Section 3.5: Summary of Curve Sketching
Calculus I
Andy Long

(* #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}]

(* #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}]

(* #23, p. 243 *)
f[x_]:=Sqrt[x^2+x-2]
Together[f'[x]]
Solve[f[x]==0,x]
Solve[f'[x]==0,x]
Together[f''[x]]
Plot[{f[x],Abs[x+1/2]},{x,-6,5}]

(* #50, p. 243 *)
f[x_]:=(1+5x-2x^2)/(x-2)
Together[f'[x]]
Solve[f[x]==0,x]
N[%]
Solve[f'[x]==0,x]
Together[f''[x]]
Plot[{f[x],-2x+1},{x,-3,5}]

(* #51, p. 243 *)
f[x_]:=(x^3+4)/x^2
Together[f'[x]]
Solve[f[x]==0,x]
N[%]
Solve[f'[x]==0,x]
Together[f''[x]]
Plot[{f[x],x},{x,-3,5}]