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AP Calc 4.4b: The Fundamental Theorem of Calculus

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hello everybody Welcome To Part 2 chapter 4.4 recovering right now next team let's talk about what the second fundamental theorem of calculus second fundamental calculator Planet Asia I'm an open interval hey man, or Every Act in the interval we can say the phone Beacon Theater DVD from a tack confunkshun are the different variables is equal to that function evaluated at X example let's look at something kind of simple the derivative media the integral from 0 to a radical what is DP Is Not underneath the radical I want you to understand at first of all a team is equal to keep their plus one that robbed do you know the exact same everything except the square I'm so into it we can say that and then the DJ of the integral from 0 to act where to go 1 what was a simple example but let's look at the question bigger equal to the integral from 5 to 2X who is the importance of attendance info on what we can do here first is recognized something at the derivative that integral of 2x cubed of cosine I want you to notice some differences between this example and we aren't just given an act like a Class Act what is recall in order to do that derivative of Big Apple. Aditya radical Pirates you to excuse me I don't even think about it all the time big , back what I can find by 30 doing the payroll is what I mean by that I can the derivative of big after that so I can say DJ Big app find the derivative of whatever's inside right time the derivative of whatever I'm going to call you and here's what we're going to call you we're going to go to YouTube the app because we don't have a way to handle. BFDI look right what we're kind of used to normally expect I made a mistake earlier here it makes it to you because that's what's helping us integrate red payroll here it doesn't do you because I'm leaving that you there until the very end calling that are inside a function until the very end The BFG you is going to be of the app the integral from crying over to you to ask you DT but I still have you ever Here and Now but we're going to call you in here we're going to call the you the xq because that's what we don't know how to handle it too so we're going to say and we're going to call this thing do you have different color changing here we're not calling an excuse anymore do you still have her find you still have time do you get back and again remember you is at xq and now play the a cube with a single variable calculus that if we have a function inside of a variable thank you. Other variable the cool thing is that I'm not happy but who you who signs of you send it over here or the derivative of you to bring me three what's the derivative of x cubed now let's not forget about you that you was what it was supposed to be a cube x 3 and not our final answer from there I want Peppa Pig. Matt Cain serum I said that if Fantasia on the closed interval from age and big app is an anti derivative Cynthia I need an eighth then the integral from A to B yeah equals big app that that can be Rewritten integral from A to B big F Prime same thing big  MP - big is just as big as Prime right they got crimes in the derivatives of that an example with the information call Mike OK Google at a rate of 180 + 3 liters per minute and 60 Minutes interview find the amount of the chemical first 20 minutes what we're going to do. I using who wants a rate of are we know by now that we're talking about great talking about. rate of change given a rate atomism function of Kira Prime and what we also know is that we have that net change theorem what says we can set up an interview using a derivative and they also gave us an interval of a 20-minute that means I meant to say 20 + 0 2820 and I can go from here and I'm going to find it first of all of the antiderivative of 100 and then I have 3 squared over 2 we should make that three up there next to the teacher and again I said from zero they're going to do that my bad from there is there any B20 can't talk about the 20 I have 120 x 20 3 times 20 squared over 2 I'm not going to become 3600 + 604200 remember in the context of the problem we're talking about Jaden that 20 minutes the paint 4200 I'll leave you alone real life example of vertical motion problem I really want to I want you Matt Cain famous saying and I want you to know the word map and I like positive and mine together plus and minuses together right all the good and all the bad but not me I think I'm going to pay that great your net pay is everything after only what you work for hourly Frozen and everything negative let's go into the example Santa velocity in feet per second a particle -10 squared + 2090 - 20 - 2nd answer this question displacement and the particle the interval from 1 to 5 okay so first things first play Smack That the derivative of talking about it's a backward with the last name so what we can do that great from 1 to 5 not take care of it and it will write from key from me from 1 to 5:50 the intersection and again we doing that over the whole interval because we're interested right now and placement is net all the positive all of that area is I don't want to sign of the whole person - 10 squared + 319 - 20 and I can take the antiderivative of all of these lovely and I get for my antenna keep me up 29 squared / 2 10 - 20 on the interval from one I'm interested in right out and short Google Now with all this would be and then I'm going to talk to you later at 5 to the force for minor 5 to the third of 3 + 20895 times 5 squared over 2 minus 20 times and that would be I did have one for the 4th that would be 1/4 - 10 x 110 and over here 29 again x one kind of simple over to and then everything I do get the basic easier to read that a fraction 25 over - + -103 / 12 will find out 128 / 12 / 3 certain one more thing that a positive answer in this case a positive position that we moved to the right along that line so we put our answer we say that 32 / 3 not my sister one more question about this example let's also answer what is the total distance traveled no I do understand the difference between example if you take one step forward and one step back and you didn't have to go anywhere I have zero The Good the Bad the Ugly hover distance traveled is only going to be the positive result only going to be in trouble how to give yourself a picture of yourself that dear can coordinate plane go in and I'm going to be going from 1 to 5 1 2 3 4 5 468 again we are doing here graphic vehicle we start at 1 Batman we have A3 maximum happening at about 6 Robert / 2 and then we have an accent 4 and I never act Interceptor 5 and so it's important for us to understand at the end of April in this area here get the total distance traveled I'm going to have to subtract the area Orange area right here and I can do that but I'm going to from here to here and I'm going to subtract the integration from here to here beginner bowls are from 1:40 to 5 the integral from 1 to 4 weeks and I'm going from 4 to 5 Anna Kendrick the integral from 1 to 4 a again - 10 squared + 20 - same thing - 10 + 2090 - 20 and so I can start to write out my antiderivative of the 4th 10/3 / 3 + 29 squared / 2 - 20 and not from 1 to 4 dropping an exact same anti Forest / 4 - 10/3 / 3 + 29 squared / 2 - 25 that is so what I still out here Alaska before Ford Sports Authority Uber for and then I would go for 40 / 3 29 x 16 right 4 Square to 1629 * 16 is going to get me 464 with the 101 / 4 521 * 1 / 2 20 times 1 and I say my everything in here when I it's going to look like a 625 / 4 - 125 write 5 cubic 125 125 * 10 is 1250 3 + 25 * 29 gets me 725 / 225 x 900 kind of running put a room but we did already and I'm going to write a minor / 4 - 640 / 3 464 / 2 - 80 and what time easier to work with 45 / 4 - + -7/12 turns out to be 71 / 6 answer in terms of not at all

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AP Calc 4.4b: The Fundamental Theorem of Calculus

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