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1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 | 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 |