| align:start position:0% |
| |
| welcome back to recitation today we're |
|
|
| align:start position:0% |
| welcome back to recitation today we're |
| |
|
|
| align:start position:0% |
| welcome back to recitation today we're |
| going to work on an optimization problem |
|
|
| align:start position:0% |
| going to work on an optimization problem |
| |
|
|
| align:start position:0% |
| going to work on an optimization problem |
| so the question I want us to answer is |
|
|
| align:start position:0% |
| so the question I want us to answer is |
| |
|
|
| align:start position:0% |
| so the question I want us to answer is |
| what point on the curve y equals square |
|
|
| align:start position:0% |
| what point on the curve y equals square |
| |
|
|
| align:start position:0% |
| what point on the curve y equals square |
| root of X+ 4 comes closest to the origin |
|
|
| align:start position:0% |
| root of X+ 4 comes closest to the origin |
| |
|
|
| align:start position:0% |
| root of X+ 4 comes closest to the origin |
| I've drawn a sketch of this curve the |
|
|
| align:start position:0% |
| I've drawn a sketch of this curve the |
| |
|
|
| align:start position:0% |
| I've drawn a sketch of this curve the |
| scale in this direction the each hash |
|
|
| align:start position:0% |
| scale in this direction the each hash |
| |
|
|
| align:start position:0% |
| scale in this direction the each hash |
| mark is one unit in the X Direction each |
|
|
| align:start position:0% |
| mark is one unit in the X Direction each |
| |
|
|
| align:start position:0% |
| mark is one unit in the X Direction each |
| hash mark here is one unit in the y |
|
|
| align:start position:0% |
| hash mark here is one unit in the y |
| |
|
|
| align:start position:0% |
| hash mark here is one unit in the y |
| direction just want to point out two |
|
|
| align:start position:0% |
| direction just want to point out two |
| |
|
|
| align:start position:0% |
| direction just want to point out two |
| easy places to figure out the distance |
|
|
| align:start position:0% |
| easy places to figure out the distance |
| |
|
|
| align:start position:0% |
| easy places to figure out the distance |
| to the origin over here where the curve |
|
|
| align:start position:0% |
| to the origin over here where the curve |
| |
|
|
| align:start position:0% |
| to the origin over here where the curve |
| starts at40 the distance to the origin |
|
|
| align:start position:0% |
| starts at40 the distance to the origin |
| |
|
|
| align:start position:0% |
| starts at40 the distance to the origin |
| is 4 units and here at 02 the distance |
|
|
| align:start position:0% |
| is 4 units and here at 02 the distance |
| |
|
|
| align:start position:0% |
| is 4 units and here at 02 the distance |
| of the origin is two units so probably |
|
|
| align:start position:0% |
| of the origin is two units so probably |
| |
|
|
| align:start position:0% |
| of the origin is two units so probably |
| we could safely say further away here so |
|
|
| align:start position:0% |
| we could safely say further away here so |
| |
|
|
| align:start position:0% |
| we could safely say further away here so |
| we're anticipating that somewhere along |
|
|
| align:start position:0% |
| we're anticipating that somewhere along |
| |
|
|
| align:start position:0% |
| we're anticipating that somewhere along |
| the curve in this region is where we |
|
|
| align:start position:0% |
| the curve in this region is where we |
| |
|
|
| align:start position:0% |
| the curve in this region is where we |
| should find our our place that's closest |
|
|
| align:start position:0% |
| should find our our place that's closest |
| |
|
|
| align:start position:0% |
| should find our our place that's closest |
| to the origin the only reason I point |
|
|
| align:start position:0% |
| to the origin the only reason I point |
| |
|
|
| align:start position:0% |
| to the origin the only reason I point |
| that out is that when you're doing these |
|
|
| align:start position:0% |
| that out is that when you're doing these |
| |
|
|
| align:start position:0% |
| that out is that when you're doing these |
| problems on your own you should always |
|
|
| align:start position:0% |
| problems on your own you should always |
| |
|
|
| align:start position:0% |
| problems on your own you should always |
| try and anticipate roughly where thing |
|
|
| align:start position:0% |
| try and anticipate roughly where thing |
| |
|
|
| align:start position:0% |
| try and anticipate roughly where thing |
| should happen in what kind of region so |
|
|
| align:start position:0% |
| should happen in what kind of region so |
| |
|
|
| align:start position:0% |
| should happen in what kind of region so |
| that you don't you don't start thinking |
|
|
| align:start position:0% |
| that you don't you don't start thinking |
| |
|
|
| align:start position:0% |
| that you don't you don't start thinking |
| if you if you do something wrong and you |
|
|
| align:start position:0% |
| if you if you do something wrong and you |
| |
|
|
| align:start position:0% |
| if you if you do something wrong and you |
| get x equals 100 and then you come back |
|
|
| align:start position:0% |
| get x equals 100 and then you come back |
| |
|
|
| align:start position:0% |
| get x equals 100 and then you come back |
| and look at the curve you realize right |
|
|
| align:start position:0% |
| and look at the curve you realize right |
| |
|
|
| align:start position:0% |
| and look at the curve you realize right |
| away well that doesn't make any sense so |
|
|
| align:start position:0% |
| away well that doesn't make any sense so |
| |
|
|
| align:start position:0% |
| away well that doesn't make any sense so |
| we would always be thinking as we're |
|
|
| align:start position:0% |
| we would always be thinking as we're |
| |
|
|
| align:start position:0% |
| we would always be thinking as we're |
| solving the problem does my answer make |
|
|
| align:start position:0% |
| solving the problem does my answer make |
| |
|
|
| align:start position:0% |
| solving the problem does my answer make |
| sense so I'm actually going to give you |
|
|
| align:start position:0% |
| sense so I'm actually going to give you |
| |
|
|
| align:start position:0% |
| sense so I'm actually going to give you |
| a little bit of time to work on this |
|
|
| align:start position:0% |
| a little bit of time to work on this |
| |
|
|
| align:start position:0% |
| a little bit of time to work on this |
| yourself and then I'll come back and |
|
|
| align:start position:0% |
| yourself and then I'll come back and |
| |
|
|
| align:start position:0% |
| yourself and then I'll come back and |
| I'll work on it as |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| well well welcome back hopefully you |
|
|
| align:start position:0% |
| well well welcome back hopefully you |
| |
|
|
| align:start position:0% |
| well well welcome back hopefully you |
| were able to get pretty far into this |
|
|
| align:start position:0% |
| were able to get pretty far into this |
| |
|
|
| align:start position:0% |
| were able to get pretty far into this |
| problem and so I will start working on |
|
|
| align:start position:0% |
| problem and so I will start working on |
| |
|
|
| align:start position:0% |
| problem and so I will start working on |
| it |
|
|
| align:start position:0% |
| it |
| |
|
|
| align:start position:0% |
| it |
| now so the again the question is that we |
|
|
| align:start position:0% |
| now so the again the question is that we |
| |
|
|
| align:start position:0% |
| now so the again the question is that we |
| want to we want to optimize uh in this |
|
|
| align:start position:0% |
| want to we want to optimize uh in this |
| |
|
|
| align:start position:0% |
| want to we want to optimize uh in this |
| in this case minimize distance to the |
|
|
| align:start position:0% |
| in this case minimize distance to the |
| |
|
|
| align:start position:0% |
| in this case minimize distance to the |
| origin from this curve and so what we're |
|
|
| align:start position:0% |
| origin from this curve and so what we're |
| |
|
|
| align:start position:0% |
| origin from this curve and so what we're |
| really trying to do is we have we have a |
|
|
| align:start position:0% |
| really trying to do is we have we have a |
| |
|
|
| align:start position:0% |
| really trying to do is we have we have a |
| constraint the constraint is we have to |
|
|
| align:start position:0% |
| constraint the constraint is we have to |
| |
|
|
| align:start position:0% |
| constraint the constraint is we have to |
| be on the curve and then we also have um |
|
|
| align:start position:0% |
| be on the curve and then we also have um |
| |
|
|
| align:start position:0% |
| be on the curve and then we also have um |
| something we're trying to minimize and |
|
|
| align:start position:0% |
| something we're trying to minimize and |
| |
|
|
| align:start position:0% |
| something we're trying to minimize and |
| the thing we're trying to minimize is |
|
|
| align:start position:0% |
| the thing we're trying to minimize is |
| |
|
|
| align:start position:0% |
| the thing we're trying to minimize is |
| distance and so we have to we have to |
|
|
| align:start position:0% |
| distance and so we have to we have to |
| |
|
|
| align:start position:0% |
| distance and so we have to we have to |
| make sure that we understand the two |
|
|
| align:start position:0% |
| make sure that we understand the two |
| |
|
|
| align:start position:0% |
| make sure that we understand the two |
| equations that we need the optimization |
|
|
| align:start position:0% |
| equations that we need the optimization |
| |
|
|
| align:start position:0% |
| equations that we need the optimization |
| or the constraint equation and the |
|
|
| align:start position:0% |
| or the constraint equation and the |
| |
|
|
| align:start position:0% |
| or the constraint equation and the |
| optimizing equation so to optimize we |
|
|
| align:start position:0% |
| optimizing equation so to optimize we |
| |
|
|
| align:start position:0% |
| optimizing equation so to optimize we |
| need to know how to measure distance in |
|
|
| align:start position:0% |
| need to know how to measure distance in |
| |
|
|
| align:start position:0% |
| need to know how to measure distance in |
| in two-dimensional space and one point I |
|
|
| align:start position:0% |
| in two-dimensional space and one point I |
| |
|
|
| align:start position:0% |
| in two-dimensional space and one point I |
| want to make is that if you want to |
|
|
| align:start position:0% |
| want to make is that if you want to |
| |
|
|
| align:start position:0% |
| want to make is that if you want to |
| optimize distance you might as well |
|
|
| align:start position:0% |
| optimize distance you might as well |
| |
|
|
| align:start position:0% |
| optimize distance you might as well |
| optimize the square of distance because |
|
|
| align:start position:0% |
| optimize the square of distance because |
| |
|
|
| align:start position:0% |
| optimize the square of distance because |
| it's much easier so let me justify that |
|
|
| align:start position:0% |
| it's much easier so let me justify that |
| |
|
|
| align:start position:0% |
| it's much easier so let me justify that |
| briefly and then we'll go on so I want |
|
|
| align:start position:0% |
| briefly and then we'll go on so I want |
| |
|
|
| align:start position:0% |
| briefly and then we'll go on so I want |
| to optimize the distance squared to the |
|
|
| align:start position:0% |
| to optimize the distance squared to the |
| |
|
|
| align:start position:0% |
| to optimize the distance squared to the |
| origin it's well distance you remember |
|
|
| align:start position:0% |
| origin it's well distance you remember |
| |
|
|
| align:start position:0% |
| origin it's well distance you remember |
| first in general between two points x y |
|
|
| align:start position:0% |
| first in general between two points x y |
| |
|
|
| align:start position:0% |
| first in general between two points x y |
| and ab is is something in this |
|
|
| align:start position:0% |
| and ab is is something in this |
| |
|
|
| align:start position:0% |
| and ab is is something in this |
| form distance squared is the difference |
|
|
| align:start position:0% |
| form distance squared is the difference |
| |
|
|
| align:start position:0% |
| form distance squared is the difference |
| between the x value squared plus the |
|
|
| align:start position:0% |
| between the x value squared plus the |
| |
|
|
| align:start position:0% |
| between the x value squared plus the |
| difference between the y- value squared |
|
|
| align:start position:0% |
| difference between the y- value squared |
| |
|
|
| align:start position:0% |
| difference between the y- value squared |
| this is should remind you of the |
|
|
| align:start position:0% |
| this is should remind you of the |
| |
|
|
| align:start position:0% |
| this is should remind you of the |
| Pythagorean theorem |
|
|
| align:start position:0% |
| Pythagorean theorem |
| |
|
|
| align:start position:0% |
| Pythagorean theorem |
| ultimately so in this case in our case |
|
|
| align:start position:0% |
| ultimately so in this case in our case |
| |
|
|
| align:start position:0% |
| ultimately so in this case in our case |
| distance is to the origin is x^2 + y^2 |
|
|
| align:start position:0% |
| distance is to the origin is x^2 + y^2 |
| |
|
|
| align:start position:0% |
| distance is to the origin is x^2 + y^2 |
| the distance squ is x2+ y^2 I just told |
|
|
| align:start position:0% |
| the distance squ is x2+ y^2 I just told |
| |
|
|
| align:start position:0% |
| the distance squ is x2+ y^2 I just told |
| you that instead of trying to optimize |
|
|
| align:start position:0% |
| you that instead of trying to optimize |
| |
|
|
| align:start position:0% |
| you that instead of trying to optimize |
| distance we can optimize distance |
|
|
| align:start position:0% |
| distance we can optimize distance |
| |
|
|
| align:start position:0% |
| distance we can optimize distance |
| squared Why is that well remember that |
|
|
| align:start position:0% |
| squared Why is that well remember that |
| |
|
|
| align:start position:0% |
| squared Why is that well remember that |
| when you optimize what you're looking |
|
|
| align:start position:0% |
| when you optimize what you're looking |
| |
|
|
| align:start position:0% |
| when you optimize what you're looking |
| for is a place where the derivative of |
|
|
| align:start position:0% |
| for is a place where the derivative of |
| |
|
|
| align:start position:0% |
| for is a place where the derivative of |
| the function of of interest is equal to |
|
|
| align:start position:0% |
| the function of of interest is equal to |
| |
|
|
| align:start position:0% |
| the function of of interest is equal to |
| zero so what I want to point out is that |
|
|
| align:start position:0% |
| zero so what I want to point out is that |
| |
|
|
| align:start position:0% |
| zero so what I want to point out is that |
| when you take the derivative of distance |
|
|
| align:start position:0% |
| when you take the derivative of distance |
| |
|
|
| align:start position:0% |
| when you take the derivative of distance |
| squared and find where that's zero it's |
|
|
| align:start position:0% |
| squared and find where that's zero it's |
| |
|
|
| align:start position:0% |
| squared and find where that's zero it's |
| going to be the same as the place where |
|
|
| align:start position:0% |
| going to be the same as the place where |
| |
|
|
| align:start position:0% |
| going to be the same as the place where |
| uh where the derivative of distance is |
|
|
| align:start position:0% |
| uh where the derivative of distance is |
| |
|
|
| align:start position:0% |
| uh where the derivative of distance is |
| equal to zero so let's notice that so |
|
|
| align:start position:0% |
| equal to zero so let's notice that so |
| |
|
|
| align:start position:0% |
| equal to zero so let's notice that so |
| this is a little |
|
|
| align:start position:0% |
| this is a little |
| |
|
|
| align:start position:0% |
| this is a little |
| sidebar |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| justification notice d squ Prime is |
|
|
| align:start position:0% |
| justification notice d squ Prime is |
| |
|
|
| align:start position:0% |
| justification notice d squ Prime is |
| equal to 2D D Prime where did that come |
|
|
| align:start position:0% |
| equal to 2D D Prime where did that come |
| |
|
|
| align:start position:0% |
| equal to 2D D Prime where did that come |
| from that's this is implicit |
|
|
| align:start position:0% |
| from that's this is implicit |
| |
|
|
| align:start position:0% |
| from that's this is implicit |
| differentiation with respect to X and |
|
|
| align:start position:0% |
| differentiation with respect to X and |
| |
|
|
| align:start position:0% |
| differentiation with respect to X and |
| this is the chain rule so if I want D |
|
|
| align:start position:0% |
| this is the chain rule so if I want D |
| |
|
|
| align:start position:0% |
| this is the chain rule so if I want D |
| Prime to equal zero I can also find |
|
|
| align:start position:0% |
| Prime to equal zero I can also find |
| |
|
|
| align:start position:0% |
| Prime to equal zero I can also find |
| where D2 Prime equals zero I'm assuming |
|
|
| align:start position:0% |
| where D2 Prime equals zero I'm assuming |
| |
|
|
| align:start position:0% |
| where D2 Prime equals zero I'm assuming |
| notice the distance is never at the |
|
|
| align:start position:0% |
| notice the distance is never at the |
| |
|
|
| align:start position:0% |
| notice the distance is never at the |
| origin so distance is never zero so I |
|
|
| align:start position:0% |
| origin so distance is never zero so I |
| |
|
|
| align:start position:0% |
| origin so distance is never zero so I |
| don't have to worry about that so that's |
|
|
| align:start position:0% |
| don't have to worry about that so that's |
| |
|
|
| align:start position:0% |
| don't have to worry about that so that's |
| a small sidebar but just to justify why |
|
|
| align:start position:0% |
| a small sidebar but just to justify why |
| |
|
|
| align:start position:0% |
| a small sidebar but just to justify why |
| we can do that now let's come back into |
|
|
| align:start position:0% |
| we can do that now let's come back into |
| |
|
|
| align:start position:0% |
| we can do that now let's come back into |
| the problem at hand what is our |
|
|
| align:start position:0% |
| the problem at hand what is our |
| |
|
|
| align:start position:0% |
| the problem at hand what is our |
| optimization problem um |
|
|
| align:start position:0% |
| optimization problem um |
| |
|
|
| align:start position:0% |
| optimization problem um |
| equation that we want to minimize we |
|
|
| align:start position:0% |
| equation that we want to minimize we |
| |
|
|
| align:start position:0% |
| equation that we want to minimize we |
| want to minimize this |
|
|
| align:start position:0% |
| want to minimize this |
| |
|
|
| align:start position:0% |
| want to minimize this |
| equation with respect to a certain |
|
|
| align:start position:0% |
| equation with respect to a certain |
| |
|
|
| align:start position:0% |
| equation with respect to a certain |
| constraint what's the constraint the |
|
|
| align:start position:0% |
| constraint what's the constraint the |
| |
|
|
| align:start position:0% |
| constraint what's the constraint the |
| constraint is what Y is y depends on on |
|
|
| align:start position:0% |
| constraint is what Y is y depends on on |
| |
|
|
| align:start position:0% |
| constraint is what Y is y depends on on |
| X and so when I solve these problems I'm |
|
|
| align:start position:0% |
| X and so when I solve these problems I'm |
| |
|
|
| align:start position:0% |
| X and so when I solve these problems I'm |
| I'm going to have to substitute in my |
|
|
| align:start position:0% |
| I'm going to have to substitute in my |
| |
|
|
| align:start position:0% |
| I'm going to have to substitute in my |
| constraint so y^2 is the square root of |
|
|
| align:start position:0% |
| constraint so y^2 is the square root of |
| |
|
|
| align:start position:0% |
| constraint so y^2 is the square root of |
| X+ 4 quantity squared so I just get X+ |
|
|
| align:start position:0% |
| X+ 4 quantity squared so I just get X+ |
| |
|
|
| align:start position:0% |
| X+ 4 quantity squared so I just get X+ |
|
|
| align:start position:0% |
| |
|
|
| align:start position:0% |
| so now I have my optimization equation |
|
|
| align:start position:0% |
| so now I have my optimization equation |
| |
|
|
| align:start position:0% |
| so now I have my optimization equation |
| how do I find a minimum or a maximum I |
|
|
| align:start position:0% |
| how do I find a minimum or a maximum I |
| |
|
|
| align:start position:0% |
| how do I find a minimum or a maximum I |
| take the derivative and set it equal to |
|
|
| align:start position:0% |
| take the derivative and set it equal to |
| |
|
|
| align:start position:0% |
| take the derivative and set it equal to |
| zero so let me come give myself a little |
|
|
| align:start position:0% |
| zero so let me come give myself a little |
| |
|
|
| align:start position:0% |
| zero so let me come give myself a little |
| more room and do that over |
|
|
| align:start position:0% |
| more room and do that over |
| |
|
|
| align:start position:0% |
| more room and do that over |
| here so d^2 |
|
|
| align:start position:0% |
| here so d^2 |
| |
|
|
| align:start position:0% |
| here so d^2 |
| Prime now I get derivative of X2 is |
|
|
| align:start position:0% |
| Prime now I get derivative of X2 is |
| |
|
|
| align:start position:0% |
| Prime now I get derivative of X2 is |
| 2x the derivative of x is one and the |
|
|
| align:start position:0% |
| 2x the derivative of x is one and the |
| |
|
|
| align:start position:0% |
| 2x the derivative of x is one and the |
| derivative of four is zero this will be |
|
|
| align:start position:0% |
| derivative of four is zero this will be |
| |
|
|
| align:start position:0% |
| derivative of four is zero this will be |
| optimized when this is equal Al to 0 so |
|
|
| align:start position:0% |
| optimized when this is equal Al to 0 so |
| |
|
|
| align:start position:0% |
| optimized when this is equal Al to 0 so |
| 0 = 2x + 1 so X is equal to |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| -2 does this pass as we would say maybe |
|
|
| align:start position:0% |
| -2 does this pass as we would say maybe |
| |
|
|
| align:start position:0% |
| -2 does this pass as we would say maybe |
| the smell test does it smell okay to us |
|
|
| align:start position:0% |
| the smell test does it smell okay to us |
| |
|
|
| align:start position:0% |
| the smell test does it smell okay to us |
| the answer will be yes because it |
|
|
| align:start position:0% |
| the answer will be yes because it |
| |
|
|
| align:start position:0% |
| the answer will be yes because it |
| remember we said somewhere in this x |
|
|
| align:start position:0% |
| remember we said somewhere in this x |
| |
|
|
| align:start position:0% |
| remember we said somewhere in this x |
| region is where we expect that we will |
|
|
| align:start position:0% |
| region is where we expect that we will |
| |
|
|
| align:start position:0% |
| region is where we expect that we will |
| have a distance closest a point closest |
|
|
| align:start position:0% |
| have a distance closest a point closest |
| |
|
|
| align:start position:0% |
| have a distance closest a point closest |
| to the origin and so we're right here on |
|
|
| align:start position:0% |
| to the origin and so we're right here on |
| |
|
|
| align:start position:0% |
| to the origin and so we're right here on |
| the x value now we have to find what the |
|
|
| align:start position:0% |
| the x value now we have to find what the |
| |
|
|
| align:start position:0% |
| the x value now we have to find what the |
| Y value is to finish the problem but |
|
|
| align:start position:0% |
| Y value is to finish the problem but |
| |
|
|
| align:start position:0% |
| Y value is to finish the problem but |
| this this is not so far very surprising |
|
|
| align:start position:0% |
| this this is not so far very surprising |
| |
|
|
| align:start position:0% |
| this this is not so far very surprising |
| it seems like maybe the right thing now |
|
|
| align:start position:0% |
| it seems like maybe the right thing now |
| |
|
|
| align:start position:0% |
| it seems like maybe the right thing now |
| we have X so now how do we find y well |
|
|
| align:start position:0% |
| we have X so now how do we find y well |
| |
|
|
| align:start position:0% |
| we have X so now how do we find y well |
| we know what Y is y is equal to the |
|
|
| align:start position:0% |
| we know what Y is y is equal to the |
| |
|
|
| align:start position:0% |
| we know what Y is y is equal to the |
| square root of x + 4 so it's equal to |
|
|
| align:start position:0% |
| square root of x + 4 so it's equal to |
| |
|
|
| align:start position:0% |
| square root of x + 4 so it's equal to |
| theun of -2 + |
|
|
| align:start position:0% |
| theun of -2 + |
| |
|
|
| align:start position:0% |
| theun of -2 + |
| 4 which simplified is three and a half |
|
|
| align:start position:0% |
| 4 which simplified is three and a half |
| |
|
|
| align:start position:0% |
| 4 which simplified is three and a half |
| which I think uh is seven |
|
|
| align:start position:0% |
| which I think uh is seven |
| |
|
|
| align:start position:0% |
| which I think uh is seven |
| halves so the point is - |
|
|
| align:start position:0% |
| halves so the point is - |
| |
|
|
| align:start position:0% |
| halves so the point is - |
| one2 comma square root of |
|
|
| align:start position:0% |
| one2 comma square root of |
| |
|
|
| align:start position:0% |
| one2 comma square root of |
| sevenes and then you just want to double |
|
|
| align:start position:0% |
| sevenes and then you just want to double |
| |
|
|
| align:start position:0% |
| sevenes and then you just want to double |
| check and make sure did I ask for the |
|
|
| align:start position:0% |
| check and make sure did I ask for the |
| |
|
|
| align:start position:0% |
| check and make sure did I ask for the |
| distance or did I ask for the point and |
|
|
| align:start position:0% |
| distance or did I ask for the point and |
| |
|
|
| align:start position:0% |
| distance or did I ask for the point and |
| right now we have the point so let's |
|
|
| align:start position:0% |
| right now we have the point so let's |
| |
|
|
| align:start position:0% |
| right now we have the point so let's |
| come over and make sure what point on |
|
|
| align:start position:0% |
| come over and make sure what point on |
| |
|
|
| align:start position:0% |
| come over and make sure what point on |
| the curve is comes closest to the origin |
|
|
| align:start position:0% |
| the curve is comes closest to the origin |
| |
|
|
| align:start position:0% |
| the curve is comes closest to the origin |
| so now we know that we've answered the |
|
|
| align:start position:0% |
| so now we know that we've answered the |
| |
|
|
| align:start position:0% |
| so now we know that we've answered the |
| correct question so again it was a maxim |
|
|
| align:start position:0% |
| correct question so again it was a maxim |
| |
|
|
| align:start position:0% |
| correct question so again it was a maxim |
| maximize Min sorry it was a minimizing |
|
|
| align:start position:0% |
| maximize Min sorry it was a minimizing |
| |
|
|
| align:start position:0% |
| maximize Min sorry it was a minimizing |
| problem it was an optimization problem |
|
|
| align:start position:0% |
| problem it was an optimization problem |
| |
|
|
| align:start position:0% |
| problem it was an optimization problem |
| where we wanted to minimize distance we |
|
|
| align:start position:0% |
| where we wanted to minimize distance we |
| |
|
|
| align:start position:0% |
| where we wanted to minimize distance we |
| had a constraint equation we had the |
|
|
| align:start position:0% |
| had a constraint equation we had the |
| |
|
|
| align:start position:0% |
| had a constraint equation we had the |
| thing we wanted to minimize then we took |
|
|
| align:start position:0% |
| thing we wanted to minimize then we took |
| |
|
|
| align:start position:0% |
| thing we wanted to minimize then we took |
| the derivative of the minimizer set it |
|
|
| align:start position:0% |
| the derivative of the minimizer set it |
| |
|
|
| align:start position:0% |
| the derivative of the minimizer set it |
| or of the optimizing equation set it |
|
|
| align:start position:0% |
| or of the optimizing equation set it |
| |
|
|
| align:start position:0% |
| or of the optimizing equation set it |
| equal to zero solved for x and then |
|
|
| align:start position:0% |
| equal to zero solved for x and then |
| |
|
|
| align:start position:0% |
| equal to zero solved for x and then |
| found the answer to the specific |
|
|
| align:start position:0% |
| found the answer to the specific |
| |
|
|
| align:start position:0% |
| found the answer to the specific |
| question by then finding the Y value and |
|
|
| align:start position:0% |
| question by then finding the Y value and |
| |
|
|
| align:start position:0% |
| question by then finding the Y value and |
| I think I'll stop there |
