Chapter 1, Section 2: Preview Activity

a. Determine the values g(−2), g(−1), g(0), g(1), and g(2), if defined. If the function value is not defined, explain what feature of the graph tells you this.

g[t_]:=
If[t==0,1,
If[t≤1,4-t^2,
If[t≠2,3-t]
]
]
p1aa=Plot[g[t],{t,-2.3,-0.05},PlotStyle->{Thickness[0.005]}];
p1ab=Plot[g[t],{t,0.05,1},PlotStyle->{Thickness[0.005]}];
p1b=Plot[g[t],{t,1.05,1.95},PlotStyle->{Thickness[0.005]}];
p1c=Plot[g[t],{t,2.05,3.5},PlotStyle->{Thickness[0.005]}];
p2=ListPlot[{{0,4},{1,2},{2,1}},PlotStyle->Large,PlotMarkers->{O}];
p3=ListPlot[{{0,1},{1,3}},PlotStyle->Large];
Show[p1aa,p1ab,p1b,p1c,p2,p3,PlotLabel->"Discontinuous Function of Preview 1.2.1",AxesLabel->{"t","y"},AspectRatio->1,PlotRange->{{-2.3,3.5},All}]
MatrixForm[Table[{x,g[x]},{x,{-2,-1,0,1,2}}]]

b. For each of the values a = −1, a = 0, and a = 2, complete the following sentence : "As x gets closer and closer (but not equal) to a, g (x) gets as close as we want to______."

f[x_]:=g[x]
Manipulate[
p2m=ListPlot[{{x,0},{x,f[x]},{0,f[x]}}];
p3m=ListPlot[{{x,0},{x,f[x]},{0,f[x]}},Joined->True,PlotStyle->Dashed];
Show[p2m,p3m,p1aa,p1ab,p1b,p1c,p2,p3,PlotLabel->"Discontinuous Function of Preview 1.2.1",AxesLabel->{"t","y"},AspectRatio->1,PlotRange->{{-2.3,3.5},All}],
{x,-2.3,3.2,.01}
]

a=-1: g(-1)=3
a=0: 4
a=2: 1

c. What happens as x gets closer and closer (but not equal) to a=1? Does the function g(x) get as close as we would like to a single value?

No: it’s split. On the left of a=1, it’s approaching 3; on the right, it’s approaching 2.

Out of curiousity, what does this function sound like? :)

g[t_]:=
If[t==0,1,
If[t≤1,4-t^2,
If[t≠2,3-t]
]
]
Plot[g[x],{x,-2.3,3.5},PlotRange->{{-2.3,3.5},All}]

gg[x_]:=Piecewise[{{4-x^2,x≤1},
{3-x,x>1}}]
Play[(gg[t]+1.5)*Sin[440 *2 Pi t],{t,-2.3,3.5}]

Chapter 1, Section 2: Worksheet problems

Here’s a picture of the graph you’ll encounter in your worksheet for this section. Let’s animate it to illustrate what we’re  talking about when we talk about limits....

h[x_]:=2^(1/x)
Manipulate[
p12=Plot[{h[x],1},{x,-3,3}];
p22=ListPlot[{{x,0},{x,h[x]},{0,h[x]}}];
p32=ListPlot[{{x,0},{x,h[x]},{0,h[x]}},Joined->True,PlotStyle->Dashed];
Show[p12,p22,p32],
{x,-3,3,.01}
]