorry a squared plus 144 is 225 a squared is 81 a is 9. for those of you who are on your game you know its a 3 4 5 triangle 3 times 3, 4 times 3, 5 times 3 okay, so now i can plug in 2 times 9
40:01times da/dt which is 2 plus 2 times 12, times db/dt which is what we are looking for is 0 so you get 36 plus 24 db/dt equals 0 or db/dt
40:32is -36/24 also known as negative 3/2 of a foot per second thats how fast the top is coming down pretty fast cause you dont want to hit the ground it will hurt everybody understand what i did?
okay were gonna do one just like it
41:22two ships leave port at the same time
41:39the ship a is traveling due north at 5 km/h ship b is traveling
42:03due east at 4 km/hr no, change that number eh never mind ill do the 4, they dont always have to work out so well how fast
42:31is the distance between them growing after 2 hours okay so two ships leave port at the same time ship a and ship b
43:02ship a is traveling due north straight north at 5 km/h ship b is traveling due east at 4 km/hr how fast is the distance between them growing after two hours take a couple minutes see if you can figure it out two ships leave port at the same time ship a is going due north ship b is going due east
43:31and this is the distance between them what letter can we sign that? how bout c?
wow, awesome, do we know a way to relate a b and c a squared plus b squared equals c squared everybody's favorite theorem that i keep pointing out we also know that da/dt is 5
44:00km/hr and db/dt is 4 we want to find dc/dt when 2 hours have pasted so when 2 hours have pasted a has gone 10 and b has gone 8 okay 5 km/hr for 2 hours, 10 km.
44:31around 4 mph for 2 hours 8 km not so hard right?
lets take the derivative 2a da/dt plus 2b db/dt equals 2c dc/dt heres something that will make your lives easier, cancel the 2's
45:04makes the arithmetic a little easier a da/dt plus b db/dt equas c dc/dt thats not so difficult, not we just plug in. we know a we know da/dt, we know b when know db/dt we need to find c but we can go back here a squared plus b squared equals c squared
45:3310 squared plus 8 squared is c squared we do a little magic and you get c is the square root of 164 or 2 radical 41 if you care to reduce, certainly dont need to alright lets plug in 10 times 5 plus 8
46:00times 4 is the square root of 164 dc/dt so you get 82 over the square root of 164 km/hr anybody get that?
yay, 2. you guys still have to take that exam thing though alright you get close?
46:31we can do one more lets do one more, let make sure everyone understands the concept.
eh now we can stop we can do it on wednesday.