### Stony Brook MAT 123 Fall 2015 Final Review: December 7, 2015

 Start okay solve for x this is a quadratic in disguises so we did ones like this before so we see this and multiply by e to the x that becomes e to the 2x so we get 2x e to the x squared 0:32 minus 12 equals 4e to the x now subtarct 4e to the x so you get e tot he 2x mijus 4e to the x minus 12 equals 0 you cqn do a substitution at this point if e to the x bothers you youj have y equals e to the x which makes this y squared 1:00 minus 4y minus 12 everyone see that okay? okay that factors very nicely y minus 6 y plus 2 an thats means that y =6 y=-2 but y is really e to the x 1:35 so e tot he x equals 6 or e to the x equals -2, e to the x always positive so you can throw that out and of e to the x equals 6 x equals natural log of 6 thats not very hard right? 2:00 well do you think thats a minimum competence or thats a part 2? who votes for part 2? there you go its a part 2 lets try something similar what if i had 2:30 what about that this factors into sinx-3 sinx-1 again you can do distribution if you have y equals sinx here sinx equals 3 the sin will equla 1 sinx cant equal 3 3:00 x is between 1 and negative 1, including 1 and negative 1 you can throw that out they just need to know where does x equal 1 on this interval you are suppose to know x equals pi/2 the review sheet i posted on a couple days ago was actually for the 126 class so must of that also applies to this final and stuff feel free to look at that i put that uo on saturday, friday 3:32 i forget lets do another ond lets do another thing you guys should be able to do 4:10 ill think of another one, in the meantime ill give you something else to do 5:21 alright a half life problem its a half life problem so what about setting up the half life equation 5:46 anything thats got exponential growth you write it a y equals a times b to the x so it says initially we have 40 grams so thats when we know that aequlas 40 6:01 or if you want to show it that means that 40 equals a times b to the 0 so b to the 0 is 1 so this is just 40 to the a so now we can go to our equation and you can substitute in 40 for a now we just to find out b, with a full equation we can find the half life 6:33 so i know that 6 hours later i have 30 grams 30 equals 40 b to the 6 you divide both sides by 40 you get 3/4 b to the 6 you need to solve b b equals 3/4 7:00 to the 1/6 that means i can now plug this in here and y equals 40 times 3/4\ to the x/6 now i just have to find the half life whats the half life, divide by 1.6 times x so half life, i want to get 20 7:33 20 equals 40 times 3/4 to the x/6 divide both sides by 40 you get a half 3/4 to the x/6 i take the log of both sides take the log of both sides i can take any log i want the natural log i can do common log log base anything 8:03 bring the x/6 in front you get log 1/2 equlas x/6 log of 3/4 do a little algebra and solve for x 8:32 you get 6log of 1/2 over log of 3/4 howd we do on that? 9:01 wy is b to the 0, initially the time is 0 next question no lets do something else 10:09 if you has f of x equals 6x squared minus 24x minus 30 and the bottom you have 2x squared minus 2x minus 12 find the vertical asymptote, the horizontal asymptote and the x intercept y intercept lets find the various pieces of this thing 10:32 we need the verticle asymptote the horizontal asymptote, the vertical asymptote is also known as the the curve and acorss the horizontal asymptte is the end behavior thats what happens when x goes to infinity you can cross the vertical asymptote but you cant cross the horizontal asymptote the x intercept which are the values where you cross the x axis and the y intercept 11:01 is your values when you cross the y axis well if you want to do this the first thing youy want to do wis demostrate something by facoring that becomes 6 times x minus 5 x plus 1 11:31 this is 2 times x-3 x plus 2 ots not my job to tell you how to factor suppose to know that so far so good? vertical asymptote look for when the denominator is 0 the denominator is 0 in 2 places x=3 and x=-2 the horizontal asymptote well now we are looking for the end behavior, looks at what happens to thei 12:00 as x goes out to infinity so you look at the power on top, the degree of the top is x squared the degree of the botto is x squared so this is 6x squared and this is 2x squared so 6/2 is 3 so this is going to approach y=3 remember thesE? x intercepts are where the function is going to equal 0 or easily where the numerator is equal to 0 12:31 so the numerator will equal to 0 at x equals 5 or x=-1 or you can write that as coordinates okay you can do either way doesnt matter and y intercept is what you get when you plug in 0 so when you plug in 0 on top what do you get? -30 when you plug in 0 in the bottom you get -12 13:02 so its -30/-12 you dont need to simplify that if you dont want to but if you do you get 5/2 we would not ask you to graph that how do i find the y intercept? y intercept is when x equals 0 i plug in 0 on top i get 0,0,-30 and the bottom you get 0,0, -12 13:30 so its -30/-12 good lets do some other type of stiff okay lets do the domains for a minute whats the domain of sinx 14:07 all real numbers, you get sin anything you want how bout cosine x also all reals to give a funtion to have sinx or cosinx in the numerator or theres not denominator then that part you can plug in anything you want what about e to the x 14:33 dont confuse domain and range you can take e to any value you want you can do e to the anything is the domain is all real but an important thing to remember about e is the range is only positive numbers so e to the someting is always positive sin and cosin range from -1 to 1 e to the x always comes out a positive number and you plug in anything you want 15:04 lets see whats the domain of log of x, any log it doesnt matter the domain of log of x is positive numbers also known as x is greater then 0, remember the log is the inverse of e so log always comes out a positive number so when you are doing natural log you can only plug in a positive number 15:34 you cannot take the log of 0 and you cannot take the log of a negative number what about 1/x the domain of x connot be 0 because you cant have 1/0 in fact it can be any number on top, any positive and last what about square root of x 16:02 x has to be greater then or equal to 0 because you cannot take a square root of a negative number alright now lets put it all together and lets say i have f of x is whats the domain 16:31 so if you want find the domain of this this is what you should pay attention to when you take the square root you take the square root of a positive number so x+5 has to be greater then or equal to 0 so x has to be greater then or equal to 5 and x squared - 3 cant be 0 17:00 x cannot equal plus or minus the square root of 3 and thats good enough lets do another one suppose i had 17:48 this is a little tricky first look at the numerator the numerator can do anything it wants im not to concern about the numerator so the key is just 18:01 then you look at the denominator and say well i have a log so x-2 has to be greater then 0 x has to be greater then 2 but we have another problem the problem is you can have 0 in the denominator so this cant come out 0 so when is log x-2 equal to 0 well when you do the log of 1 18:35 the log of 1 is 0 so that would be when x-2 equals 1 x cant equal 3 it can be greater then 2 but ca be 3 thats probably a little over tricky but i thought id throw that in tomake sure you guys know all those things 19:00 alright lets do some other kind of stuff im sort of going back and fourth for part 1 and part 2 stuff 19:30 just so youre prepared for everything alright thats not to hard so if you want to find the equation of a line you need a point and the slope but we already have 2 points so we basically want 20:02 use the form y-y1 equals the slope times x-x1 we already have a point so you can either use 5,10 ad 12,7 either it doesnt matter now youre just gonna find slope the slope is 10-7 over 5-12 so this -3/7 3/=7 20:32 so the equation is y-10 =3/7 times x-5 or you could use y-7 and -3/7 times x-12 doesnt matter if you simplify you can arrange and end up in the same spot so you also have to find the equation of a line 21:41 so we need the equation of a circle with the center of h,k and the radius of r x-h squared plus y minus k squared equals r squared so for example 22:02 the centers at -3,2 the radius is 5 and it will be x+3 squared y-2 squared equals 5 squared that wasnt very hard how can we make this harder? lets see suppose i gave you 22:36 suppose i gave you that whats the center? whats the radius? 23:00 i recommend you factor it dont need to complete the square you just need to factor it 24:27 x squared plus 8x plus 16 24:31 is x plus 4 squared right thats x plus 4 times x plus 4 how do you know its the square root of this and the square root of this y squared minus 6y plus 9 is y minus 3 squared cause its the square root of y and square root of 9 so now you know that the cente is at -4,3 25:01 and the radius is at 10 lets do something slightly messier well you have the equation the equation either this or this so if i said whats the center and whats the radius when you see it in this form 25:30 you have to be able to transform it into this form so if you had something like x squared plus 3x plus 225 how do you know how to factor that well this is a perfect square and this is a perfect square and this number in the middle is doubled this number then it factors into this 26:04 the key is the middle number has to be double that square root or generally x squared plus 2ax plus a squared the factors of x plus a squared so square root of a double that number is 2a 26:33 and if you have a minus, so if you had square root of 36 is 6 12 is double 36 so it owuld be x minus 6 squared if you have that pattern okay binomial thats the square root 27:06 how about some inverse syuff you like the inverse 27:49 alright lets find the inverse of this y=8x-4 over 7 you switch x and y 28:04 x is 8y cubed minus 4/7 you dont have to switch the x and y you cans witch it at the end doesnt really matter e is going to isolate the other variable now what do we do? you multiply both sides by 7 you get 7x 8y cubed minus 4 and we add 4 divide by 8 and so on so 7x 28:31 plus 4 is 8x cubed divide by, 8y sorry divide by 8 and take the cubed root and heres something very important you get so excited you get to this part youd be so happy youd be so happy you then forget to write f 29:00 inverse of x is the cubed root of 7x plus 4 alright dont forget the last step okay all you did hear was find y you didnt demonstrate you know the inverse were we able to find the inverse okay lets move onto the next stuff 29:59 lets see you do those 3 30:01 things f of cubed x we take g of x and we plug it inside of x so everywhere i see an x i put in an x/7 and you can sort of leave that alone i dont care if you simplify that but if you really wanted to you get x squared 30:30 over 49 minus 4x over 67 but if you really want to show how great you are x squared minus 28x over 49 but you know they are all the same to me you demonstrate which ones goes inside which one what about the last part, f of x plus h well what is f of x plus h 31:02 f of x plus h is x plus h squared minus 4 times x plus h what is x plus h squared? i put that on the instagram thats x squared 31:30 plus 2xh plus h squared i think i put that on that, i recommend you remember that if you cant just foil it out, i recommend you remember it minus 4x minus 4h so if i wanted to find f of x plus h minus f of x f of x plus h minus f of x 32:00 is this minus f of x which is this thats x squared plus 2xh plus h squared minus 4x minus 4h thats f of x plus h, minus x squared minus 4x the clue whenever you have a polynomial 32:31 when you have f of x plus h minus f of x all the polynomial terms will cancel the terms of the left side, the f of x plus h so everything that will be left will have an h in it so you would cancel and get 2x h plus h squared minus 4h, how did we do on that able to do that part? thats a good sign that makes you very happy because thats what part 1 stuff 33:06 lets do something else was that hard? 34:17 now we are going to give you on the test the sin of a plus b and the cosin of a plus b ill write that down for you pay attention its on the cover sheet 34:33 the sin of a plus b is sin a cosinb plus cosina sinb and cosin of a plus b is cosina cosinb minus sin a 35:02 sin b you dont know this youre not gonna know the others if you can figure out the sin plus b you can find out sin minus b alright so lets do these we have a couple of stages with this so first\ you just have to figure out missing pieces 35:33 sin of a is 9/10 the sin of a is 9/10 36:00 and here in quadrant 2 so that means this is 9 and this is 10 and you use pythagorean theorem and thats the square root of 19 and negative you should just know that in the 2nd quadrant cosin and sin are negative the cosin of b 36:31 is negative 7/8 and were in the third quadrant thats negative 7 and thats 8 thats the square root of 8 square minus negative 7 squared which equals minus 49 thats the square root of 15 and thats gonna be negative found in the 3rd quadrant 37:11 so we want to find sina cosin b and this is minus so its gonna be minus cosina sinb well sina is 9/10 37:32 cosb is -7/8 cosa is - radical 19/10 and sinb is -7/8 i dont really care to simplify 38:33 how do you do cisn2b again you dont have to know the double angle formula cuase if we tell you cosina plu b is cosacosb minus sinasinb to find the cosin of 2b you can write this like b plus b so cosb cosb minus sinb 39:00 sinb thats all you need to do so to find the cos2b its gonna be -7/8 -7/8 minus negative radical 15/8 39:31 negative radical 15 over 8 40:18 couple other things how do we feel about these? good? im gonna erase this whole mess 41:28 do this one 41:34 something like this is very straight forward basically you do the two parts seperately so where does 5x squared minus 2x equal to 0 well 5x plus 2x squaed 5/2 equals x squared x is plus or minus squared root of 5/2 42:04 but that is the piece for when x has to be 0 so you can throw out the positive answer so you can only use x equals negative square root of 5/2 you look at the other part of the function and set it equal to 0 3x plus 6 equals 0 or x equals -2 but thats in the branch where x has to be greater then or equal to 0 42:32 so you throw that out the only solution is negative square root of 5/2