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