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2011 Calculus AB Free Response #1 parts b c d

Video transcript
now as do part be part be fined in the average velocity of the particle for the time period between times euro and time 60 the easy way to think about this average velocity is that your average or when we think about the distance that you travel sometimes I used for distance or you could use a spur sometimes for displacements use but your distance in general is equal to your average velocity times time or your average velocity is equal to the distance you travel divided by time and so how do you figure out the distance that we travel over that period especially because they didn't explicitly give us the position function well should think about it let's think about the velocity function right over your state did give us the T and I don't know exactly what it looks like we just draw a general velocity function right over here and we care so this is Artie axis and this is our velocity access right over here and we care about between zero between times euro and time six and let's say our function does something like this we don't need doesn't really matter I'm just this is just a general driving over here how do we figure out the total distance over that time well if you take any small time right over here and you multiply it times the roughly constant velocity over that time period than that will tell you the area the area of this little column will tell you approximately will tell you approximately the distance traveled over this very small interval of time because you have time for time time times in approximately constant velocity will give you a distance and so in general if you take the area under this curve the area under the curve from 0 to 6 that will give you the total distance traveled Saudi is equal to the definite integral from 0 to 6 of GFT of this thing right over here VFT GTE and so if you want the average velocity just find the distance which is this definitely Negro and then you divide that by the time or the elapsed time the elapsed time in this case I could actually right times delta T right over here I change in time right over here is going to be six we started your own we finish at six so we just have to evaluate this definitely a role which is not that easy to evaluate analytically and this lucky for us to be do allow us to use calculators in this part so we'll evaluate it using the calculator and then we just have 2/6 let's first find out the distance traveled between times euro and time six and to do that we can use the T 85 we can use will go to catalog tells us all of the different functions they have and to evaluate definite integrals we use the just go straight to the ass straight to the F this one page down so this right here FN into this is definite integrals so I used FN into you could just type that in this way we know we got the case right and everything so the definite integral of instead of instead of output VFT here but alright at his view of actions because axes and easier key to get too so we're just get this is we want to definitely love to sign of to sign of E to the two-year her for so we actually put up another print this year so it's two times a sign of E E to the TV over for awhile to write excellent for our tax instead of T just easier to get to and I'll tell the the calculator that we're integrating with respect to x and not T X divided by four X divided by four clothes that parentheses and then closed at parenthesis that closes this print is our clothes this print is right over there and then I have a plus one so that is our velocity function expressed as a function of acts and I'll sit here that the variable of integration that we're integrating with respect to is ax I could have put their interior and that would've been fine and their bounds are from 0 to 60 six and haven't made a mistake with a parenthesis the should evaluate to something so what does that evaluate 285 is just taking away eleven-point 696 so this is equal to this raid over years eleven-point 6962 the total distance traveled as 11.69 60 change in time is 6 we just have to divide this by that we have to buy the total distance divided by the change in time and so we just take the total distance and we divided by this is just our previous answer divided by our change in time so it's 1.9 494 average velocity average velocity as you go 1.9 4909 4910 the unit didn't give us any units here so that's part B now as do part C artsy find the total distance traveled by the particle from time t cozier at Eagle six lucky for us we actually figure that out as part of part B we figure that out of his 11 points 696 so we just have to write that down eleven points 696 and then finally we're on Part D hard to do this in a way to do this in this blue-collar Part D 40 between forty between 0 and six the particle changes direction exactly once find the position of the particle at find the position of the particle at that time so let's think about the time when the particle changes direction and then we will have to figure out how we can evaluate its position I we can figure out its position at that time so the particle changes direction that means that the velocity that means that the velocities going from either a positive velocity or negative aww city or negative velocity to a positive velocity switching direction so if that would be a point of this is to me drive so this is our time axis and this is our velocity access it must be a point in time so we're switching directions so far velocity function looks like something like that then this would be a point in time when we're switching direction we're going from a positive velocity to a negative outlaw city or or it could be something like or it could be something like it could be something like this we're going from a negative velocity to a positive vote loss but the bottom line is a point where the velocity hits zero and then on either side of that we have kind of switched sides it wouldn't be a point like this we're just going for positive 20 and then positive get this economy something that just going in the same direction after pausing so let's figure out let's figure out where between 0 and six are velocity function is equal to 0 and if we have and they're telling us that the particle changes directly X changes direction exactly once if we find the zero of the velocity function their details of the the particle change direction you could test it on either side to make sure that is changing direction but it's probably a pretty safe bet that if you find the zero that that will be the the time when you change direction so essentially just have to figure out when the velocity is equal to 0 here so the velocity function is to sign to the TV over 4+1 and we want to set this equal to we want to find out when this equals when this equals Iran you want to get a time that's between 0 and six and so we can subtract one from both sides you get to sign of heed to the TV over for is equal to negative one we can divide both sides by to sign heed to the TV over for is equal to negative one half and we can use calculators year but or we could just we could just think about it a little bit in terms of the unit circle which I always like to do if you're short on time you might want to use your calculator but the unit circle since you know exactly what what you're doing when you're taking the inverse sine it's a little bit more useful because there's obviously multiple there's multiple angles right over here that you can put in here that would give you a sign of what happened you want to make sure that what ever answer you get it gives you a team that in that interval that we care about so let's think about the unit circle of the sign of something is negative one half there's two points on the unit circle where the sign is equal to negative one half its right over here where the sign is equal a negative one half and you can view this single right over here negative one-half you could use his name as negative pie over six or you can view this as 2-2 pie minus piracy if you want to view it going all the way around there and then the other point on the unit circle where where you are at the sign is negative one half is right over here israeli renso to get over there we would have gone pie first you go potty and then you'll go another pie over 60 this angle right over here the single right over here is high plus high over six which is equal to 6 pie plus high over six or seven PI over 67 by over six this angle right over here this would be it would be eleven by 11 pile 462 pie minus pie over six if you want to go all the way all the way around there and I and you can add you can continue to add to PI do either one of these two get multiple angles where the sign of an angle is equal to negative one-half malicious try missus tried this over here let's assume let's just go with this first angle this kind of the first angle that gives us a sign of negative one half and see if we solve for t we get a tree that in our interval so we could say if sign of this business over here is equal to or negative one half then this business in here than E to the TV over for could be equal to 7 pile over six could be equal to 7 pie over 60 logic we just did over here and then to solve for t we could take the natural log of both sides so we could say that he over for is equal to the natural log 7 pie over six or that T is equal to he is equal to four times the natural log of seven pie over 60 let's hopefully this gives us a tee that inside of our intervals so let's try this out out let's try this out so we get we want to multiply four times the natural log of 77 hi 7/6 let's see what we get we get 5.19 so lucky for us it is in our internal so this is the team that matters the five-point 1960 say so this is equal to 5.19 six approximately so that's the T word changes direction but that's not just what they're asking us to the particle change directions exactly once find the position of the particle at that time is this is interesting cuz they haven't explicitly given us they haven't explicitly given us the position function but we know that the position function is really just the integral of is just the interval of the velocity function we know we know that the position function X St is just going to be the integral from 0 to 50 VA 50 det + plus some constant and they tell us they give us an initial condition they tell us that X 0 is equal to two they told us that they told us that over over here access 00 222 we actually know what the constant is because if you evaluate this is exit 0 you get X 20 is equal to the integral from 0 to 0 of 250 det + see and this is going to be equal to 20 this part right over here value is 20 so we know that our constant is equal to two so we know that X St XST is equal to the definite integral from 0 to T T GTE plus two and you might say well how do we know the position if we get at a given time well we luckily have a calculator that can calculate definite integrals quite well so if we want our position at 5.19 six seconds we have to take the definite integral from 0 to 5.19 six of our velocity function of our velocity function had to do that and so let's get the calculator back out and let's actually calculate that so once again let's go to our catalog of functions I want to go down to the acts so I'll go down to the ass down one more page no i didnt wanna go down the page down two pages actually so one more ok so there we go this is my definite integral function and i wanna take the definite integral of the definite integral of our velocity function actually let me get it the velocity functional have a green over here actually we've already done it up here this is our velocity functions to let me write it again so the velocity function was to sign two times the sign of E and others user access my variable of integration X divided by four X divided by four close one parentheses closed another parenthesis that gets us there I want to close out one and then I want to do a plus one plus one right over there and then i want to sale my variable of integration is ax and i'm taking the integral from 0 instead of from 06 and taking the integral from 0 to 5.19 six rushes my previous answer so I can just say my previous answer my previous answer and so this will evaluate this will evaluate my definite integral evaluate this first part right over here and then I just have to add to to it I have to add too so it's had to get my answer the calculator think about it for a little bit and then why law fourteen-point 1344 final position is equal to 14.13 45 rounded off a little bit