Center Of Math Live Ap Calculus Exam Study Session (2013)
This is a YouTube Live via Google+ Hangouts On Air study session for the AP Calculus Exam
hi um I’m Kelsey and I’m here to turf for tonight so welcome to the center of mass first worldwide tutoring session hang out we’re so glad you could attend and we look forward to spending the next hour with you before we get started I just like to explain a few things about this hangout so only the first ten people who join will be able to ask questions verbally we don’t think this is great so we’re gonna be monitoring the chat so if you have any questions Lucas our moderator will try to get them to me and we’re really gonna try our best to get through all of them but we’re kind of on our short time schedule we’ve only got an hour so we’re gonna try to focus on questions for people that are taking the AP exam you know beginning of May this year so we can go ahead and get started I’ve got a few response bustin ready the way from our trip and then after I go through it if anyone has any questions put them in the chat or speak up okay so this is a released problem off of the BC AGM from 2009 and it comes with the graphic so let me go and draw that so I’ve got a parabola that creates this region R in the first quadrant and the problem tells me that a baker is creating a birthday cake and the base of the cake it’s in this region are in the first quadrant under the graph of y equals 20 sine of pi X over 30 from 4 X between 0 and 30 and both x and y are measured in centimeters and then I’m also given the derivative of this so that derivative is 2 PI over 3 times cosine of pi okay Part A says that the region is cut out of a 37 meter by 27 meter rectangular shaped cardboard and the remaining cardboard is discarded find the area of the discarded cardboard so let me draw that piece of cardboard that we’re going to cut it out from I’m gonna say 20 by 30 so I can see you might read in our touches the top of the cardboard and then I have these two kind of triangle ish shapes that are gonna be discarded so it’s pretty easy to find the area of the cardboard that I’m cutting out from that would just be to multiply that you sidelines so 20 times 30 60 but the area of the region R is a little bit more difficult and for that I’m going to want to integrate so area under the curve it’s fun integrate a function over my boundaries for X so when you go and set that up so this is my area of our and I’m gonna subtract that from 60 to get the area of the discarded hard work and to do that this is question number one so graphing calculator is allowed I’ve got a ti 83 here and I’m gonna go ahead and use a tfn inch command to evaluate this integral so I’m just going to type this into the calculator so I’ve got 16 minus FN int is it math and then you scroll down to nine if you’re not familiar with that and then type in my function haha and then they eat variable and I’m integrating with respect to Y so X near the lower bound and an upper bound Oh problem okay so type this in and I get 218 point 0 to 8 and X cubed so that’s my answer for Part A and let me go ahead and check they’ve only got one board to work with so I’m gonna check one part at a time and per day you get three points two for the integral which is the integral of 20 sine of pi X number 30 divided from x equals 0 to x equals 30 and then subtracting that from 16 and 17 squared but are you intervention serious so Part A is good you know I’m going to write that answer off to the side and then let’s get started on my eight today so Part B says the cake is a solid base R we know that in cross sections of the cake perpendicular to the x axis are semi circles so tendency it and then if the Baker uses five one hundreds of gram of unsweetened chocolate for each cubic centimeter of cake how many grams of unsweetened chocolate do we need to make this cake well to find out how much chocolate we need I need to find the volume of that cake and I can do it with these particular cross sections so I want to think about integrating the area of one of those semi circles so for area of a spoil of a circle I just need to know its radius so to do that I need to find the length right here right here or here as a function of X and the easiest way to do that is to use this function that I’m given twenty sine of pi X over thirty so that’s my radius well that’s actually my dad either if I put that in half of your radius so ten sine of pi X over thirty and then the area of the circle is PI R squared so the area of a semicircle is going to be 1/2 PI R squared okay so this is my area of one semicircle and if I integrate that over my region X from 0 to 30 I can get some volume of the region and the cake and again I’m going to use my graphing calculator the of N in command and I get two thousand three hundred fifty six point one nine four seven meters cube and then that gives me the volume of the cake but I want to know how much chocolate I used so and if I have point oh five grams of chocolate per cubic centimeter so if I’m multiplying the amount of talking for one cubic centimeter by all of the cubic centimeters then I want to talk what I need so I’m gonna put this in my calculator two three five six point one nine four multiplied by 2005 I get one hundred seventeen point eight one zero grams so my answer for Part B is to make this cake I’m gonna be a hundred seventeen point eight one zero grams of chocolate and I can check it and again Part B is three points two for the integral which matches what I’ve got and then one for the actual answer which is not the boiling of the cake but and let’s check out Part C so C asked to find the perimeter of the base of the cake so that had darker region yeah so the easy part is its base right there which is thirty centimeters and then for the top part I need to find the arc length of this parabola and this is a good formula to know for the exam but the arc length of a function Rabelais it’s the integral over some region so here we have x from zero to thirty of the square root of one plus the first derivative of the function squared here I’m given the first derivative so I’m gonna go and plug that in okay and again this is something that I need to put in my calculator again Evan and is our friend for this problem and this is a little tricky to put in so just make sure that you’ve got you know parentheses everybody need are my calculators evaluating this integral and I get eighty one point eight zero four row excuse me fifty one and then I need to go back in and add this base and once I do that I’ll get 81 my perimeter of the cake let’s check this with you so they have one expression they have 30 plus this integral equals anyone would 8: 03 or eight zero four depending if you truncate the decimal or wrap okay so everything’s all set with this ever cube does anyone have any questions about it excuse me while our lovely moderator Lucas monitors the chat let’s just give him a second are we good we’re all set so I pulled a couple multiple choice questions um there various sites on the internet that have released multiple choice questions available they’re a little bit harder to find than the free response questions but I found a really great one online that had bunch so I think got some good ones that we can work off of and they’re off of the Navy ensign so we cut the diagram and go ahead and take a look at the first question so in us what is the x coordinate of the plantar flexion on the graph of y equals one third X cubed plus 24 okay and I’m given five choices I was writing for you on the side I’ve got five zero negative ten thirds negative five and negative 10 okay so then we’re gonna remember how to find my reflection any volunteers on how to do it quite a bunch today so to find a point of inflection you would have set the second derivative of a function equal to zero so first I would find that second derivative to the question thank you who was that Susan shout out to Susan good job okay so let’s go ahead and find the first derivative of this function so we can find these signs and this is the power rule so remember multiply by the exponent and then subtract one from the exponent so I’ve got 1/3 X cubed I’m gonna multiply that 133 1 from the exponent and get x squared C with 5x squared multiply that by 2 to get 10 and subtract one exponent is X and 24 is a constant so when I can’t get to riveted thousand times 0 that’s my first derivative and let’s go ahead and find the second ok x squared again multiply by the exponent subtract I wanna catch X 10x it’s gonna be time so now we want to set this equal to 0 find my points of inflection so I know someone’s got the answer for this one we’ll get struck the track negative 5 that’s good thing I know Susan Thank You Susan that’s right so D is our answer okay what’s this okay so this is a graph of a piecewise linear function f which this picture looks a little bit prettier on the paper I’ve got this is my best for X between negative 1 & 4 and they asked us to find the integral of f of X that mineral from x equals negative 1 to x equals 4 well I don’t actually have a function here to integrate so does anyone else know another way that I can do this without actually giving any integration interact into the group chat or those who recently joined us oh yes okay I’m the area area uh. what under the curve perhaps yes okay so area under curve will give me a definite an area above the x axis is considered positive and then below the x axis is considered negative so I see you kind of three regions I want to split this into one is this trapezoid guys another is his little orange rifle and the third is a little yellow square and my little triangle and little square are both going to be negative since they’re below the x axis never getting this trap so it’s going to be positive I have a base of three and a k2 and the base is one so Erica trapezoid is one half piece one plus base two I’m so high it’s gonna call those things in times one plus three times two so the one half of the two cancel each other out and I’ve got a positive four for the area of this craftsman and now we’ve got this triangle and the square to worry about so let’s do the triangle it has a base and a height of one so remember area of a triangle is 1/2 base times height so that’s just 1/2 times 1 which is 1/2 and then we’ll see us on the square these square has a side length of one so the area of the square is gonna be 1 squared they’re not gonna combine these three areas I’ve got 4 minus 1 minus 1/2 3 minus 1/2 negative and then I can see that that’s one of my choices our religion have given me the others but they were one for five point five and eight so we got this one right anybody have any questions we do it up a couple of seconds to see if anyone had any questions plus two point five yes oh this probably looks like a you’re supposed to be in equals yeah thank you very much and also nikki poulos hell says that it’s positive this correct thank you for your help yes thanks nice this track this doesn’t want to write there any questions okay so the next question I’ve got is just a definite in a row really straightforward okay so it is the indention rule of 1 over x squared DX from 1 to 2 and my choices are negative 1/2 7: 24 1/2 1 or 2 Ln of 2 okay so this can be rewritten a little bit easier that 1 over x squared does anybody know how I can rewrite it you yes we are multiple answers accident they gave two perfect ice everyone’s in there good job and now I can use the power rule great so I want to add one to the exponent so I’ve got an X to the negative one and then so negative x to the negative one which judging by your last answer seems like you guys all know that it’s like whatever X so first I’m going to plug in x equals one so I have negative 1 over 1 which is 1 then I’m going to subtract plug in x equals 2 so I am subtracting a negative 1/2 which is like adding one hat so negative 1 plus 1/2 is negative 1/2 so that’s should be a dollar it guys Susan says put the chew in first nice catch nigga got your painted veterans you’re right definite integral’s always the top in first and then subtract the bottom telematics even says yes you’re doing great thanks okay so like that more I can do but does anybody eat how many specific things that they want to cover any questions they’ve got maybe you forgot different Taylor Maclaurin series any suggestions you know slogan says solving in a differential equation I did thank you okay first order and second order first order okay I brought my textbook with me let’s see if we can’t find some exercises on that is that a world like differential calculus book I’m just wondering system it is actually oh this is our AP Barbary people fitting for this hangout okay let’s check out the 141 is that we gotta run so it looks good perfect okay so I’ve got one question for you do we know how to integrate or if you want to do this another way separation of variables question yeah so separation of variables I want to rewrite my rhyme as dy/dx a little too easy so then I want the dy/dx on one side so I can integrate both sides right I’m looking for yes yes okay so you get that the XT other side or he’s going to multiply both sides by so cancels over here so now I’ve got dy equals 8 and x squared plus 7x all times DX now integrate both sides so I’ll do the easy one know the integral of dy is just going to be Y even can you help me out on the integral for the other side fair I’m the easy one okay so I can help you up so 8x squared we’ll just go turn back to firm I’m gonna add one to the exponent so I’m going to get X to the third and I want to divide by that so we’ve got 8/3 X to the third plus let’s do the same thing for seven x add one to the exponent which is an invisible one to get x squared and then divide by that you alright Tommy do it out of let’s see what’s up yes both or just one yeah both says David you can do both I was talking that when you’ve been great you down that constant let me make it a little bit easier all of these I was thought to just add BC to the wealth to the left side and then you can always think about it adds like a capital C and then state somewhere but that equals little see mines FBI got in the other side we good on that one is your harder one and it’s entangle devotees on this okay I’m gonna step out for a second check our second computer about if we have some initial conditions and a little bit harder okay I can do harder and I can do an initial the harder we were looking for her own says nice snakes okay so I want to group the dy with the variables with respect to Y that are y and I want to group the DX just we have a question false alarm okay I want to group the DUI with the Y function area and I want to group the DX with the function of X did somebody tell me how to do that both sides by DX yes sir I want to group the dy with okay okay now I’ve got the DX with the text friends but I still need the y squared goes forward to head over to the dy so I multiply both sides by Y on this side okay was born d y equals x sine of X DX and now I’m going to be integrate both sides all right well so I’m giving me the answer to go right yeah can someone give me the energy to the left while we’re working me on them the answer to the right side the only thing that’s the right way we’ve got Y cubed divided by 3 plus 4y that was answered by multiple participants even I believe I’m gonna add a plus c1 and now I will do you all the honor of toggling this integral but can someone tell me what technique is product role is their response from Natalie by parts no response okay controller usually by differentiation techniques but they do have kind of their opposites in the integration world and when I think of kind of the opposite the product rule I think of by parts by parts be pool responded that right before you say oh nice okay so at least you guys might use different variables so let me go ahead and break out with it so I have a mural of you easy equals UV my zero video does that look like we’ve worked on the original microscopy you seem to be agreeing okay good so what should my view be X yes because then my D you over here so I want to differentiate D I want to differentiate you to find D U which is just one DX and then for DV to find V I want to integrate can’t even give me the integral of sine of X negative cosine X what was that Susan Susan I’m going today and I don’t know about you but sometimes I get a little iffy on these trig integrals differentiation they’re doing so many times I forget where the negatives are so when you always just check this by differentiating V to make sure I get that you so the derivative of cosine is negative sine x n 1 and I’ve got sign next you then I trusted you compromise that was for me okay so now let’s just write everything back and I’m going to plug some stuff in so for UV I get negative X cosine of X and then for v vu I’ve got negative cosine of X DX no that’s right right everyone no complains it’s a good time alright so that could have been we’ve already learned that sometimes I’m a little over crosses on my negatives but I’m subtracting a negative cosign I was going to cancel out those two negatives it makes it easier and then I don’t risk losing track of that negative liquor so X cosine of X is just gonna hang out it’s good then I want to integrate cosine of X DX I bet someone can give me the answer for that one anyone sine X Y s o+ c everybody home car guests without long integrals I’m very impressed sure does if we want them in a neutral condition can anybody kind of throw one out there and then work on solving Bertie you I think we got one Tama negative for the coordinate okay I can work with Adam David it I’m gonna 1 everywhere I see an X and a negative 4 everywhere I see a Y right ok if it make it a little easier on me how about 0 negative 4 we compromise on that one he says it might be a little rough because he just made it up so I think yes he’s definitely okay with it ok all right so these are pretty easy to figure out you got your pocket third is negative 64 and then somebody cosine of 0 is follow up I’d also like the sine of 0 1 yes that’s equal and son 0 is 0 okay so I’m gonna have this sign together notice all the minus and the negative on the target are two different buttons so this adds up to be negative one whole over 3 plus C 1 plus C 2 minus 1 so I’m gonna add 1 to this side and I’m going to subtract C 1 from both sides and move it to the other so I’ve got negative 1 and I number 3 equals C 2 minus C 1 and solving for C 1 C 2 individually I can get this me tune to the other side or C 1 excuse me so I can subtract C 1 from both sides and then this cancels out C 3 david says yes I can call C 3 C 2 minus C 1 so you had mine nicely done yes gotcha all right does anyone else have any suggestions we’ve got 10 more minutes left we’ve got a question yes Natalie what about this where did the X in front of the negative cosine X go okay use I forgot it that was a great catch big catch I am impressed so this should have been a zero right oh it’s better right and then that Li says that’s great she’s usually the one who’s doing that so well let me tell you no matter how advanced you are you still make mistakes the author of the cocktails book actually made a great error happens to everyone all right any more questions music Justin that’s the room now do you know my dad you really miss jessie’s I mean I have more to work on but I would I prefer to meet the needs of the community if you guys have anything else it can be about anything car feels really good any questions probably in Sam’s life I have taken the all right how about a couple area and volume David just asked area and boy we do have one question from Susan that mr: quick did you put your answers in a box on the free response questions I suggest you I mean AP graders they’re gonna read your all of your work obviously but a lot of stuff to look at if you can put a box around it it’s just easier for them to look at and you want to worry about then be like what are your interests so yes we’ve got our suggestion of area and volume from David and Natalie asked for cross sections okay so I’m gonna take airing volume yeah I bring you here for our first FRU because that covered a little bit of cross sections David just confirmed cross sections so you can send a variant volume but okay whatever works I think I didn’t know did Natalie miss that one sir oh yeah I’m pulling on the wrong section sorry guys looking for a good one all right I know you a boy but I’m having trouble finding cross sections I didn’t do crossing okay so let’s say I’ve got this region the first quadrant yeah and it’s bound by the gravel and sex in new line and I have a solid constructed from it whose cross sections perpendicular to the x axis our rectangles with a height of three times service that I can point together that’s y equals x square oh yeah well I’m gay guys thank you okay okay so I’ve got this region and I’ve got cross sections perpendicular to the x axis that are rectangles with a height of three times their base so I’m going to integrate the area of one of these rectangles and recall area of a rectangle is base times height so can anyone give me an expression for the base of one of those rectangles you rectangle of equals height or Kangas height equals three times base yes okay and I know it’s base different ones with looks like this obviously I’m no artist that’s why I major but I have feet like rectangles that sit like this and point out of the board so they’re beasts sits in this region so what would the length from there to there be no help basically equals x squared yes David good job then I try you can also tell me the pipes for I think it’s probably for at some point in here but I do have a variable height since it’s three times the base the base expands as I go up then you wouldn’t think of an expression that involves X that gives me the high number these rectangles DX cutie Marty asked there’s gonna be X in the interval but it’s the height of one of these rectangles gonna be three times the base we got 3x squared that’s height Peter just answered that Peter nice job so that means the area of one of these rectangles is going to be 3x to the fourth great is multiplying these you’re not going to integrate that over my region doesn’t even know what my bounds of integration are going to be I’m leaving for bounds on X yes I’m going right start my first point of intersection zero zero then I’m cut by the way x equals two okay so three X to the fourth it’s pretty easy to integrate so I want to add one to the exponent and then divide by that exponent does anyone have an answer for me I’m with the you’ll have to do the evaluation part just what the integral that’s going to be in general not only says 3x the fifth divided by five perfect okay who dosed a fifth power of two off the top there thirty two I know yes so I plug in the upper bound first two I’ve got three times thirty Coomer four five plug in the lower bound x equals zero so this whole term second term drops out I off with three times thirty two over five which you don’t need my calculator 496 yes now let’s say this is on a network you and this sum what I’m making is like a new children’s toy or a cake like it was in the first problem I tell you that x and y are in meters what would the units be for this someone mentioned that it’s a bit over nineteen like nineteen point two day or just a ten three meters can I’m sorry yes yes nice carrot and that is one de that I can give you all on the exam it’s to watch for units if all of those ever use they know that you can do is integral they want to see then you can apply to the problem good units are always a big plus for anything graders okay well we’re about out of time does anyone have one quick how do I do this very good tomorrow you to stream related rates are pretty time consuming um but we do have a video on our website I’ve done a related rates problem for the center um you actually have a series of our communities and think we’ve got five so far and one of them covers related rates so once we finish up all that who gets comment with the link to that video and that is a release question I believe from the 2005 it’s yeah but it’s definitely Todd forward one and hopefully that video will help you out it goes through a whole overview and checks answers at the end take your time but don’t wait don’t drag on forever some people are like oh it’s not going to be enough time but if you kind of think through everything it totally can be and make sure you understand how much time you have for each section I don’t want to scare anyone but in my efforts the proctor actually cut us ten minutes short on the first section but then someone had read all the instructions at the beginning and raise their hands like look you get 30 minutes you only gave us 20 so there are a lot of instructions at the beginning and they tell you to go over it I know you can be like this but do this important stuff and never ever talk about the multiple choice until it’s released I mean you sign your life away on that one okay starting we’re all set thanks so much for tuning in guys about having one more quick question yes is there a topic the exam focuses on most I wouldn’t actually be able to tell you got off the top of my head but the College Board does publish some data I believe in here it’s kind of like a course guide and there I’m pretty sure there’s a distribution of how much show up in the multiple choice and I know I’ve seen one for the are accused so again if we can find a link for that which I think I know where it is will comment either was that someone on the chat yes we will to share it on Google+ and I think we can mention that this is going to be hosted on our our page so if you wanted to go over something again in the future that’ll be available for everyone all right thanks for being a great audience guys and thanks to everybody and answered my questions when I asked for you coulda done it without you guys..
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