Section 3.5 homework
Calculus I
Andy Long

(* #1, p. 242 *)
f[x_]:=x^3-12x^2+36x
(* It's going to be symmetric about the point of inflection. *)
Together[f'[x]]
Solve[f'[x]==0,x] (* That's a double root at x=6 *)
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],f'[x],f''[x]},{x,-1,7}]

(* #9, p. 242 *)
f[x_]:=x/(x-1)
(* It's going to be hyperbolas! *)
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],1},{x,-3,5}]

(* #12, p. 242 *)
f[x_]:=x/(x^2-9)
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],0},{x,-6,6}]

(* #13, p. 242 *)
f[x_]:=1/(x^2-9)
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],0},{x,-6,6}]

(* #26, p. 242 *)
f[x_]:=x*Sqrt[2-x^2]
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x]},{x,-Sqrt[2],Sqrt[2]}]
Plot[{f[x],f'[x]},{x,-Sqrt[2],Sqrt[2]}] (* Notice that the slopes are becoming infinite at the endpoints. *)

(* #44, p. 243 -- think about Newton's "F=ma" *)
f[x_]:=k(-1/x^2+1/(x-2)^2)
k=1
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],0},{x,-2,4}]

(* #46, p. 242 *)
f[x_]:=(2x^3+x^2+x+3)/(x^2+2x)
s[x_]:=2x-3
Plot[{f[x],s[x]},{x,-8,5},PlotRange->{-30,40}]

(* #48, p. 242 *)
f[x_]:=(5x^4+x^2+x)/(x^3-x^2+2)
s[x_]:=5(x+1)
Together[f'[x]]
Solve[f'[x]==0,x]
Together[f''[x]]
Solve[f''[x]==0,x]
Plot[{f[x],s[x]},{x,-3,5},PlotRange->{-20,40}]