Visual Guide for Lab 10Our goal is to determine a delta-epsilon definition of continuity at a point x_0. Loosely speaking, we will do this by looking at an arbitrary horizontal band centered about f(x_0). The goal is to determine if there is a corresponding vertical band centered about x_0, such that domain values in the vertical band map to function values in the horizontal band.Enter the Plotting Package.with(plots):
with(plottools):Defining the Function.First we define the function.f:=x->abs(x);We include an example of a piecewise-defined function.g:=x->piecewise(x<4,(x^2-16)/(x-4),x=4,7,x>4,(x^2-16)/(x-4));Now we name the point at which we wish to investigate continuity.x0:=0;We compute the function value at this point.f(x0);Graphing the Function.Use the graph below or some other method to decide if the function is continuous at x_0. We include an example of plotting a piecewise-defined function. You can change the range on the domain values if you want or need to.plot(f(x),x=x0-3..x0+3);What follows are the directions for plotting a piecewise-defined function. There are two parts. The first part is the graph of the function up to the point where the definition splits and the second part is the graph after the point where the definition splits. pt1:=plot(g,0..4,style=point):
pt2:=plot(g,4..6,style=point):
display([pt1,pt2]);
Setting up the Horizontal epsilon-band.We set the width of the horizontal band.epsilon:=1;Plotting the Horizontal Band.We define an interval centered at x_0 as the domain of the graph. For the radius of the interval, the value 2 was chosen arbitrarily. You may need to adjust it. xrange:=x0-2..x0+2;Now we will plot the function and the horizontal band.plot({f(x),f(x0)-epsilon,f(x0)+epsilon},x=xrange,color=green);Save the above graph.gra:=plot({f(x),f(x0)-epsilon,f(x0)+epsilon},x=xrange,color=green):Determining the Vertical Band.Now we will show you how to experiment with different vertical lines. Suppose our guess for the width of the band centered about x_0 was 1.5. Then we would graph two vertical lines as shown below. We need to set some y limits. You can determine reasonable values for these from your graph above.implicitplot({x=x0-1.5,x=x0+1.5},x=xrange,y=-2..2);guess:=implicitplot({x=x0-1.5,x=x0+1.5},x=xrange,y=-2..2):Now we add the vertical lines to the graph of the function and the horizontal band.display({guess,gra});This was a bad guess. Did you expect us to give you a good one?!restart;