|Start||Okay, so if we just ask where sine of x is a half, that would be a minimum competence question and you are suppose to know where sine of x is a half
Okay what if we do sine 3x equals a half?
and we tell you that x is between zero and pi Well half as we say that
|0:30||so if it's between zero and pi
well you just have to play with it a bit
you say to yourself, well first of all
3x has to be pi over 2. Lets do it in degrees though.
If you first do it in degrees, okay?
Then convert in the end so you say to yourself well where is sine equal to half, whats equal to have in the first quadrant?
30 degrees then you go over here to the second quadrant and its also
|1:01||equal to a half because right we know what they say about the albany students so its true in the second quadrant and we can keep going around so instead of staying true to 30 degrees, well put 150 degrees and its going to be true to more places in a matter of fact its true there's an infinite number of them so if we add 360 degrees so let's just test this out for now so if we divide|
|1:31||so this is where 3x is equal so if I divide everything by 3
you get 10 degrees
So those are all less then a 180 degrees because pi is a 180 degrees right?
so all of those work Are there more answers? well lets see I have all the way round here and that was 390. We keep going Thats 510 degrees, how did I get these numbers?
I add 360 I add 360
|2:00||Divide that by 3 and I gonna get 170 degrees
Thats still less then 180 right?
The next one I come up with is gonna more than 180 Because I'm already at 170 So now I don't need degrees, I just convert it into radians. How do you convert 10 degrees into radians?
Have any ideas?
Multiply it by pi/180 Take 50 and multiply it by pi/180 Take 130 times pi/180
|2:31||and 170 times pi/180 you can stop there thats a hundred perfect answer You dont have to reduce those if you don't want to so one of the ways you can make your life easier is to do the degrees first and switch it into radians at the end Well you have to find all angles until you get to 180 degrees So you just going until you get 180 and stop Its not gonna be that many of them. We're not gonna have 20 of them|
|3:02||20 is a lot
however you shouldnt assume there is only 1
you should assume -- there's 3x
the clue is 3x
Normally if its 360 degrees, its two answers
because 3, 2 times 3 is 6x
So for 180 degrees its not necessary half of that
Can i do another one of these? yeah!
|3:32||how do I know where to start off? Where does sine equal a half?
Sine is the a half at 30 degrees, thats how I know to start there, right?
okay, we understand what i did? So you say to yourself where does sine equal 1/2 30 degrees 150 and keep going okay? and then when you divide by 3 then you keep going until you run out of numbers less than 180 degrees
|4:00||Go ahead -- if its worth it, i'll do it
sure lets do that one
2pi? now im just gonna -- okay
we have cosine of 3x
equals square root of 3/2
x is between, this is one of the one from my practice stuff right?
So lets try it for 2 minutes, do it in degrees
|4:31||so ask yourself, wheres cosine, uh where is cosine equal to radical 3/2 in degrees Try to find all the places the clue is thats 3x so 3 times 2, that 6 so 6 answers alright i think that long enough so cosine of 3x, so wheres cosine equal radical 3/2 well lets see cosine is positive in the first quadrant|
|5:02||and again down here on in the fourth quadrant cosine of 30 degrees is the square root of 3/2 and now youre going to go all the way to the fourth quadrant so thats 330 degrees so i have 3x equals 30 degrees or 330 degrees so if i was doing graphs|
|5:30||okay ill show it to you have to graph another way in a second so now lets go around this circle the full time, 360 degrees and and then go another 30 and you go all the way around to here and thats 690 degrees we all have paper to work on theres lots of blank space and lets divide by 3 and see how we do, thats 10 degrees 110 degrees 130 degrees|
|6:00||230 degrees so far i have all these possibilities, have to go all the way to 360 i mean we might spot a pattern but lets add another 300 degrees to each of these. so i get 750 and 1050 I divide by 3 and i get 250 and 350 thats what you get how do i know im done? because the next one is going to be more than 360 degrees and now i want to convert these into radians|
|6:32||I multiply by pi and divided by 180
or you can make it pi/18
okay and now i get 110pi/180
also known as 11pi/18 and so on
isnt that a nice fast way to do these?
now why is that happening?
|7:02||so what does cosine3x look like?
cosine3x looks just like cosinex except I squished it by a factor of 3 so thats 2pi so thats 2pi/3 and 4pi/3 so when i want to find what equal to the square root of 3/2. i have 1,2,,4,5,6 solutions. Thats the theory behind it
|7:38||the rest of the theory you are thinking will it show on the midterm
but thats why you get 6 answers, because you cross the graph 6 times
okay so lets do something else
some other type of question. notice i covered that back up, i dont want you spending so much time thinking about it
do a graphing problem, sure!
|8:06||im gonna make one up you could but im gonna make one up lets just start off with something like that|
|8:34||alright lets do this one so i have some changes coming here do you get your test back, oh thats a good question. You get your midterms back in recitation you dont go to recitation you dont get your midterm back i end up with a box at the end of the semester, you know what i do.. so i have two things going on here. ive go this 4, i got the 2 and i got pi/6|
|9:12||so this tells me, move my curve up two and this tells me stretch the curve by 4 so you can stretch it by 4 or move it up 2 thats really what happens you start at|
|9:30||you start going up 4 and down 4 and then you move it up2 and now you are here okay?
so you go up 4 from here and down 4 from here this tells me move pi/6 to the right i havent shrunk the curve at all so the length of the curve is gonna be 2pi+6 its perfectly fine leaving it like that you can turn that into 13pi/6 why not? i dont care. if you do degrees and then you convert back at the end
|10:03||you just take the 30 and add 60 to it alright so your sine curve its gonna look like that okay. it should cross in the middle, dotted line all you gotta do thats half way, which would be 7pi/6|
|10:31||you want to find half way between the 2 numbers take a 1pi/6 and the 13, so you take the 1 and the 13 add them together. that makes 14. divide it by 2. you get 7 anytime you want to find the half of 2 numbers, you add them and dived them by 2 thats all you had to do, thats the whole curve isnt it easier well im either gonna graph first. so i dont want to start avariation on that yet|
|11:19||alright so how do we do this well first very important, we have to pull the 3 out|
|11:34||okay so you have to do that and notice when you take the 3 out of the x you are taking the 3 out of pi. and that makes pi/3. and how do i know thats right 3 times pi/3, will give you pi dont over think these things now we can do the same kind of thing we did before were still shifted up 2|
|12:02||still go up 4 and down 4 we start pi/3 to the right and you add 2pi, to get to the end of the curve so pi/3 plus 2pi is 7pi./3 so it right on 7pi/3|
|12:32||here comes the fun part okay lets say the middle is 4pi/3 now what about this 3? oaky take your values at the x axis and dived them by 3 so this now becomes pi/9 4pi/9, 7pi/9|
|13:00||thats how the factor you pressed the curve by 3 all the numbers get divided by 3 that is your horizontal compression do you want me to make it harder? cause i can make it harder i thought this was pretty clear no the 3 does not cancel out the 2/3 either way we just need to be multiply by x because whats happening is that is in parenthesis|
|13:40||when youre told, you have a 3 multiply by x you c=are compressing it by 3. We did this when we talked about transformation because you need, you take pi/9 and multiply it by 3 and you get pi/3 so thats why you can divide a number by 3. because this multiplies in that|
|14:03||good so lets do a piecewise function
it was 40 divided by 4
you would like a word prolblem? sure could we do the ones that were in the packet?
the packet problems? you do yes
|14:35||then ill make one up not easy to make up actually alright do you want to do senor, or do we wnat to do like light that does the electricity bill|
|15:03||do we want rodrego or do we want or do we what the electricity one?
we will do rodirgo, alright he is driving from lancing to columbus lancing is a street in michigan, colombus is in ohio just in case you were wondering
|15:37||rodrigo drives at 50 miles per hour for 2 hours my god speed up we have to figure out how far Rodrigo goes so those of you who love physics, remember your 7th grade word problems|
|16:00||rate times time equals distance or if you read it the other direction, distance equals time times rate youll get drt so to figure out how far rodrigo goes, you take how fast hes going and figure out how long he goes for. 50 mph is 50 miles, fifty miles times 2 hours is 100 miless and so on so for the first phase of the journey you want to find out where rodrigo is, you just multiply 50 by how long he has been driving|
|16:35||and you keep this up for 2 hours but then in 2 hours, something happens it shifts locations how do you have 2 different equal signs. dont worry about it for now, ill explain there are 2 hours he magical stomps on the brakes, so he is only goign 30 mph|
|17:05||now hes on 25a at 5 oclock he would beat long island record anytime now hes only driving 30 mph but be careful, how far has rodrigo gone after 2 hours? hes gone a 100 miles so he starts a 100 mile mark and now 30 mph its not t hours its t minus 2 hours, because you want to take away those first 2 hours|
|17:33||because at 2 and a half hours you dont wan 2 1/2 you want a half here
plus the original 100
and thats at dawn
for 2 more hours
how i do know for 2 more hours?
he has to get 160 miles hes already gone a 100 miles so he has 60 left to go 30 miles an hour to go 60 miles takes 2 hours now i couldve put an equal sign here
|18:02||or here it doesnt matter its sort of customary to put the equal sign that one but it doesnt matter cause he instantly shifts or stops on the break. now he stops at preise for a while so now we have a 160 miles, so if you want to know where is he you would say hes gone a 160 miles and that last for an hour. an hour|
|18:30||i dont know, I guess shes worth an hour I dont know how attractive she is she have a good conversation blah blah blah think they are having a good meal, maybe its a happy meal from mcdonalds now he says oh boy am i late so he has to get through the remain distance to columbus and he is now is live at 65 mph just so you know the speed limit is the limit|
|19:01||so you good under the speed limit, its not a suggestion its a law well i think its a suggest its like hey how about 65, you go sure 65 sounds good and for that you have to take away the first 5 hours and you can keep that up until he gets there so he has to go those last 65 miles, im sorry 90 miles|
|19:32||thats 65 mph plus another 5 miles, you can leave it like that.
that is an equal sign and hes done the key to a lot of these problems you have to think about what is going on in each space tell yourself of story, you want to break up every different interval how did i get the 90/65, alright so you got 160 miles he has to go 250 miles and he 90 miles left to go
|20:05||and remember rate times time equals distance. so 65
t equals 90 miles so the time it is going to take is 90/65
plus 90/65 passed the 5 hours
you are going to do this in physics but its going to be much harder
|20:32||yes i put the 5 hours and added 90/65. thats the remaining time add the 5 hours so 3 times so in four hours, he stop at preise for an hour, is 5 hours and now he drives from ther|
|21:05||alright i can try and make up a word problem but its hard to do on the spot
can we do the electrcity one?
|21:49||the electricity is sold for residential customers for a following rate plan for the first 250kmph, 1 hour you use of energy|
|22:00||the first 250kwatt hours that a customers use cost 24 cents per kwatt hour if we want to find the cost look at this wave and you say to yourself the structure the 3 sets of things for h starts out at 0 till it gets to 250 illl move it down a bit then from 250 to the next 500 so thats up to 750|
|22:40||thats the 3 times. the 3 structures so the first piece is 24 cents times the number of hours and the after 250 kwatt hours|
|23:00||how much money does that cost, well youre gonna have to do a little multiplication and youll get 60 bucks so for the second piece hes already spent 60 dollars and its gonna be 26 times hours after 250|
|23:43||so far so good?
wheres this come from?
so 60 dollars is 24 cents times the first 250 kwatt hours so you just do 24 times 250 and you work the decimal place thats 60 dollars then its 26 cents
|24:02||times the number of hours after the 250 kwat hours you can just add it you dont actually have to figure out what 60 is but be careful. youll have lost of scrap paper you should be able to do it i dont thin the numbers are particular hard that we came up with|
|24:31||and now what happens so how much is the second time it cost you well its gonna be 500 kwatt hours at 26 cents a kwatt hour thats a 130 dollars so now its a 130 bucks a 190 dollars because you spent 60 and then you spend another 130 and get a 190 dollars plus 28 cents|
|25:01||times everything minus 750 how come it never equal 750 now it can equal too you know theres one in every crowd pointing out all my flaws this is terrible goo thing were not married|
|25:30||alright next one I can try and make one up but there hard to make up on the spot well how about a inverse? are we ready to do some? lets practice one|
|26:08||there you go find f inverse of x
alright lets do the inverse of this
so what do we do?
we write this as y and then you switch x and y and now we just have to isolate y
|26:31||we can cross multiply and get 7x, 4y cubed minus 1 add 1 to both sides and you get 7x plus 1 equals 4y cubed dived by 4 and take the cubed root|
|27:05||that is your inverse function
how do we know its an inverse?
ask yourself what you are doing x first thing you is you cube it then you multiply by 4, then you subtract 1, last you divide by 7 so first you multiply by 7, second you add 1 and third divide by 4 and last you get the cubed root
|27:31||so you reverse the order and you reverse the operations lol
thats how you know you are doing it correctly
you can also check, make up a number and plug it in and see what happens
this comes out 0. sorry not true, well 3/7 so we plug in 3/7 here and you get 1
3/7 times 7 is 3. plus 1 is 4 divided by 4 is 1 cubed 1 is 1.
|28:02||so you can always check but putting in a number and getting an answer and putting an answer in the other one
how do you know if the inverse does not exist?
if you have an even power of x, youre going to havea problem okay?
the only need of a problem is when you have x squared then when you do the inverse you have to plus or minus the square root of x
|28:32||how do you get 2 values that would be bad these are cubed roots there wont be an issue lets do another one, lets do a slightly harder one|
|29:04||lets find the inverse
what do we do? we switch x and y
okay what are our steps?
first you cross multiply and we get 3xy minus 7x
|29:30||equals 2y plus 11 now we are going to group, factor and divide first we group, everything with a y goes on one side, everything without a y goes on the other side its 3xy minus 2y, 7x plus 11 now we are gonna factor out the y|
|30:05||y, 7x plus 11/3x minus 2 and that is that is the inverse okay some people will have the 3xy on this side and the 11 on that side so thats why when i wrote the answers up, i gave both possibilities there both correct you might of had minus 7x minus 11, everything over 2minus 3x same thing|
|30:41||either form is fine. you either this or this either it doesnt matter|
|31:11||lets practice domain|
|31:44||find the domains of both of those i dont know if you would need to use notation but you might, i dont remember if we require it or not my advice, if youre not sure then first write the answer without interval notations so you can just demonstrate what you are doing even if you screw up with the interval|
|32:04||make sure its clear that you know what you are doing okay whats the domain of the square root of 3 minus 5x you cant take a square root of a negative number 3 minus 5x has to be greater than or equal to 0 move the 5x to the other side divide by 3|
|32:30||the domain you can write either x is greater then or equal to 3/5
or you can right it in interval notation like that
so far so good?
you can not take the square root of a negative number so the domain inside has to be greater then or equal to zero okay what about the second one?
you can take the cube root one out of 1
|33:09||so you can either right down all real numbers, all reals, you can use that fun r symbol or you can right infinity and minus infinity i can take a cubed because itd be 3/5|
|33:30||you take the square root of 0, thats 0 alright so you can include that for square root i didnt i moved the x to the other side alright lets make it slightly harder|
|34:19||how bout that|
|36:21||so we look at this, its just like before we got a square root so 2x minus 1 needs to be greater then or equal to 0 0 we have 2x minus 1 greater or equal to 0|
|36:32||2x is greater then one x is greater then or equal to a half the denominator can not be zero so x minus 10 can not be 0. so x cannot equal 10 so that could be your answer or if you want to do interval notation, you start at a half go to 10 and then you stop at 10 and start again|
|37:02||without including 10, you can do it that way the inverse is fine with me we do not specify interval notation, so you should not feel obligated to use interval notation but you should be doing a lot of webassign interval notation, you should be able to do this by now|
|37:41||okay i will do a couple more. then i will kick you guys out and go and sleep|
|38:22||how bout this one?
sinx equals 8/9, and this tells me im in the second quadrant
|38:36||so i draw a triangle in the second quadrant i put in an x and i do sine now i need to figure out the tangent tangent requires opposite and adjacent and i only have the opposite and i need the adjacent so i am going to need to figure out the missing side so i use the pythagorean theorem 8 squared plus 2 squared equals 9 squared|
|39:01||do a little arithmetic and ill get b square root of 17 because 8 plus 9 equals 17 funny how that works works everytime plus or minus square root of 17, works everytime whats 10 squared, its 100, whats 11 squared 121. whats 10 plus 11, 21 works everytime okay so i want to find tangent, tanx|
|39:35||well is it 8/the square root of 17? not is negative 8/square root of 17 why is it negative? were in the second quadrant|
|40:27||do a composite function|
|40:33||you want to do f(gx) you take gx and you put it inside f so fog of f, if you like to call ot that square root of2/x cubed minus 1 you dont have to do anything to that, you can just leave it as 2/x cubed minus 1 if i want to do g(fx) I take fx and put it inside g|
|41:03||so g(fx) is i take the square root of x minus one and put it in ther heres what i want you to do. I want you to relax, memorize what you need to memorize good luck on the exam and see you next week|