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We're asked to multiply 65 times 1. So literally, we just need to multiply 65, and we could write it as a times sign like that or we could write it as a dot like that but this means 65 times 1. And there's two ways to interpret this. | áá˝áźáášáąááŹášáááŻá áá
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As we begin our journey into the world of economics, I thought I would begin with a quote from one of the most famous economists of all time, the Scottish philosopher Adam Smith. And he really is kind of the first real economist in the way that we view it now. | economics ááᯠ|
So welcome to the Lean Launchpad class. If you're interested in the logistics on how to take this class, this lecture zero will describe it. But if you want to just get started, click on lecture one. | Lean Launchpad áá˝ááááłáááŻááŤááášâá |
We're told to factor r squared plus 4r, minus 45. And there are simpler ways to factor this, because this has 1 as a leading coefficient on the r squared right here, but what we're going to do is we're going to factor this by grouping. And when you factor it by grouping, what you want to do is find two numbers whose pr... | r²+4r-45 ááᯠáááźá˛áááŤáááš áááźá˛áááš áááŻá¸áá˝áášá¸áąáᏠááášá¸ááášá¸ááşáŹá¸ ááá˝ááášá áĄááášáąážááŹáášááááŻáąááŹáš r²ááźááš áąáťááŹáášáąááŹášááášáášá¸á 1 áťáá
ášááášá ááŤáąáááášá áá˝áźáášáąááŹášáááŻá ááášáááŻáąááź ááášááźáŹá¸ááᲠáááŻáąááŹá áĄáŻáášá
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ášááŻáá˛á áąáťáá˝áŹáášáąááŹášáááášá¸ááᯠáá˝áŹááşáášááášá ááŤáááŻáá°áááŻá áááášá¸áąááźá a áá˛á b ááŹá¸áááŻáášááášáááŻááášá ... |
So let's look at all of the different combinations that goes to negative 45, and see if when you add them, you ever get to 4. So let's take the combinations. Let's look at their different a's and b's, and let's see what happens when you sum them. | -45 áťáá
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ášááá˛á 1+(-45) áá˛á áąááŤáášá¸ááášááᯠáá°áááŻáášáááš 4 áá˛á ááśáŻá¸ááááŽá¸áááŻáášááŤáá°á |
So I could try 1 and negative 45, but if you take their sum, that's not going to get you anywhere close to 4, even if you swapped them, negative 1 and positive 45. Let's see, if you do 3 and 15, 3 and negative 15, or negative 3 and 15, those are way more than 4 apart. | -1 áá˝áášá (+45) áá˛á áąááŤáášá¸áááš áááŻáąáťááŹáášá¸áá°ááášáąááŹáášáá˝ áááŽá¸áááŻáášááŤáá°á¸ áá˝áźáášáąááŹášáááŻáá 3+(15), 3 (-15) ááŤáá˝áááŻááš |
We're going to have to find numbers that are closer together. So let's see, 3, 4, 5 and 9, that looks interesting. 5 and 9 are 4 apart. | - (3+(15)) áąááźáĄáŹá¸ááśáŻá¸á 4 áá˛á ááźáŹáąáááŤáááš áá˝áźáášáąááŹášáááŻáá áááźáŹáá˛ááááášá¸ááᯠáĄáá°áá˝áŹážááááš ááŤááᯠ3,4,5 áá˛á 9 áááŻážááášá ážáááąáĄáŹáášá 5 áá˛á 9 á 4 áá˛á ááźáŹáąáááŤáááš áááášáááŻá 5+(-9) ááááŻáááš áá°áááŻááá˛á áąááŤáášá¸ááášá -4 áťáá
ášáááš áááášáááŻá -5+9 áááŻááášáąááŹá áąááŤáášá¸ááášá 4 áťáá
ášááźáŹá¸áťááŽá ááŤááᯠ-5 áá˛á +9 á áĄááŻáášááŻáášááŤáťááŽá ááŤááᯠáá˝áźáášáąááŹášááŻááááŽáá˝áŹ ááŹááŻáášá... |
And the second group right here, has a common factor of negative 5. | -5 áááŽáááášá¸ ááᯠááśáŻá¸áá˛á ááášáááŻáášáąááááš |
So let's factor it out. | -5 ááᯠáĄáťááášááŻáášáááŻáášááŤá
ááŻá |
Negative 5r divided by negative 5 is just an r. | -5r áĄá
áŹá¸ -5 á r áťáá
ášáááš |
Negative 45 divided by negative 5 is a 9. And then we have an r plus 9 in both terms. r times r plus 9, negative 5 times r plus 9. So we can factor that out. | -45 áĄá
áŹá¸ -5 á 9 áťáá
ášáááš r+9 á áá˝á
ášááŻááśáŻá¸áá˝áŹááá˝áááš r(r+9), -5(r+9) ááŤáááŻáááš factor áááŻááŻáášáááŻááááŤáťáᎠááŤááᯠ(r+9) ááᯠááŻáášáááŻáášáááš r(r+9) áĄá
áŹá¸ (r+9) áááŻáááš r áťáá
ášááźáŹá¸ááášá áťááŽáąááŹá -5(r+9) áĄá
áŹá¸ (r+9) á |
And then negative 5 times r plus 9, divided by r plus 9 is negative 5, and we're done! | -5 áťáá
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(Applause) | (áááşááŻááşááśááťáŹá¸) |
(Music) (Applause) | (ááŽáááś) (áááşááŻááşááśááťáŹá¸) |
Human beings have always observed that is you have an object that is moving, so this is a moving object, traveling to the right here, that it seems to stop on it's own. That if you do nothing to this moving object, on it's own, this object is going to come to a stop. It is going to come to rest. | áá°ááŹá¸áá˝áą áĄááźá˛ááᯠááźááşáá°á¸ááŹá¸ááŹá áá˝áąáˇááťáŹá¸ááąáá˛áˇ áĄááŹááášááŻáá
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áşáááşáážáŹ ááźááşáˇááźááşááąáŹáˇáá˛á áá˝áąáˇááąáá˛áˇ áĄááŹááášááŻáá
áşááŻááᯠáááşáá˝áąáˇá
áąááťááşááťážááş áĄáŹá¸áá
áşáᯠáááşáááŻááş áááş... |
you have to apply some type of a net force to it. And once again, this is completly consistent with everyday human experience, this is what we've all experienced our entire lives. But then these gentlemen show up in the 1600s, and you might be surprised to see three gentlemen here, because this is about Newton's first ... | áĄáŹá¸áá
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áşááąáŹááşá ááŹ... |
So the completly opposite way of thinking. For over 2000 years, objects tend to stop on their own, if you want to keep the movement, apply a force. These guys say, | ááŽááąáŹáˇ áá
áşáááşáˇ áááşáˇááťááşáááş áĄáá˝áąá¸ááŤáá˛á áážá
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áşááŻáááşá¸ááąáŹáááşá¸á áááşááąáŹááşáážáŻááźáŻ (áááŻáˇ) áĄáážáááşáááŻá¸ááąá¸ (áááŻáˇ) áĄááťááşááźáąáŹááşá¸ááźááşá¸ áááŻáˇááźááşáˇ ... |
Many thanks to Sal! :) | Many thanks to Sal! :) |
In the last few videos we wrote a nice little program here that asks the user for some type of input and then computes the factorial of that number and then prints it out. And thats nice, but you could imagine a world where you would want to calculate the factorial in multiple places or in multiple different programs o... | áĄáááš ááŽáźááŽáááŻáąááźááŻáášá¸á ááááŻáááášáąááŹáášá¸áąáá¸áąááźááᯠáá˝áźáášáąááŹášáááŻááááŻá ááŽáá˝áŹ áąáá¸áá˛áˇáżááŽá¸áżáᎠáĄááśáŻá¸áťááłáá°áąááźááᯠinput áĄááşáłáá¸áĄá
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ášááŻááᯠáąááźá¸ážááášáˇááášáˇáááš áĄá˛ááŽá
ááŹá¸ááśáŻá¸... |
The keyword in Python for defining a function is "def". So I'm going to define a function, I'm going to call it the factorial function. | Python áá˝áŹ function áá
ášááŻááᯠááášáá˝áášááášáááŻáááš "def" áááŻáá˛áˇ keyword ááᯠááśáŻá¸áááŤáááš áá˝áźáášáąááŹášá function áá
ášááŻááᯠááášáá˝áášáá˝áŹáťáá
ášáá˛áˇáĄááźááš áĄá˛áᎠkeyword áąáá¸ááᯠáąáá¸áááŤáááš áżááŽá¸áąááŹáˇ ááŤá áĄááŹáááłáłáąááź ááŹááŻáášááášáááŻáᏠáŤááášáťááąáá¸áááŻáášáá˛áˇ á
ááášáá°á¸áąáá¸áá
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áąáá¸áá˛áˇáá°áąááźáá˝áŹ áťáá
ášááášááŹá |
Sometimes beginning programmers have a habit of naming things like "x245" and someone who comes in later will have no clue what variable or that function is all about. So you definitely somehow name it so it gives an idea what is supposed to do. My function is called factorial and the user when they call factorial, the... | " x 245 " ááᯠááŹááášáąáá¸ááŹáąááź áťáá
ášááŤáááš áżááŽá¸áąááŹáˇ áááşáłáááá°áąááźá áááš variable ááᯠááśáŻá¸áááášá áááš function ááᯠááśáŻá¸áááášáááŻááŹáąááźááᯠáááážáááŤáá°á¸ ááŤáąážááŹáášáˇ ááášááşáŹá¸ááŹááášáąáá¸áá˛áˇáĄááŤáá˝áŹ ááášááşáŹá¸ááŹááŻáášááášáááŻááŹááᯠáťáááŻááášááášáˇ ááŹááášáááŻáąáá¸ááášáˇáááš áá˝áźáášáąááŹáˇášáá˛á function ááąááŹáˇ factorial áááŻá áąááááŤáááš áżááŽá¸áąááŹáˇáĄááśáŻá¸áťááłáá°áąááźá factorial áááŻá áąáá... |
So returns the factorial of the argument "number". And sometimes you will hear the word parameter and sometimes you'll hear the word argument. When you are making a definition of a function, this is more formally called a parameter. | "number" áááŻáá˛áˇ arugment áá˛á function ááᯠreturn áťááášááááš áżááŽá¸áąááŹáˇ áá
ášááŤááąááá˝áŹ "parameter" ááŤáá˝áááŻááš "argument" áááŻáá˛áˇá
ááŹá¸ááśáŻá¸ááᯠááášááşáŹá¸ážááŹá¸áááášáˇáááš ááášááşáŹá¸ function áá
ášááŻáá˛á áĄáááá¸áŤáášááᯠáťááłááŻáášááŹááᯠ"parameter" áááŻááąáááżááŽá¸ |
"Number" is one of the parameters to the factorial function. The actual number that someone else passes to it, so let's say someone calls factorial with a "3" here, then more formally that 3 would be the argument. So returns the factorial of the argument "number". | "number" ááŻááᏠfactorial function áá˛á parameter áąááźáá˛á áá
ášááŻáťáá
ášááŤáááš áá
ášá
áśáŻáá
ášáąááŹáášá "number" áá
ášááŻááᯠpass ááŻáášááŻááášááášá áżááŽá¸áąááŹáˇ áĄá˛áᎠnumber ááᯠ"3" áááŻá ááŹá¸ááŻááášá፠áááŻááášá¸á
áá
ášááşááş áąáťááŹáááášáááŻáááš "3" á "argument" áťáá
ášááášáˇááŤáááš ááŤáááŻáááš "number" áááŻáá˛áˇ argument áá˛á áąáťáááŹáášáąááŹášáááášá¸ááᯠreturn áťááášááááš ááŽáá˝áŹáá˝áááŹááąááŹáˇ ... |
And then with the same logic we did before, "for i in range(number)", now we are not getting the number using the input function it is just being passed to the function, then "product" is equal to "product" x ("i"+1). Same logic as we had before. | "for i in range (number)" áááŻáá˛áˇ á
áŹáąážááŹáášá¸áąáá¸áá˝áŹ áĄááŻáááŻáááš áá˝áźáášáąááŹášáááŻáá input function áąááźááᯠááśáŻá¸áżááŽá¸ áááášá¸áąááźááᯠááąáá¸áąáá¸ááŤáá°á¸ function á
áŽááᯠpass ááŻáášáąáá¸ááŻááášááśáŻááŤáᲠáżááŽá¸áąááŹáˇ "product" áᏠ"product" x ( " i " + 1) áá˛á ááŽáááš áá˝áźáášáąááŹášáááŻá áĄááášáááŻáášáá˛áˇáá˛áˇ logic áĄááŻááášá¸ááŤáᲠááŤáąážááŹáášáˇ ááášááşáŹá¸ for-loop ááᯠáąáá¸áżááŽá¸ááźáŹá¸áá˛áˇ... |
So after you go through this for-loop, after you go through it, you are essentially going to go through it "number" times. "Product" will have in it the factorial of "number". And instead of outputting it directly, what we wanna do is return, we are going to return it to the calling program and in the next video I'm go... | "product" áᏠ"number" áąááźáá˛á functional áá˛áá˝áŹ áá˝ááąááá˝áŹá፠áżááŽá¸áąááŹáˇ ááŤááᯠáááŻáášáááŻááš output ááŻáášáąáá¸ááášáˇáĄá
áŹá¸ return ááŻáášáąáá¸ááşáášááŤáááš return ááŻáášááášáááŻááŹááąááŹáˇ function ááᯠáťááášáąááááŹá፠áżááŽá¸áąááŹáˇ áąááŹááš ááźáŽááŽáááŻáá˝áŹáááŻáááš áá˝áźáášáąááŹášá ááŤááᯠááśáŻááźá˛áťáááŤáááš ááášááşáŹá¸ áááášáąáááášáááŻáááš áááŻáá˝áášá¸áąáĄáŹáášáááŻáá፠ááŤáááŻáááš product ááᯠreturn áťáááš... |
You define "product" is equal to 1 and then you go "number" times. | " for i in range (number)" áá˝áŹáááŻáááš |
So "for i in range(number)", we have explained the logic of that in the previous video. | logic ááᯠáĄáááš ááźáŽááŽáááŻáąááźáá˝áŹ áá˝áášá¸áťááżááŽá¸ááźáŹá¸ááŤáżáᎠááášááşáŹá¸ áá
ášáá˛áá
áá˛áˇáĄááşááášááŻááášá¸ |
Everytime you start with 1, then you do 1 x 1 is going to be 1, then "product" is going to be 1 but then "i" is going to be 1. "i" starts off at zero, so it is going to be 1 x (0 + 1) gives you 1, then "i" is 2, so it's going to be 1 x 1, sorry, then "i" is 1 it is going to be 1 x (1 + 1), which is 1 x 2, so it is goin... | " 1 x 1 = 1 " áááŻáżááŽá¸ áťáá
ášááźáŹá¸áááš áżááŽá¸áąááŹáˇ áąáťáááŹáášááášá 1 áťáá
ášááźáŹá¸ááŤáááš i áᏠááŻáááąáá
áá˛áˇáĄááźááš 1 x (0 + 1) = 1 áááŻáżááŽá¸ ááźáášáᏠáťáá
ášááŤáááš i á 2 áááŻáááš (1x1) áᏠáá˝áŹá¸ááźáŹá¸áááŻáááŤá i á 1 áááŻáááš 1 x (1 + 1) áᏠ1 x 2 áťáá
ášááźáŹá¸áżááŽá¸ áĄáąáťáááąááŹáˇ 2 ááźáášááŹááŤáááš áżááŽá¸áąááŹáˇ ááŽááášá¸áĄááŻááášá¸ áááŻá¸ááźáŹá¸ááŤáááš ááŤáááŻáąááŹáˇ áąááŹáášááśáŻá¸ ááźáŽááŽáááŻáá˝áŹ áĄáąáá¸á
ááášá... |
What we could do is, we could say, we still have the input, so this is our function definition but then along our main program, we have defined our function and now we could just say: "Look, from the user get a non-negative integer to take the factorial of" | "ážááášáˇááŻááášá
áášá¸ áąáťáááŹáášáąááŹášáááášá¸áąááźááᯠnon-negative integer" á áąááᲠuser áąááź áá°ážááááš ááŤááᯠááŽááŻá ááŹá¸áááŻáášá
ááŻá |
And let's put that into, lets call that a variable called "user_input". | " user-input" áááŻááąáááá˛áˇ variable áąáá¸ááᯠáąááááŻááášá፠áĄáᯠáá˝áźáášáąááŹáš ááŹááŻáášáąáááŹáá˛áááŻáąááŹáˇ |
And then what I'm going to do is I'm going to define another variable to called "factorialofuser_input" and now this is going to be interesting. I'm going to call this function up here, so this is going to be equal to factorial of whatever the user had input. So the factorial of "user_input", the factorial of this vari... | "factorial of user-input" áááŻáá˛áˇ áąááŹááš variable áá
ášááŻááᯠdefine ááŻáášáąáááŹá፠ááŤá á
ááášááášá
áŹá¸áááŻááąááŹáášá¸ááŹáżáᎠáá˝áźáášáąááŹášá áᎠfunction ááᯠááŽáĄáąáááá˝áŹ áąááááŻááášáááš ááŤá user input áąáá¸ááŻááášáá˛áˇ factorial áá˛á ááŽáá˝áşááŤáááš ááŤáąážááŹáášáˇ "user-input" áá˛á factorial áááŻáᏠááŽáąáááŹá variable áąáá¸áá˛á function ááŤáᲠuser input áąáá¸ááŻááášáá˛áˇ factorial |
The factorial of user input. So now "factorialofuser_input" will be storing, I guess we should say, it would be pointing to the actual answer and now we just have to print it out. | "factorial of user-input"áááŻáᏠááśáŻáťááášááášáąááááášáˇáááš áá˝áźáášáąááŹášáááŻá áąáťááŹááášáˇááŹá ááŤá áĄáąáťááá
ášááŻááᯠáŤááášáťááąááżááŽá¸áąááŹáˇ áá˝áźáášáąááŹášáááŻá áĄáᯠááŤááᯠprint ááŻáášááááš áĄáᯠáá˝áźáášáąááŹášáááŻá "factorial of user-input" ááᯠprint ááŻáášááŻááášááŤáżáᎠáá°áááŻáá ááŤááşáąááźáááŻáᲠááŤáąááá˛áˇ áá°áááŻáá variable name áąááźá፠á፠ááŹááášáąáá¸ááŹá¸áá˛áˇááášá¸áĄááŻááášá¸ áĄááŻááᯠvariabl... |
So at least nothing broke so far, so once again, so I want to be clear the program started here but up here all we did is defined this so it is not creating any interaction with the user, it is actually not processing anything just yet, so it just defines this function then it goes down here and says: "Okay, get some i... | "Okay, get some input from the user" áááŻá áąáťááŹááŤáááš ááŤá áá˝áźáášáąááŹášáááŻá ááŹááŻáášáąáááášáááŻááŹááŤáᲠááŽáá˝áŹ input áĄááşáłáááąáá¸áżááŽá¸áá˛áˇáĄá፠áᎠfunction ááᯠnumber áá˛á áĄáá° áąááááŻááášáááš áá˝áźáášáąááŹášáááŻá input áąáá¸ááŻááášáááš input áąáá¸ááášáˇ number á "user-input" áá˛ááᯠáąááŹáášááźáŹá¸áááš áżááŽá¸áąááŹáˇ ááŤááᯠfactorial áááŻááąáááżááŽá¸ |
So let's try with the number 3. And it did not work! Oh, and I see why it did not work because I had this leftover from the previous program which actually makes no sense now. | "user-input" ááᯠargument áĄáąááá˛á ááŻáášáąáá¸áááš áá˛ááŤááᯠnumber "3" áá˛á á
áášá¸ááášážááášáˇááąáĄáŹááš ááŤá áĄááŻáášáááŻáášáá°á¸ á፠ááŹáááŻá áĄááŻáášáááŻáášáᲠáá˝áźáášáąááŹášáááżáᎠáá˝áźáášáąááŹáš ááŤááᯠáĄáááš program áá˛áá˝áŹ ááşáášáá˛áˇáááŻááᲠááŹáĄáááá¸áŤáášáá˝ ááá˝á áťáá
ášááźáŹá¸áááš ááŤááᯠááášááŻááášáááš ááŤá áááááśáŻá¸ ááášááášá¸á
áŹáąááŹáášá¸ áááŻááášááŹáᲠáĄááŻáášáżááŽá¸áąáĄáŹááš áĄáá ááááłá¸á
áŹá¸á፠ááŤááᯠááášááŻá... |
Then you could just replace this later on and then any program that calls it, as long as it still works, it will work for that program and it will just work that much better. | Defining a factorial function |
Hello everyone, my name is Thomas Suarez. I've always had a fascination for computers and technology, and I made a few apps for the iPhone, iPod Touch, and iPad. I'd like to share a couple with you today. | áĄáŹá¸ááŻáśá¸ááᯠáážáŻááşáááşááŤáááşá ááťá˝ááşááąáŹáˇáĄáááşá Thomas Suarez ááŤá ááťá˝ááşááąáŹáşáᏠáĄááźá˛áááşá¸áᲠáá˝ááşááťá°ááŹááťáŹá¸ áážááˇáş áááşá¸áááŹáá˝áąááᯠá
áááşáááşá
áŹá¸áá˛áˇááŤáááşá iPhoneá iPod Touchá áážááˇáş iPad áááŻáˇáĄáá˝ááş áĄááşááşáĄááťááŻáˇáááŻááąáŹááş ááźáŻá
áŻáá˛áˇááŤáááşá ááŽááąáˇ áážá
áşááŻááąáŹááşááᯠáááşááťáŹá¸áááŻáˇááᯠááááşáááşááąá¸ááťááşááŤáááşá ááťá˝ááşááąáŹáşáá˛áˇ ááááŚá¸ááŻáśá¸ áĄááşááşá ááá°áá°á¸áá˛áˇ ááąáááş ááąáŹáá... |
My favorite and most successful app is Bustin Jieber - (Laughter) - which is a Justin Bieber Whac-A-Mole. I created it because a lot of people at school disliked Justin Bieber a little bit, so I decided to make the app. So I went to work programming it, and I released it just before the holidays in 2010. | Bustin Jieber ááŤ- (áááşááąáŹááśááťáŹá¸) - áĄá˛ááŤá Justin Bieber Whac-A-Mole ááąá ááťáąáŹááşá¸áá˛á áá°ááąáŹáşááąáŹáşááťáŹá¸ááťáŹá¸á Justin Biebe áááźááŻááşáá˛áˇááźáááŻáˇ ááťá˝ááşááąáŹáşá áá°áˇááᯠáĄááşááşáĄááźá
áş ááŻááşáááş ááŻáśá¸ááźááşáá˛áˇááŹááŤá áĄá˛ááŤáá˛áˇ ááťá˝ááşááąáŹáşáᏠáĄá˛ááŤááᯠááááŻááááş ááąá¸áá˝á˛ááŹáááŻááşáᏠáááá ááŻáážá
áş ááťáąáŹááşá¸ááááşáááş ááááŻááşááŽáážáŹ ááźááˇáşááťááááŻááşáá˛áˇááŤáááşá áá°ááąáŹáşááąáŹáşááťáŹá¸ááťáŹá¸á... |
First of all, I've been programming in multiple other programming languages to get the basics down, such as Python, C, Java, etc. And then Apple released the iPhone, and with it, the iPhone software development kit, and the software development kit is a suite of tools for creating and programming an iPhone app. This op... | Pythoná Cá Javaá á
ááŹáá˝áąááᯠááąáˇááŹááąáá˛áˇáááşá áĄá˛ááŽááąáŹááşáážáŹ Apple á iPhone ááᯠááŻááşááąáŹááşá¸ááŹáᏠáĄá˛ááŤáá˛áˇáĄáá° iPhone áá˝áśáˇááźááŻá¸ááąá¸ ááąáŹáˇáᲠáá
áŻáśááŤááŹáááşá áĄá˛áᎠáá˝áśáˇááźááŻá¸ááąá¸ ááąáŹáˇáᲠáĄá
áŻáśáá˛áážáŹ iPhone áĄááşááşáá˝áąááᯠáááşááŽá¸áááş áá˛áˇ ááááŻááááş ááąá¸áá˝á˛ááąá¸ ááááááŹáá˝áą ááŤáážáááŤáááşá áĄá˛ááŤáᏠááťá˝ááşááąáŹáˇáĄááŤááş áĄááŹá¸áĄáᏠáĄáá
áşáá˝áąáá˛áˇ ááášááŹááźáŽá¸ááᯠáá˝ááˇáşááąá¸áááŻááşááŤáá... |
I've gotten a lot of inspiration from Steve Jobs, and I've started an app club at school, and a teacher at my school is kindly sponsoring my app club. Any student at my school can come and learn how to design an app. This is so I can share my experiences with others. | Steve Jobs ááśáážááąááźáŽá¸ ááťá˝ááşááąáŹáşáᏠáážáŻáśáˇááąáŹáşááťááşáá˝áą áĄááťáŹá¸ááźáŽá¸ ááá˛áˇááŤáááşá áĄááŻááąáŹáˇ ááťá˝ááşááąáŹáşáᏠááťáąáŹááşá¸áážáŹ áĄááşááş ááááşááᯠá
áááşáá˛áˇááŤáááşá ááťá˝ááşááąáŹáˇ ááťáąáŹááşá¸áááŹááťáŹá¸ááááşá¸ ááťá˝ááşááąáŹáˇ áĄááşááş ááááşááᯠááąáŹááşááśáˇááźááŤáááşá ááťá˝ááşááąáŹáˇ ááťáąáŹááşá¸áá˛á áááşááťáąáŹááşá¸ááŹá¸áááᯠááŹááźáŽá¸ áĄááşááşááᯠááŻáśá
áśááąá¸áá˝á˛áážáŻááᯠáááşáá°áááŻááşááŤáááşá ááŽáááŻáááşá¸ááźááˇáş ááťá˝ááşááąáŹáˇ áĄáá˝... |
So - (Laughter) - sorry - (Laughter) - so this is a resource to teachers, and educators should recognize this resource and make good use of it. I'd like to finish up by saying what I'd like to do in the future. First of all, I'd like to create more apps, more games. | ááŽááąáŹáˇ- (áááşááąáŹááśááťáŹá¸) - ááąáŹááŽá¸áᲠ- (áááşááąáŹááśááťáŹá¸) - ááŤáᏠáááŹááťáŹá¸áĄáá˝ááş áĄáááşá¸áĄááźá
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áşáá˝áą ááááşá¸áĄáá
áşá... |
I think now might be a good time to address another question that is probably circulating in your head-- and that is, what happened to the gold? Remember when we started this reserve bank, all of these national banks or whatever we want to call them, they had gold as a reserve. And then at some point, they said, well, ... | áĄááŻáĄááşááášáááášáááŻáá
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In case X percentage of people wanted their gold back. So they would come back to the central bank and say, give us our gold. But the gold fundamentally-- it had no other function. | X% áááŹááá˝ááá˛áˇáá°áąááź áá°áááŻááąáááąááźáťááášáááŻááşáášáá˛áˇáĄááŤáá˝áŹá፠ááŽáąááŹáˇáá°áááŻáááááŻááášááŽááŹáťááŽá¸ áá°ááŻáááąáááąááźáąááŹáášá¸áááŻáááááš áĄáąáťáááśáĄáŹá¸áťááášáˇáąáááá˝áŹáĄáťááŹá¸functionááá˝áááŤáá°á¸ ááášá¸áá
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In order to increase the money supply with GDP because people are inventing computers and railroads and cars and highways are being built and we're all becoming more efficient-- in order to keep the money supply up with that extra economic activity, if we stay on the gold standard and if we want to keep these ratios be... | GDP áá˛á áąááźááášááášáááášá¸áááŻáťáá˝áášáˇááášáááŻáááš áá°áąááźáááźáášáťáá´ ááŹáąááźá ááŽá¸áááŹá¸ááśááášá¸áąááźá ááŹá¸ááášá¸áážááŽá¸áąááźááŽááźáášáá˛áˇáá˛áˇáĄááźáášáĄááşá áł á¸áááąááŹáášááŤáááš áąááźááášááášáááášá¸áááŻá
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The U.S. government-- what are they good for? They're a bunch of crooks. They lie, cheat, and steal. | U.S. áĄá
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It's backed by people's trust in the ability of the U.S. economy to support debt to pay off the value of this currency. We'll talk more about that in the future. I don't want to get too circular in my conversations. | U.S. economy ááĄáąážááźá¸áąááźáááŻáĄáąááŹáášáĄáá°áąáá¸ááŹá¸áááš ááŤáááŻááĄá˛ááŤáááŻáąááŹáášáá˝ááášáąáťááŹážáááŤáááš ááŤáááŻáá
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I'd like to talk about my dad. My dad has Alzheimer's disease. He started showing the symptoms about 12 years ago, and he was officially diagnosed in 2005. | ááťá˝ááşááá˛áˇ áĄááąáĄááźáąáŹááşá¸ ááźáąáŹááŤáá
áąá áĄááąáˇáážáŹ áĄááşááşáááŻááşá¸ááŹá¸ááąáŹá፠áážáááŤáááşá ááąáŹááŤááášáááŹáá˝áą ááźááŹá áá˝ááşáá˛áˇáá˛áˇ áá áážá
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(Applause) Hi. My name is Cameron Russell. | (áááşááŻááşááśááťáŹá¸) áááşášáááŹááŤá ááťá˝ááşááááąáŹáˇ áááşááá˝ááş ááŹáááşááşááŤá áá˝ááşáá˛áˇáá˛áˇááŹáááąáŹáĄáá˝ááşá¸ ááťá˝ááşá ááąáŹáşáááşáá
áşááąáŹááşááźá
áşáá˛áˇáá°á¸ááŤáááşá ááááşááąáŹáˇ áááážá
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áŻáśáááŻáááşáá˛áˇáááááŻáˇááŤá (áááşááśááťáŹá¸) ááśááąáŹááşá¸ááŹá áá˛áááŻáˇáĄáááşáá
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áŻáśááŤááŹááŤáááşá ááŤáááááŻáśá¸ Ted á
ááşááąáŤáşáážáŹ... |
And I always just say, "Oh, I was scouted". But that means nothing. The real way that I became a model is I won a genetic lottery, and I am the recipient of a legacy. | "ááťá˝ááşáááąáŹáˇ áĄáá˝áąá¸ááśáááŹááŤá"áááŻáˇ ááźáąáŹáááşááŤáááşá ááŤááąáá˛áˇ áĄá˛ááŤá ááŹáážáááŻááşááŤáá°á¸á ááááˇáş ááźá
áşááŹááŻáśáááąáŹáˇ ááťá˝ááşá ááťááŻá¸áááŻá¸ááŽáááŽááąáŤááşáá˛áˇáááŻáˇááŤá ááťá˝ááşá ááŽáĄáážáááŹá¸áĄáá˝áąáááŻáááşááśááážááá˛áˇáááŻáˇááŤá áážááşáááŻáˇáááąáŹáˇ áĄáá˝áą áááşááśáááşáááŻáᏠááŹáá˛áááŻáˇ ááááťááşááąáážáŹááąáŤáˇá áá˝ááşáá˛áˇáá˛áˇááŹá
áŻáážá
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Naomi, Tyra, Joan Smalls, Liu Wen | "ááąááŤáŚá¸á Naomi. Tyra. Joan Smalls. |
And first I commend your model knowledge, very impressive [Laughter] But unfortunately, I have to inform you that in 2007 A very inspired NYU PHD student counted all the models on the runway | Liu Wená" áááááŻáśá¸ áážááşáááŻáˇáá˛áˇ ááąáŹáşáááşáĄáááááŹááᯠááťáŽá¸ááťá°á¸ááŤáá
áąá áĄáááşááźáŽá¸ááąáŹááşááŤáááşá (áááşááśááťáŹá¸) ááŤááąáá˛áˇ ááśáááąáŹááşá¸á
á˝áŹáᲠáážááşáááŻáˇááᯠáĄááááąá¸ááťááşááŹá áááá áážáŹ áĄáááşá¸áááşááźá˝áá˛áˇ NYU ááŤááá° ááťáąáŹááşá¸ááŹá¸ áá
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áŽáááŻááąáŤáˇá áĄá˛áᎠáááˇáşáĄááşááŹá¸... |
"No no Cameron, I want to be a model" Well then I say, "Be my boss" Because I'm not in charge of anything | "áááá°á¸ áááá°á¸ Cameron ááťá˝ááşá ááąáŹáşáááşáá˛ááźá
áşááťááşáááşá" ááŤáááŻáááş ááťá˝ááşááááźáąáŹáážáŹááąáŤáˇ "áááşá¸ááŹááŹááŻáśá¸ááźááşá" áĄááźáąáŹááşá¸á ááťá˝ááşá ááŹáááŻááž áááźáŽá¸ááźááşáá°á¸ááąá áážááşá American Vogue áá˛áˇ ááąáŤááşá¸ááąáŹááşáááşá¸ááźááşáá° ááŤáážáááŻááş H&M áá˛áˇ CEO ááŤáážáááŻááş ááŻáááSteven Meisel ááźá
áşáááŻááşááŤáááşá áážááşááźáŽá¸ááŹáááş ááąáŹáşáááşááźá
áşááťááşáááşáááŻáˇ ááźáąáŹááŹáᏠááźáŽá¸ááŹáááş Powerb... |
And when we're back stage we give an answer that maybe makes it seem like that We say it's really amazing to travel And it's amazing to get to work with creative inspired passionate people | "áááş áááşááąáŹááşáá
ášá
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áşááŤáááşá áĄá˛ááŤáá˝áąá ááťá˝ááşááááŻáˇáááźáąáŹááťááşáá˛áˇ áĄááŹáá˝áąáááşá¸ááźá
áşááŤáááşá ááťá˝ááşáá Cambridge áážáŹ ááźáŽá¸ááźááşá¸ááŹááŤá áá
áşááąáˇ ááŻááş... |
And I think that it's great I got to come You know, before ten or twenty or thirty years had passed And I'd had more agency in my career | "á፠áááşáážá
áşááźáááş áĄáááşá¸ááśáááá˛" "áááşáĄááťáááş áĄáááşá¸ááśáááá˛" á ááááˇáşááąá¸áá˝ááşá¸áá˝áąááŤá ááťá˝ááşá ááŽááąáŹááźáąáŹáá˝á˛áĄáá˝ááş ááąáˇááŹááąááŻááşá¸ áĄááąáááááşáážáŹ áĄáááş áá áážá
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áşáááşááźááŤáá°á¸áááŻáᏠáááážááááŻááşáááŤáááşá áĄá˛áĄááąáĄáá˝ááş áá°áááŻáˇ áááážá
áş ááąáŹááşáá˛áˇáĄááťáááş áá%áĄáá ááźááˇáşáááşáá˝áŹá¸ááŤáááşá ááąáŹááşááŻáśá¸ááąá¸áá˝ááşá¸áááąáŹáˇ |
Um, if there's a take away to this talk I hope it's that we all feel more comfortable acknowledging the power of image | "ááąáŹáşáááşáá
áşááąáŹááş ááźá
áşááŹáááşáááŻááŤáá˛" ááŤá ááťá˝ááşááááşááŹááąáŹáˇ áá°áááŻáˇáááŻááťááşáá˛áˇáĄááźáąá |
In our perceived successes and our perceived failures Thank you [Applause] | "ááááşáááŻáˇ áááşá áááşá¸áááşá¸áááŻááááşáááş ááśáááşá áááŻááąáŹááşááááşáááŻáááş ááąáŹáşááąáŹáşááťáąáŹáşááźáŽá¸ áááşá¸ááŹáááşááąáŤáˇá" ááťá˝ááşááááŻáˇ á
ááşááąáŹááşáááşááąáŹááşáááş |
We need to calculate 9.005 minus 3.6, or we could view it as 9 and 5 thousandths minus 3 and 6 tenths. Whenever you do a subtracting decimals problem, the most important thing, and this is true when you're adding decimals as well, is you have to line up the decimals. So this is 9.005 minus 3.6. | 9.005 áĄááŻááš 3.6 áááŻááźáášážáááąáĄáŹááš 9 áá˝ááˇáš 0.005 áĄááŻááš 3 áá˝ááˇáš 0.6 ááášááá ááŻá
áŹáŚááŻáášáťááášá¸áááᯠáĄááá áĄáąáá¸áááŽá¸ááŹá áááááᯠáąááŤáášá¸áá˛áˇáĄá፠ááááááášá¸áąááź ááᯠá
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ážáááąáĄáŹááš 5 áĄááááš ááŹáá˝ááá˝ááᏠ(áááŹ) 3.6 ááᯠááśáŻáąááŹášážááášáˇá፠(áááŻá) 3 áá˝ááˇáš 0.6 á... |
I had a fire nine days ago. My archive: 175 films, my 16-millimeter negative, all my books, my dad's books, my photographs. | áá˝ááşáá˛áˇáá˛áˇ á áááşá ááŽá¸ááąáŹááş ááśááááşá ááťá˝ááşááąáŹáˇáá˛áˇ ááąáŹáşáá˝ááşá¸áááŻááşááŤá ááŻááşáážááş ááá
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"King, Murray" won Cannes Film Festival 1970 -- the only print I had. That's my papers. That was in minutes -- 20 minutes. | Cannes ááŻááşáážááşáá˝á˛ááąáŹáşáážáŹ ááŻááá˛áˇááŹ... áááŻááşáážáŹáážááá˛áˇ áá
áşááŻáááşá¸ááąáŹ ááŻáśáážáááşáááşááŹá ááŤá ááááşá¸á
áŹáá˝áąá áááá
áşáááŻááşá¸áá˛á áááá
áş áá ááŤá ááááááşááźááŹá áá
áşááŻááŻá áááŻááşáááşáááŻááşááŹááŤá |
"You've got to make something good out of something bad," I started to say to my friends, neighbors, my sister. By the way, that's "Sputnik." I ran it last year. | "áĄáááŻá¸áááą áĄááąáŹááşá¸ááŻááşáááŻáˇáááŻáááşá" ááááşáá˝áąáá˝áąá áĄáááşááŽá¸ááťááşá¸áá˝áąá ááŽááááŻááźáąáŹáááş ááŤáá˛áˇááąáŚá¸á ááŤá Sputnik ááážá
áşá ááźáá˛áˇááŹá |
"Sputnik" was downtown, the negative. It wasn't touched. | Sputnik á ááźááŻáˇáá˛áážáŹá áááşááá
áşááąá á፠ááááá˝áŹá¸áá°á¸á |
These are some pieces of things I used in my Sputnik feature film, which opens in New York in two weeks downtown. | Sputnik ááŹá¸áá˛áážáŹ ááŻáśá¸áá˛áˇáá˛áˇ áĄáááŻááşá¸áá˝áąáážááááşá |
I called my sister. I called my neighbors. I said, "Come dig." | New York ááźááŻáˇáááşááąáŹááşáážáŹ á áááşáĄáá˝ááşá¸ áá˝ááşáážáŹá ááŽááááŻá áĄáááşááŽá¸ááŹááťááşá¸áá˝áąááᯠááŻááşá¸áááşá ááŹááźá áá°á¸ááźá á፠áĄáᯠá
áŹá¸áá˝á˛áážáŹá ááŤá ááąáŹááşáááŻáˇ áážá
áş áá ááťáááşáá°ááá˛áˇ á
áŹá¸áá˝á˛á áááá˛áˇáĄáááŻááşá¸á áĄááŻááşááŻáśá¸ááąáá ááŤá áááŽá¸ Jean ááąá áá°ááŹáááşá |
She came. She's a nurse in San Francisco. | San Francisco á áá°ááŹááźáŻááŤá |
"Dig it up," I said. "Pieces. I want pieces. | "áá°á¸áááŻááş" áááŻáˇááźáąáŹáááŻááşáááşá "áĄáááŻááşá¸áĄá
áá˝áąá áĄáááŻááşá¸áĄá
áá˝áą áááŻááťááşáááşá áĄá
áĄááąáá˝áąááąá ááŽá
áááşáá°á¸ááá˛áˇááŹá áĄáááŻááşá¸áááŻááşá¸áá˛áˇ áááá
áşááŻá áĄááŻáá˛á
ááŻááşááąáá˛áˇ áááąáŹááťááşááąáŤáˇá ááŤá ááŽáááŤá áááşá¸ááťáŽááŹá¸áá˝áąááᯠá
áąáŹááˇáşáážáąáŹááşááŤáááş ááťá˝ááşááąáŹáşá áĄááťáŹá¸ááźáŽá¸ááźáąáŹáááşáááŻáˇ áááşáá˛áˇ ááťážááşáá
áşááźááş ááŹááşááŻáśáá˝áą áĄááźáŽá¸á
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áąáŹááşá¸áá°áááŻáˇááŤá ááŽááŻáśáá˝áąá ááŽá¸ááąáŹááşáá˛áˇáĄáá˛á ááąáŹááş... |
"I'm going to make something out of it, and maybe next year ... " And I appreciate this moment to come up on this stage with so many people who've already given me so much solace, and just say to TEDsters: I'm proud of me. | "ááŤáááąáá
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Add and simplify the answer and write as a mixed number. So we have two mixed numbers here. | . áąáááżáááŤáááášá¸ááşáŹá¸ááᯠáąááŤáá¸ášáᏠáĄáá˝áášá¸ááŻáśá¸ááŻáśá
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லŕŻŕŽŕŽżŕŽŞŕŻŕŽ°ŕŽžŕŽľŕŽżŕŽ˛ŕŻ ŕŽŕŽŠŕŻ ŕŽŕŽ´ŕŻŕŽ¤ŕŻŕŽ¤ŕŻŕŽŕŻŕŽŕŽłŕŻŕŽŞŕŻ பயனŕŻŕŽŞŕŽŕŻŕŽ¤ŕŻŕŽ¤ŕŻŕŽŕŽżŕŽąŕŻŕŽŽŕŻ ŕŽŕŽŠŕŻŕŽŞŕŽ¤ŕŽąŕŻŕŽŕŻ விளŕŽŕŻŕŽŕŽŽŕŻ ŕŽŕŻŕŽŕŻŕŽŕŻŕŽ ŕŽŕŽžŕŽŠŕŻ ŕŽŕŻŕŽ¸ŕŽżŕŽ˛ŕŻŕŽŻŕŻŕŽŽŕŻ, ŕŽŕŽžŕŽŠŕŻ ŕŽ
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வŕŽŕŻŕŽ ŕŽŕŽżŕŽŕŻŕŽ நமŕŻŕŽŽ ŕŽŕŻŕŽłŕŻŕŽľŕŽżŕŽŕŽł ŕŽŕŻŕŽŕŻŕŽŕŽ˛ŕŽžŕŽŽŕŻ. நலŕŻŕŽ˛ŕŽ¤ŕŻ மாணவரŕŻŕŽŕŽłŕŻ ŕŽ
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பŕŻŕŽŞŕŽąŕŽŽŕŻ ABC ŕŽŕŽ´ŕŻŕŽ¤ŕŻŕŽ¤ŕŻŕŽŕŻŕŽŕŽłŕŻ ŕŽŕŽ˛ŕŻŕŽ˛ŕŽžŕŽŽŕŻ.? ŕŽŕŽŽŕŽž.... ŕŽŕŽ°ŕŽżŕŽŻŕŽžŕŽŠ ŕŽŕŻŕŽłŕŻŕŽľŕŽż தானŕŻ. நலŕŻŕŽ˛ŕŽ¤ŕŻ ŕŽŕŽ¨ŕŻŕŽ¤ŕŽŕŻ ŕŽŕŻŕŽłŕŻŕŽľŕŽż பறŕŻŕŽąŕŽż நாம ŕŽŕŻŕŽŕŻŕŽŕŽŽŕŻ யŕŻŕŽŕŽżŕŽŕŻŕŽŕŽ˛ŕŽžŕŽŽŕŻ. ŕŽŕŽ°ŕŽż, ŕŽŕŽžŕŽ˛ŕŻ ŕŽŕŽ¨ŕŻŕŽ¤ŕŽŕŻ ŕŽŕŻŕŽłŕŻŕŽľŕŽżŕŽŕŻŕŽŕŻ ŕŽŕŽŠŕŻŕŽŠ பதில௠ŕŽŕŻŕŽ˛ŕŻŕŽ˛ŕŽŞŕŻŕŽŞŕŻŕŽąŕŻŕŽŕŻŕŽ..? ŕŽ
லŕŻŕŽŕŽżŕŽŞŕŻŕŽ°ŕŽžŕŽľŕŽżŕŽ˛ ŕŽŕŽ´ŕŻŕŽ¤ŕŻŕŽ¤ŕŻŕŽŕŻŕŽŕŽžŕŽŠ தŕŻŕŽľŕŻ ŕŽŕŽŠŕŻŕŽŠ? ŕŽŕŽ¨ŕŻŕŽ¤ŕŽŕŻ ŕŽŕŻŕŽłŕŻŕŽľŕŽżŕŽŕŻŕŽŕŽžŕŽŠ பதிலŕŻŕŽŞŕŻ பல ŕŽŕŻŕŽŁŕŽŕŻŕŽŕŽłŕŻŕŽ˛ பாரŕŻŕŽŕŻŕŽŕŽ˛ŕŽžŕŽŽŕŻ. தŕŻŕŽ°ŕŽżŕŽŻŕŽžŕŽ¤ ŕŽŕŽ°ŕŻ ŕŽŕŽŁŕŻ ŕŽŕŽ°ŕŻŕŽŞŕŻŕŽŞŕŽ¤ŕŽžŕŽ வŕŻŕŽ¤ŕŻŕŽ¤ŕŻŕŽŕŻ ŕŽŕŻŕŽłŕŻŕŽľŕŻŕŽŽŕŻ. ŕŽ
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X ŕŽŕŻŕŽŕŻŕŽŽŕŻ Y ŕŽŕŻŕŽŕŻŕŽŽŕŻ ŕŽŕŽŕŻŕŽŻŕŽżŕŽ˛ŕŽžŕŽŠ ŕŽŕŽąŕŽľŕŽżŕŽŠŕŻ மŕŻŕŽŕŽżŕŽľŕŽżŕŽ˛ŕŻŕŽ˛ŕŽžŕŽ¤ ŕŽŕŽŁŕŻŕŽŁŕŻ ŕŽŕŽŕŻŕŽŕŻ ŕŽŕŽłŕŻŕŽł ŕŽŕŽłŕŽżŕŽŻ ŕŽŕŽŽŕŽŠŕŻŕŽŞŕŽžŕŽŕŻ நமŕŽŕŻŕŽŕŻ விளŕŽŕŻŕŽŕŻŕŽŕŽżŕŽąŕŽ¤ŕŻ. ŕŽŕŽ நŕŻŕŽŕŻŕŽŕŽłŕŻ ŕŽŕŽŠŕŽŕŻŕŽŕŻ X ŕŽŕŽŕŻ ŕŽŕŻŕŽŕŻŕŽŕŻŕŽŕŻŕŽŽŕŻ பŕŻŕŽ¤ŕŻŕŽ˛ŕŻŕŽ˛ŕŽžŕŽŽŕŻ நŕŻŕŽŕŻŕŽŕŽłŕŻ ŕŽŕŽŠŕŽŕŻŕŽŕŻŕŽŕŻ ŕŽŕŻŕŽŕŻŕŽŞŕŻŕŽŞŕŽ¤ŕŻ மŕŻŕŽŠŕŻŕŽąŕŻ, ŕŽ
தŕŻŕŽ¤ŕŻŕŽŕŽŠŕŻ ŕŽŕŽŠŕŻŕŽąŕŻŕŽŕŻ ŕŽŕŻŕŽ°ŕŻŕŽ¤ŕŻŕŽ¤ŕŻ நான௠ŕŽŕŽŕŻŕŽŕŽłŕŻŕŽŕŻŕŽŕŻ நானŕŻŕŽŕŻŕŽŕŻ ŕŽŕŻŕŽŕŻŕŽŞŕŻŕŽŞŕŻŕŽŠŕŻ ŕŽŕŽŠŕŻŕŽŞŕŽ¤ŕŻ மறŕŻŕŽąŕŽľŕŽ°ŕŻŕŽŕŽłŕŻŕŽŕŻŕŽŕŻŕŽ¤ŕŻ தŕŻŕŽ°ŕŽżŕŽŻŕŻŕŽŽŕŻ. நŕŻŕŽŕŻŕŽŕŽłŕŻ ŕŽŕŽŠŕŽŕŻŕŽŕŻŕŽŕŻ ŕŽŕŻŕŽŕŻŕŽŞŕŻŕŽŞŕŽ¤ŕŻ 7.1, ŕŽ
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Welcome back. Now that we hopefully have a little bit of an intuition of what a limit is, or finding the limit of a function is, let's do some problems. | ááźááşááŹáᏠáááşášáááŹá፠ááŤáááŻáˇlimitáááŻááŹááŹáᲠáááşáážááşáá
áşááŻáá˛áˇ limitááᯠáááşáááŻáážáŹáááá˛áá˛áˇ áááşáááşáááŻáˇ áá˛áá˛ááąáŹáˇ ááááąááźáŽáááŻáˇ ááťážáąáŹáşáááşáˇáááş ááŻá
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áşááťááŻáˇ ááŻááşááźááĄáąáŹááş ááŽáá˛á áá
áşááťááŻáˇááąáŹáˇ á
áŹááąá¸áá˝á˛áážáŹ ááąá¸ááŹáá˝áąáˇááááş ááąáá°ááť limit ááŻá
ášááŹáá˝áąááᯠááźáąáážááşá¸áá˛áˇáĄááŤáááşá¸áá˝áąáˇááááş |
So let's say what is the limit-- once again, my pen is not working. What is the limit as x approaches-- let's say negative 1. | limit ááŹáá˛áááŻáᏠáááşááźáąáŹááźá
ááŻáˇ áĄá˛.. ááąáŹáááşá áĄááŻááşáááŻááşááŤáˇááŹá¸ x á ááťááşá¸áááşááąáá˛áážáŹ limit á ááŹáááşáááŻá¸ áááŻááŤáááŻáˇ |
And let me see, what's a good-- let's say my expression is-- I'll put it in parentheses so it's cleaner. It's 2x plus 2 over x plus 1. | -á ááᯠááťááşá¸áááşáá˛áážáŹ ááźááşáˇáááŻááşáŚá¸áááş áĄáŹ.. ááąáŹááşá¸áááŻááşáá˛áˇ ..... áááşáážááş ááᯠááźáąáŹááááşáááŻáááş áááşáááşá¸áá˝ááşá¸áá˛áááşáˇááŹá¸áááŻááşáááş ááźááşáˇááážááşá¸áĄáąáŹááşáááŻáˇ áĄá˛ááŤá 2x áĄááąáŤááşá¸ 2 áĄá
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So here's a case, just like what we saw in that first video, where the limit actually can't equal what the expression equals when you substitute x for the number you're trying to find the limit of because you get an undefined answer. | LimitáááŻáážáŹáááŻáˇ xááᯠáĄá
áŹá¸áá˝ááşá¸áááŻááşáá˛áˇáĄá፠|
So let's see if, using the limit, we can come up with a better answer for what it's approaching. Well, since we're just starting with these limit problems, let me draw a graph. | Limit á áĄá˛áᎠáááşáážááşáá˛áˇ áá
áşáááş ááá°ááŽááąáŹáˇáá˛ááźá
áşááąááŹááą ááŹáááŻáˇáá˛áááŻááąáŹáˇ áĄáááášááŤááşáááşáážááşááťááşáááşá¸ áá˛áˇááąáᏠ(undefined) áááąááŹáááŻá¸ áᲠááźááşáˇááźáŚá¸á
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So you'll see, and I'll make sure you see it visually here, that the limit as approaches negative 1 of 2x plus 2 over x plus 1, it equals 2. Let me draw a line here so you don't get messed up with all of it. | Limit áááş á áá˛áˇ ááŽáááşáááŻáᏠááŤááŽáážáŹ ááťááşá¸áá
áşááźáąáŹááşá¸ áá˝á˛áĽáŽá¸áááş áĄá˛ááąáŹáˇáááşá¸ áĄááąáŤáşáááŹáá˝áąáá˛áˇ áááąáŹááąáŹáˇáá°á¸ááąáŤáˇ ááŻááźáááşáˇ áááşááąáááąáŹáˇ ááááááááźáŽá¸ááąáŹáˇ ááŻááşáá°á¸ááąáŤáˇ |
And I'm not formally, I guess, proving here that the limit is 2, but I'm showing you kind of an analytical way, and this is actually how it tends to be done in algebra class, is that you tend to simplify the expression so that you say, oh, if there wasn't a hole here, what would the f of x equal, right? And then you'd ... | limit á á áááŻááŹááą ááŤááąáá analytical áááşáááşááąá¸ááąáŹáˇ ááźááąá¸áááş ááááşááąáŹáˇ áĄáášáááŹáááşášááťáŹáááşá¸áážáŹ ááŻááşááááŻááťááŻá¸ááąá¸á፠áááşá¸á ááŽááąáŹáşááźááťááşááᯠáĄáááŻá¸ááŻáśá¸ááŻáśá
áś áá˝ááşá¸áááş ááźáŽá¸ááąáŹáˇ ááźáąáŹáááŻááşáááş áĄáááşáááťáŹá¸ áĄá˛ááŽáážáŹ áĄááąáŤááşáá
áşááąáŤááş ááážááááŻáˇáááąáŹáˇ f(x) á ááŹáá˛áˇááťáŹá¸ ááŽááŤááá˛áááŻáˇ ááŻááşááźáŽááŹá¸ ááźáŽá¸ááąáŹáˇ áĄá˛ááŽáĄáážááşáážáŹ áááşáážááşáááşáááŻá¸ááᯠáá˝ááşáááŻááşááąáŤáˇ ááŤá... |
A line has a slop of negatave 3/4ths, and goes through the point (0,8). What is the equation of this line in slope-intercept form? So any line can be represented in slope-intercept form as y=mx+b. | ááťááşá¸áá
áşááźáąáŹááşá¸áá˛áˇ ááťážáąáŹá
áąáŹááşá áĄááŻááş 3/4 ááźá
áşááźáŽá¸ áĄáážááş (0, 8) ááᯠááźááşáá˝áŹá¸áááşá ááŽááťááşá¸áá˛áˇ ááŽááťážááźááşá¸ááᯠslope intercept form áážááşáˇ ááźááŤá áááşááťááşá¸ááźáąáŹááşáˇ áááᯠslope intercept form y=m x + B áážááşáˇ ááźáááŻááşááźáŽá¸ áĄá˛ááŽáážáŹ (ááąáŹááŽ) m á ááŽááťááşá¸áá˛áˇ ááťážáąáŹá
áąáŹááş ááźá
áşááźáŽá¸ ááŽá b áááąáŹáˇ ááťááşá¸áá˛áˇ y ááŻáśááťááşááŤá áááşá¸áááşá¸áááŻááźááşááŹáá˝áŹá¸áĄáąáŹááş ... |
We're told that Jared is twice the age of his brother, Peter. | Jared á áá° áˇáᎠPeter áááš áĄááášáá˝á
ášáážááŽá¸ááášááᯠáˇáąáżááŹááŹá¸áááš |
So, Jared is twice the age of his brother, Peter. Peter is four years old. | Jared á peter áááš áĄááášáá˝á
ášáážááŽá¸áżááŽá¸ Peter á áĄáááš ááá˝á
ášáá˝áááášáá˛áˇ |
Peter is four years old, and Jared is twice the age of his brother, Peter. So, Jared is twice Peter's age of four. And then they tell us Talia's age is 3 times Jared's age. | Peter á á áá˝á
ášáżáá
ášáżááŽá¸ áá° áĄá
ášááᯠJared á áĄááášáá˝á
ášáážááŽá¸áááš ááŽáąááŹáˇ Jared á PeteráĄáááš á áá˝á
ášáᲠáˇáá˝á
ášááąááŤáˇ áąááŹáášáżááŽá¸ Talia á Jared ááášáĄáááš ááážááŽá¸áááš ááŤááᯠTalia áᲠáˇJared á áĄááášááášáąááŹáášážááŽá¸ážáááᲠááŽáąááŹáˇ Talia á Jared ááášáĄáááš á áážááŽá¸áááš áĄáášá¸... ááşáąáá¸ááąáĄáŹááš ááŽáąááŹáˇ Peter áĄááášá ááá˝á
ášá |
Jared is twice the age of his brother, Peter. So Jared-Let me do that in yellow-So Jared- That's not yellow. Jared- Jared is twice the age of his brother. | Jared á áá°áᎠPeter ááášáĄááášáá˝á
ášáážááŽá¸áááš ááŽáąááŹáˇ áĄááŤáąááŹáášáᲠᡠáąáá¸ááášá Jared á .. áá° áˇááŽáĄááášááášáá˝á
ášáážááŽá¸áááš áá˝á
ášááááŻáąááŹáˇ áá° áˇááŽáᲠáˇáĄááášááᯠáá˝á
ášáᲠáˇáąáżáá˝áŹáášááŹáᲠáˇáá°áá°áᲠááŽáąááŹáˇ Jared á áá°ááŽááášáĄááášáá˝á
ášáážááŽá¸áááš (ááᯠáˇ) áá°ááŽáĄááášáááŻááᲠáˇáąáżáá˝áŹáášááŹáᲠáˇááŽááášááᯠáˇáąáżááŹááᯠáˇááááš áĄáášá¸... áá° áˇááŽáĄááášá ááá˝á
áš ááᯠáˇááŽáá˝áŹáąáżááŹááŹá¸áááš ááŽáąááŹáˇ Jar... |
And then finally they say Talia's age- let me write Talia's- Talia right over here. Talia's age and they say it- they tell it to us right over here. | Talia áĄááášá..ááŽáá˝áŹáá°áąáá¸ááŹá¸ááŹá |
It's three times Jared's age. So Talia's age is three times Jared's age. Well we were able to figure out Jared's age. it is eight. | Jared áĄááášáááš ááśáŻá¸áážááŽá¸ááášááᯠáˇáąáżááŹááŹá¸ááášá áĄáášá¸ ááŽáąááŹáˇááźáşáášáąááŹášááᯠᡠJared áᲠáˇáĄááášááᯠááżááŽá¸áżááŽá áááááš |
It is three times Jared's age. If you know your multiplication tables this should maybe jump out at you. And your multiplication tables really are one of those things in mathmatics | Jared áĄááášáᲠáˇáááááŻáąááŹáˇ ... áĄááŽáááŹá¸ááŹáááááš áąááşáŹášááźáášááᯠáˇááżáᎠáąááŹáášáżááŽá¸ááášááŹáĄááŽáááŹá¸áááŻáááááš áááşá¤áŹááźáášáááŹáąááŹášáąááŹášáĄááášáąáżááááš ááŽáąááŹáˇáĄááŽáááŹá¸áááŻáĄáżááášáżááášáĄáá˝áášáá˝áášáááááš ááŤáááŻáááš ááášáˇáááááášááŹáąááŹášáąááŹášáĄááśáŻá¸ááášáááš ááŤáá˝áááŻáášááášáᲠá áĄáąáżáá˝áŹááš á áááᯠáˇáááśáŻá¸ááŤáąááŤáášá¸áááŻááš ááŽáąááŹáˇá +á+á áááŻáąááŹáˇ... á+á á áá áá+ á á áá. ááŽáąááŹáˇ á áĄáąáżáá˝áŹááš ... |
I have this rectangle here and I want to figure out its area. I want to figure out how much space it is taking up on my screen, right over here. And I encourage you to pause this video and try to figure out the area of this entire rectangle. | ááźáşáášáąááŹášáá˝áŹáąááŹáášáˇáá˝áášá
ááŻááśáá
ášááŻáá˝ááťááŽá¸ áĄá˛áˇááŤáá˛áá§ááááŹáááŻáá˝áŹááşáášáááš áĄá˛áˇááŤáááŽááŹá¸áá˝áŹáá˝ááá˛áˇááşáášáá˝áŹáťááášáá˛á ááášáąááŹáášáááŻáąáááŹáá°ááŹá¸áá˛ááááşáášáááš ááášááŽááŽáááŻáááŻááááášááŹá¸áťááŽá¸ ááŽáąááŹáášáˇáá˝áášá
ááŻááśáá
ášááŻááśáŻá¸áá˛áá§ááááŹáááŻáá˝áŹážááášáˇáąá
ááşáášááŤáááš áá˝áŹáá˛áˇáĄááŤáá˝áŹ ááŻáášáááŻáášáá˛áˇááášá¸áá˝á
ášááášá¸áááŻá
áĽášá¸á
áŹá¸á፠á
ááŻááśáá˛ááĄááşáŹá¸áá˛á áĄááśáááŻáąáťáá˝áŹáášáááŻáááášá¸ááááš ááŤááááŻáá
ášá... |
Find the greatest common factor of these monomials. And when they say monomials, that's just a fancy word for saying a one term expression. | - áĄáąáŹááşá፠áá
áşááŻááááşá¸ (monomials) ááťáŹá¸ááž áĄááźáŽá¸ááŻáśá¸ ááŻáśááá˝á˛ááááşá¸ááᯠáážáŹááŤá monomials áááŻááŹ- áá
áşáááşá¸áĄáŹá¸ááźááşáˇá (ááášááᏠáááŤáá˛áˇ) áá
áşááŻááááşá¸ áááşášááťáŹááąáŹáşááźááťááş (one term expression) áá˝áąáááŻááźáąáŹááŹá ááŤáá˝áąáá
áşááŻá
áŽáážáŹ ááááşá¸ (term) áá
áşááŻá
ᎠááŹááŤáááşá áĄáᯠáĄááźáŽá¸ááŻáśá¸ááŻáśááá˝á˛ááááşá¸ááᯠáážáŹáááşáááŻáááş áĄá˛áˇááŽááááşá¸ áá
áşááŻááťááşá¸áážáŹ ááŤáá˛áˇ ááááşá¸ááąá ááááş... |
like cd squared, break it up into the product of the most simple variable, for example, c times d times d. So let's do that for each of them and see what the greatest common factor is. Where do these overlap in terms of their factor? | "c d áážá
áşáááş" ááᯠáĄáážááşá¸ááŻáśá¸áĄááźážáąáŹááşááŻáśá
áś (product) ááźá
áşáĄáąáŹááş áá˝á˛ááŻááşáááşá áĽáááŹá á "c " áĄááźážáąáŹááş "d " áĄááźážáąáŹááş "d " áᲠáĄá˛ááŤáá˝áąáá
áşááŻááťááşá¸á
áŽááᯠáĄá˛ááŽáĄáááŻááşá¸ ááŻááşááźáŽá¸ áĄááźáŽá¸ááŻáśá¸ááŻáśááá˝á˛ááááşá¸ áááşáᏠáááŻáᏠáážáŹááĄáąáŹááşá áááşáážáŹ ááŻáśááááşá¸áá˝áą ááŤááąáá˛á ááźáŽá¸ááąáŹáˇááťá˝ááşááąáŹáşáááŻáˇ áĄááźáŽááŻáśá¸ááŻáśááááşá¸áááŻáᲠáá°áááşá áᲠáᎠááááá
áşááŻááᯠááŻááşááĄáąáŹááşá |
10 cd squared, what is that equal to? Well 10 is equal to 2 times 5. You could do a factoring tree here, but these are pretty straightforward numbers to factor into. | "áá " "c d áážá
áşáááş " á ááŹáá˛áˇááŽááá˛á ááŽááąáŹáˇ"áá"á "á" áĄááźážáąáŹááş "á
" áá˛áˇááŽáááşá ááŽáážáŹ factoring tree áááŻááŻáśá¸áááŻááşáááşá ááŤááąáááşáˇ ááŽááŹáá˝áąáᏠááá˝á˛ááááşá¸áážáŹáááŻáˇ áááşáá˝ááşáá˝ááş ááááşá¸áá˝áąááŤá áá°áááŻáˇáá˝áąáᏠááŻáášáááá˝á˛ááááşá¸áá˝áąááźá
áşáááşá ááŤáááŻ"áá" á"á" áĄááźážáąáŹááş "á
"á "c" ááᯠáá˝á˛áááŻáˇá |
So 10 is 2 times 5, c, all you can do is break that, you could just write that as a c, you can't really simplify that anymore. | "c" áááŻáˇááąá¸ááŻáśáá˛á ááŤá áĄáážááşá¸ááŻáśá¸ááŻáśá
áśáá˛á |
And d squared can be written as d times d. | "d" áážá
áşáááşááᯠ"d"áĄááźážáąáŹááş "d" áááŻáˇááąá¸áááŻááşáááşá |
So I have essentially broken 10cd squared into this, into the product of kind of the smallest constituents that I could think of. The prime factors of 10, and then c, and then d. Now let's do 5cd. | "d" áážá
áşáááşááᯠ"d"áĄááźážáąáŹááş "d" áááŻáˇááąá¸áááŻááşáááşá ááŤáááŻááťá˝ááşááąáŹáşá "áác d áážá
áşáááş" ááᯠááŽááᯠáĄááąá¸ááŻáśá¸ááĄáážááşá¸ááŻáśá¸ááźá
áşáááŻááşáááąáŹááşááᯠáá˝á˛ááźáŽá¸áá˝áŹá¸ááźáŽá ááŤá áá áá˛áˇááá˝á˛ááááşá¸ ááźáŽá¸ááąáŹáˇ c áážááşáˇ d áĄáᯠ5cd ááᯠááŻááşááźááĄáąáŹááşá 5cd áážáŹ 5 á ááŻáášáááááşá¸ááźá
áşáááşá áá°áˇáá˛áˇ ááá˝á˛ááááşá¸áážáŹááťážááş 5 áᲠááááşá c áááŻááąáŹáˇ áááşáá˝á˛áááŻáˇááááąáŹáˇáá°á¸á c áá˛ááźá
áşáááşá ... |
In fact you can't have a greater number than 5cd be a common factor, because the largest factor of 5cd is 5cd. So the greatest common factor of these three monomials, or these three expressions, is 5cd. | "5cd" áá˛áˇ áĄááźáŽá¸ááŻáśá¸ ááŻáśááá˝á˛ááááşá¸á "5cd" áá˛ááźá
áşáááŻáˇááŤá áĄá˛ááŽáˇááąáŹáˇ ááŽáá
áşááŻááááşá¸ ááŻáśá¸ááŻáá˛áážáŹ (áááŻáˇ) áᎠáááşášááťáŹááąáŹáşááźááťááş ááŻáśá¸ ááŻáá˛áˇ áĄááźáŽá¸ááŻáśá¸ááŻáśááá˝á˛ááááşá¸á "5cd" á áᎠáááşášááťáŹááąáŹáşááźááťááş ááŻáśá¸ááŻáá˛áážáŹ áĄááźáŽá¸ááŻáśá¸ááŻáśááá˝á˛ááááşá¸ á |
The largest number of factors that overlaps with all three of these expressions is a 5, one c, and one d. | "5" áá
áşááŻáśá¸ á " c " áá
áşááŻáážááşáˇ " d " áá
áşááŻááźá
áşáááşá |
We are asked to simplify a log base 5 of 25 to the "x" power over "y" So I will give you some logarithm properties and I do agree that this does require some simplification over here | ááŽááŻá
ášááŹâá 'áâá
áááşâáá˝ážááşá¸ x' áĄá
áŹá¸ 'y' áá˛áˇ log áĄááźáą á
ááᯠáážááşá¸âáááŻáˇâááźá
áşâááŤâáááşá ááŽâáážáŹ logarithm âáá˛áˇâááŻááşâááášááâáááťááŻáˇ ááᯠááŻáśá¸âáááŻáˇâáááŻâááŤâáááşá ââáážâááşá¸ââáááŻáˇâáááşá¸ ááŽâáážáŹâáááŻááŤáááş |
That this having this right over here inside of the logarithm is not a pleasant way to look at it So the first thing that we realize is that this is one of our logarithm properties So logarithm for a given base so lets say that the base is "x" of a/b | logarithm áá˛âáážáŹ ááŽâáĽá
ášá
áŹâáážáâááąâááŹâáááŻâá áĄááźááşâáâáážááşá¸âááŤâáá°á¸á áááâááŻáśá¸âááâááŹá¸ááááŹâááąáŹáˇâlogarithm âáá˛áˇ ááŻááşâááášááâááŤáᲠáĽááᏠ'a áĄá
áŹá¸ b' ááᯠlog áĄááźáą x áá°âáááşâáááŻâáááş... ááŤâá 'a áá˛áˇ log âáĄááźáą x' áĄáážáŻááş 'b áá˛áˇ log áĄááźáą x' áá˛áˇáá°áááş ááŽâáážáŹ 'áâá
áááşâáá˝ážááşá¸ x' áĄá
áŹá¸ 'y' áážáâááąâáááşá ááŤáááŻáážááşá¸áááşáááŻáááş |
Log base 5 twenty five to the "x" over "y" to this property means that its the same thing as log base 5 to the twenty five to the "x" power minus log base 5 of "y" | log âáĄááźáą á
áá˛áˇ 'áâá
áááşâáá˝ážááşá¸ x' áĄá
áŹá¸ 'y' ááᯠááŻâáâá ááŻááşâááášááâááŻáśá¸âáááŻááşâáááş |
Now this looks like we can do a little bit of simplifying It seems like the relevant logarithm property here is if I have log base "x" of a to the "p" power That's the same thing as b times log base "x" of a | log âáĄááźáą á
áá˛áˇ 'áâá
áááşâáá˝ážááşá¸ x' áĄáážáŻááş log áĄááźáą á
áá˛áˇ 'y' áááŻâáâááŤâááááşáˇâáááşá ááŽááŹâááᯠááźááşáˇâáááşâáááŻâáááşâ áá˛â áá˛âáááşâáážááşá¸âáááŻáˇ áááŻâááąâááŤâááąá¸âáááşá ááŤâáááŻâáážááşá¸âáááŻáˇáááşáˇááťáąáŹáşâáá˛áˇ logarithm ááŻááşâááášááâáá
áşâáᯠáááąáŹáˇ ááááşâáááŻáˇ log áĄááźáą x áá˛áˇ a áááşâáá˝ážááşá¸ b áážáâáááşâáááŻâáááş.... ááŹâáâáâáá˛áááŻâááąáŹáˇ b áĄááźážáąáŹááş log áĄááźáą x áá˛áˇ a ... |
What I thought I would do is I would start with a simple request. I'd like all of you to pause for a moment, you wretched weaklings, and take stock of your miserable existence. (Laughter) | ááťá˝ááşááąáŹáşááŻááşáááşáááŻáˇ ááźáśááŹá¸ááŹá áááŻá¸áááŻá¸ áááşááźáŹá¸ááťááşáá
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áŻáážáŹ St. |
Now that was the advice that St. Benedict gave his rather startled followers in the fifth century. It was the advice that I decided to follow myself when I turned 40. Up until that moment, I had been that classic corporate warrior -- | Benedict á áá°áˇáá˛áˇ áĄááąáŹáˇááᯠááťáąáŹááşááťáŹá¸ááąáá˛áˇ ááąáŹááşááŤá¸áá˝áąáááŻááąá¸áá˛áˇ áĄááźáśááŤá ááŤá ááťá˝ááşááąáŹáş áĄáááş áá áĄááąáŹááşáážáŹ áááŻááşáááŻááş áááŻááşááŹáááŻáˇ ááŻáśá¸ááźááşáá˛áˇáá˛áˇ áĄááźáśááŹááşááŤá áĄá˛ááŽááŹáááááŻááşáááşáâ ááźá
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It's particularly important -- this isn't on the World Wide Web, is it? I'm about to get fired -- it's particularly important that you never put the quality of your life in the hands of a commercial corporation. Now I'm not talking here just about the bad companies -- the "abattoirs of the human soul," as I call them. | WWW áážáŹááážááá°á¸áááŻáˇááŹá¸á ááŤáĄááŻááşááśáááąáŹáˇáááşá ááŤá áĄáá°á¸áĄááąá¸ááźáŽá¸ááŹá á
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