Calculus I: Section 4.4

Andy Long

#1,  p. 326

f[x_]:=1/(x^2Sqrt[1+x^2])
antiF[x_]=Integrate[f[x],x]
antiF[x]
Plot[antiF[x],{x,1,2}]
Together[antiF'[x]]

#5,  p. 326

f[x_]:=x^2+x^(-2)
(* I'm going to choose the anti-derivative that's 0 at 1: *)
antiF[x_]=Integrate[f[x],x]+2/3
(* The general anti-derivative should have an arbitrary constant. *)
antiF[x]
Plot[f[x],{x,1,2},Filling->0,PlotRange->{{1,2},{0,5}}]
Plot[antiF[x],{x,1,2}]
Together[antiF[x]]

#45,  p. 327

f[x_]:=2x-x^2
Plot[f[x],{x,0,2},Filling->Axis]
Integrate[f[x],{x,0,2}]

#50,  p. 327

np[t_]:=20ArcTan[x/8] (* just imagining a growth rate... *)
bees[x_]:=100+Integrate[np[t],{t,0,x}]
Plot[np[x],{x,0,15},PlotRange->{{0,15},{0,25}},Filling->Axis]
Plot[bees[x],{x,0,15},PlotRange->{{0,15},{0,600}}]