4.3 Homework
Calculus I
Andy Long

(* #4, p. 318 *)
f[x_]:=3Cos[2Pi/12*x]
Plot[f[x],{x,0,6}]
intf[x_]:=3Sin[2Pi/12*x]12/(2Pi)
Plot[intf[x],{x,0,6}]

(* #12 *)
Integrand[t_]:=Cos[Sqrt[t]]
G[x_]:=Integrate[Integrand[t],{t,x,1}]
Plot[{Integrand[x],G[x]},{x,0,2},PlotLabel-> "G and Integrand"]
Gp[x_]:=-Cos[Sqrt[x]]
Plot[{Gp[x],G[x]},{x,0,2},PlotLabel-> "G and G'"]

(* #14 *)
Integrand[z_]:=z^2/(z^4+1)
h[x_]:=Integrate[Integrand[z],{z,1,Sqrt[x]}]
Plot[{h[x],Integrand[x]},{x,0,2},PlotLabel-> "h and Integrand"]
hp[x_]:=Integrand[Sqrt[x]]*1/(2*Sqrt[x])
hp[x]
Plot[{h[x],hp[x]},{x,0,2},PlotLabel-> "h and h'"]

(* #21 *)
Integrand[t_]:=5-2 t+3t^2
Integral=Integrate[Integrand[t],{t,1,4}]
(* an anti-derivative *)
f[x_]:=5x-x^2+x^3
f[4]-f[1]

(* #25 *)
Integrand[x_]:=Sin[x]
Integral=Integrate[Integrand[x],{x,Pi/6,Pi}]
-Cos[Pi]+Cos[Pi/6]

(* #38 *)
Plot[Piecewise[{{2,-2≤x≤0}},4-x^2],{x,-2,2}]
Integrate[Piecewise[{{2,-2≤x≤0}},4-x^2],{x,-2,2}]
Int=2*2+Integrate[4-x^2,{x,0,2}]