| Start | |
| 9:19 | plug in and you get that, okay? let's do another one yes wasn't tat fun? no |
| 9:36 | we need the 20 centimeters, which tells you the base you want to find the height, you have to go back to the original equation for y. okay the derivative just tells you when things are increasing and decreasing if you want to find the dimension you need what X Y are. you plug 22 into the the derivative to determine the maxima minima okay? you're always plug in the derivative to help you figure out the |
| 10:01 | sign chart. to get your Y values it goes with your x value go back to the original equation okay? we'll so some more, let's do another one right now different type of box problem |
| 11:51 | okay were gonna take the sheet and we gonna cut it in squares out of each corner the sheet |
| 12:05 | feel like this is a magic trick don't try this at home identical squares. this is what they say a rough version of this. all right see you cut identical squares out of the corners and then you fold up the sides voila okay and you have a box with no top that's amazing now I make a pigeon |
| 12:38 | come out of the box. all right so who wants that? 20 dollars I'm going to throw that in corner and whoever gets the Box gets food for the whole year |
| 13:06 | guys are too old for hunger games
gotta keep my references very recent. alright what are we doing here?
you have a sheet.. oh and wait i forgot to give you the demensions |
| 13:32 | okay let's say the sheet is 14 by 16. okay that makes the problem doable.
before it was jus entertaining. of course you can do it without numbers but its harder we have a sheet it is 14 inches by 16 inches then were going to cut squares out of each corner, like that and that's going to give us |
| 14:08 | something that now looks like this okay and we fold on the dotted lines. and youll end up with something that looks like that I'll let you |
| 14:33 | guys guess what my grades were art class they were not the best possible grade it didn't maximize those grades, in fact minimize those grades all right so what are we doing we're cutting squares out of each corner so these squares are all X by X. which means that this was originally 16 inches you cut X have each |
| 15:03 | side that this is 16 minus 2x and if this is 14 inches and we cut an x out of each side that's 14 minus 2x so this is now 14 minus 2x by 16 minus 2x by x because you folded it up because it's 16 took away an X and an X so this is 16 |
| 15:36 | minus 2X ,the 14 you take away an X and X thats 14 minus 2x and you fold it up and that's the X okay I would spin it around but i think you go the idea. so how do we find the volume of a box. length times width time height. volume is 14 minus 2x times 16 |
| 16:04 | minus 2x times x. look at that triple product rule, no. well we can multiply it out and you get 14 times 16 is is 224, i think that's right, yep. minus 32 minus 28 is minus 60 X plus 4 x squared all times x |
| 16:36 | feel free to check my rhythmic. Or 224 X - 60 x squared plus 4x cubed |
| 17:02 | what do we do now? take the derivative this is calculus class. V prime is 224 - 120 X plus 12x squared. it's very unlikely these factor nicely. it takes a lot of work to make it come out nicely, you just make it up set it equal to 0. let's see, we can reduce. lets see 6? no i think we can do 4 |
| 17:45 | so if I divide I think a 4 i would get 56 minus 30 X plus 3x squared equals 0 and use the quadratic formula |
| 18:28 | okay which I don't really care at this point but you have two options and of course |
| 18:38 | like i said we have to give you easier numbers but heres something very important to know...what's the biggest X can be X can't be very big whats the biggest x can be? it can't be more than seven you can't cut more than 2 7 inch squares out of something thats 14 inches long |
| 19:03 | that makes sense? once it's more than seven inches that's it you used up all your thing. so this one the 30 plus radical 20 over 6 is way bigger than 7. so its going to be this one, this one is a little less than 2 and 1/2 because 228 is just bigger than 15 |
| 19:32 | how do we know it's a max? well now it's the simplest thing to take the second derivative you take the second derivative and you get negative 120 plus 24x. when you plug this in you get a negative number that makes it a maximum |
| 20:10 | how about a rolling window problem?
first derivative and took the second derivative and second derivative test is taking critical numbers you plug them in the second derivative. if you get a positive answer you get a minimum negative answer you get maximum. you take the take the |
| 20:34 | critical numbers plug it in the second derivative you get a positive answer you have maximum negative answer minimum. so the second derivative is the derivative of this times this or you can sign test it. if you really confident you can just say I'm sure it's the maximum every once awhile we get you guys on those |
| 22:34 | okay you learned two words today
anybody here speak French? what does sur mean in french?
surmount means to be mounted on top |
| 23:22 | speaking in french |
| 23:35 | speaking in spanish " word for on top" in another words, you have a window thats a semi circle on top of a rectangle kind of like the webassign problem. sort of a semi circle |
| 24:01 | so the radius of the semi circle, you got the semi circle, you got the base of the window. r and 2r, okay make sense and then we've got the lengths and the widths of the window, called X and in this piece of the window is a half of the circumference of that circle that would be PI R |
| 24:34 | does that make sense? because circumference is 2 PI R so half of 2 pi R is PI r. so far so good? what do we know? well we know how much the perimeter of the window is let's figure out what we should make r to be the maximum area. any ideas well you can |
| 25:13 | take the derivative until you have equations. we have two pieces of information we want to find the maximum area, well the area is going to be 2rx that's the area of the rectangle, plus the area the circle, semicircle, which is PI R squared over 2 and the perimeter is 12 |
| 25:40 | feet so 2x plus 2 R plus PI R is 12 what are we going to do? we're going to use this equation isolate X and plug it in the first equation which will be just in terms of r. take your derivative |
| 26:04 | and set equal 0 so I'm going to stand here for a couple mins till you guys figure it out
this isnt as bad as it looks. so we need I isolate x, so 2 x is 12 minus 2 R minus PI R or x is 6 minus R
minus PI over 2 R.
|
| 26:36 | now we go to the area formula and you take X and we substitute this. why do we substitute for x and not r? thats awful hard to square it thats annoying. so a is now 2r times this mess, 6 minus r times pi over 2 R plus Pi R squared over 2, remember pi is constant |
| 27:01 | whats the derivative of pi? 0. whats the derivative of pi to the fifth its not 5pi to the
fourth? you sure its not 0? what about e to the fifth?
whats the derivative of e to the 5th just making sure everyone can do derivatives, okay? you sure the derivative of pi is 0? you get a is 12 R minus 2 R |
| 27:32 | squared minus PI R squared plus PI R squared over 2. you can combine these if you can get the derivative
it doesn't really matter. which should we do? combined or take the derivative?
you want to combine? or you gonna raise your hand later and say i dont know where that came from. i get that a lot. Im gonna leave it alone, cause i would assume if you were doing it |
| 28:00 | you wouldnt combine you would just take the derivative. so a prime is now 12 must 4 R minus 2 PI R plus Pi R so we set this equal to 0. is 12 minus 4 R minus PI R equals 0 or 12 is 4 R plus PI R |
| 28:33 | factor out the R
and divide.
and there you go that's your radius. how do I know that that's maximum? take the second derivative. second derivative is minus 4 minus PI it's negative number |
| 29:02 | second derivative is always negative, which means this thing is always concave down means you have a maximum. okay let's do another one. okay this is the kind of stuff you're going to be able to do. youll have something like this on the final |
| 29:35 | oh let's do one of those |
| 31:49 | I know i said ice cream, youre all excited about that alright a swimmer is swimming 40 meters off a straight |
| 32:05 | shore line she's spots an ice cream stand 200 meters down shore, from this point directly opposite her. her she can swim in 4 meters per second and run in 6 meters per second where should she land to get to the ice cream in the least amount of time so you look at that and you say I have no idea what you're talking about well let's set this up. here is the shore here's the swimmer, female hence the hair |
| 32:42 | and here is the ice cream stand. with a big cone of vanilla ice cream ben and jerrys vanilla ice cream somebody has been celebrating 4/20. so notice she can go directly to the ice |
| 33:05 | cream that's the fastest path but she swims slowly. actually four meters a second is pretty fast but she only swims at 4 meters per seconds she runs and 6 meters a second she could get out of the water as fast as you can and then run all the way down the shoreline because she runs six meters per second she only swims at 4 meters per second but that's the longest possible distance |
| 33:34 | somewhere in the middle is the best place to land. that's x that's 200 minus X. so if she swims to here and then runs the rest of the way that'll be the fastest way to get to her ice cream okay so the question is what's X can you guys figure it out yes you can remember this equations from seventh grade |
| 34:03 | rate times time equals distance or if you took physics you were trying to get there in the fastest time. let's see if you can figure it out. I guess all this noise means is you solved the problem and youre waiting for me. that's the only possible interpretation |
| 34:33 | rate times time equals distance. the swimmer is 40 meters away so the swimmer is gonna swim this distance which thanks to the pythagorean theorem uses squared root sixteen hundred plus x squared because we love pythagorean theorem. in fact most of you can't name another theorem. some of you can |
| 35:02 | so first she has to swim this distance rate times time equals distance for the time is equal the distance and over the rate not minus, divided by. so how long will take her to swim this distance? 1600 plus x squared divided by her rate which is four. so that's her time but wait once shes |
| 35:37 | there she then has to run the remaining distance at a rate of 6 meters per second so that'll be 200 minus X divided by 4 and that's how long it'll take her to run this or to swim and run oops thats the square root sorry that makes me more annoying. alright now we take the derivative |
| 36:05 | oh my we may want to simplify this just a little bit this is 1/4 times the square root of 1600 plus X square plus 50 minus a quarter x oh that should be 6, good thing I brought you guys. if it's four it's easy |
| 36:35 | youd land in the middle. well I shouldn't say that's so fast that i have to work it out all right lets take the derivative because we love derivatives. so t prime i okay one quarter and then the derivative of this which is one half 1600 plus x |
| 37:00 | squared to the minus 1/2 times 2 X because that's the chain rule make sure you know your chain law it's 1/2 1600 plus x squared minus 1/2 times the derivative of the inside, the derivative of 100 over 3 is 0, minus 1/6. and we going to set that equal to 0, its not gonna be as hard as it looks. this is 1 over 8 radical 1600 plus x |
| 37:33 | squared times 2x. so its 2x over that, minus the 6 equals 0 or equals 1/6. does that look fun to solve?
sure whenever we have this kind of fractions we cross multiply 12 x equals 8 radical 1600 plus x squared square both sides |
| 38:26 | so let's see 144 x squared equals 102400 plus 64x squared that's |
| 38:38 | right. whats 64 times 1600. cant do that one in your heads? pretty sure its 2 to the 10 so you get 80 x squared equal 102400, x squared equals 1024 divided by 8 which is |
| 39:06 | 128 so x equals 8 radical 2
she should laying there okay? I might have gotten that wrong it's hard in your
head you know and I'm old thank you all right we have time for one
more problem. Should we vote? you guys want one more problem?
|
| 39:41 | alright we'll stop |