Consider the following data. data6:=[[1,1],[2,1.5],[3,3]];

NiM+SSZkYXRhNkc2IjclNyQiIiJGKDckIiIjJCIjOiEiIjckIiIkRi8=

plot(data6,x=0..4,y=0..4,style=point,symbol=box);

LSUlUExPVEc2KC0lJ0NVUlZFU0c2JDclNyQkIiIiIiIhRio3JCQiIiNGLCQiMysrKysrKysrOiEjPDckJCIiJEYsRjQtJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkRixGPEY9LSUrQVhFU0xBQkVMU0c2JFEieDYiUSJ5RkItJSVWSUVXRzYkO0Y9JCIjU0Y8RkctJSZTVFlMRUc2IyUmUE9JTlRHLSUlRk9OVEc2JCUqSEVMVkVUSUNBR0Y7LSUnU1lNQk9MRzYkJSRCT1hHRjs=

Find a linear function that approximate the three points of data6. (this involves some guessing). Meassure the error using the square of the vertical distances. Azure:=x->x;

NiM+SSZBenVyZUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJTkkRiVGJUYl

Anna:=x->x+1;

NiM+SSVBbm5hRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIkYuRi5GJUYlRiU=

Anne:=x->.9*x-.1;

NiM+SSVBbm5lRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiQiIiohIiIiIiI5JEYxRjEkRjFGMEYwRiVGJUYl

Jessica:=x->x/2+.5;

NiM+SShKZXNzaWNhRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCMiIiIiIiMkIiImISIiRi9GJUYlRiU=

Zhen:=x->.5*x+.5;

NiM+SSVaaGVuRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiQiIiYhIiIiIiI5JEYxRjFGLkYxRiVGJUYl

Jen:=x->3*x/2;

NiM+SSRKZW5HNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJDkkIyIiJCIiI0YlRiVGJQ==

Let us plot all the lines in the same graphplots[display](plot(data6,x=0..4,style=point,symbol=box, color=black),plot([Azure(x),Anna(x),Anne(x),Jessica(x),Zhen(x),Jen(x)],x=0..4,color=[blue,orange,gold,black,yellow,green]));Here is the graph of the line, with the segments representing the distances whose squares we added.L:=Jen;

plots[display](plot(data6,style=point,symbol=box, color=green),plot([L(x)],x=0..5,y=0..10,color=red),seq(plot([data6[i][1],t,t=data6[i][2]..L(data6[i][1])],thickness=3,color=blue),i=1..3));Define the error function and find its value for each of the line above.

Plot the error functionplot3d(E(data6,m,b),M=-1..6,B=-5..4,view=-1..5,axes=boxed,style=patchcontour);Find the minimum of the error funcion.expand(E(data6,M,B));

solve({diff(E(data6,M,B),B)=0 ,diff(E(data6,M,B),M)=0},{M,B});

Find the linear function that best approximate these points by the method of least squares. Plot the result wiht the points.Find the line that best approximate data6 by least squares method.Find the line that best approximate the points [1,2],[2,4],[3,5],[4,6] by the two methods above (finding the mininum of the error function and using the CurveFitting command).data7:=[[1,2],[2,4],[3,5],[4,6] ];1. Using the points of data7, define an error function that adds the squares of the horizontal distances. Find the minimum of this function and compare these new results with old ones.2. Repeat the same with the points of data6. Consider the following data. Find the line that approximate these points by LeastSquares. Do you think it is a good approximation?data8:=[seq([i/4,i/20+1],i=-1..20),[1,4]];

NiM+SSZkYXRhOEc2Ijc5NyQjISIiIiIlIyIjPiIjPzckIiIhIiIiNyQjRjBGKiMiI0BGLTckI0YwIiIjIyIjNiIjNTckIyIiJEYqIyIjQkYtNyRGMCMiIiciIiY3JCNGQ0YqRkU3JCNGPUY3IyIjOEY6NyQjIiIoRiojIiNGRi03JEY3I0ZMRkM3JCMiIipGKiMiI0hGLTckI0ZDRjdGRzckI0Y5RiojIiNKRi03JEY9IyIiKUZDNyQjRklGKiMiI0xGLTckI0ZMRjcjIiM8Rjo3JCMiIzpGKkZLNyRGKiNGU0ZDNyQjRmBvRiojIiNQRi03JCNGU0Y3I0YsRjo3JCNGLEYqIyIjUkYtNyRGQ0Y3NyRGMEYq

plot(data8,x=-1..5,y=-1..5,style=point);evalf(ln(2));

NiMkIisxPVpKcCEjNQ==

e:=(data8,m,b)->sum(ln(1+(m*data8[i][1]+b-data8[i][2])^2/2),i=1..21);

NiM+SSJlRzYiZio2JUkmZGF0YThHRiVJIm1HRiVJImJHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJHN1bUc2JEkqcHJvdGVjdGVkR0YxSShfc3lzbGliR0YlNiQtSSNsbkdGMDYjLCYiIiJGOCokLCgqJjklRjgmJjkkNiNJImlHRiU2I0Y4RjhGODkmRjgmRj42IyIiIyEiIkZGI0Y4RkYvRkE7RjgiI0BGJUYlRiU=

solve({diff(e(data8,m,b),m)=0,diff(e(data8,m,b),b)=0},{m,b});

Warning, computation interrupted

plot3d(e(data8,m,b),m=0..5,b=0..8,axes=boxed);

-%'PLOT3DG6'-%%GRIDG6%;$""!F*$""&F*;F)$"")F*X,I)anythingGI*protectedGF26"F3[gl'!%"!!#\bm":":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