Shape of a Function: Section 3.3
Calculus I
Andy Long

#9, p. 221

(*  
f is a cubic, so it's really just a question of which kind of cubic graph it's going to exhibit.
*)
f[x_]:=-2x^3+3x^2+36x
f'[x]
Solve[f'[x]==0,x]
f''[x]
Solve[f''[x]==0,x]
Plot[{f[x],f'[x],f''[x]},{x,-5,3}]
(* oh, that one!
:*)

#10, p. 221

(*  
f is a cubic, so it's really just a question of which kind of cubic graph it's going to exhibit.
*)
f[x_]:=4x^3+3x^2-6x+1
f'[x]
extremes=Solve[f'[x]==0,x]
f[-1]
f[1/2]
f''[x]
Solve[f''[x]==0,x]
Plot[{f[x],f'[x],f''[x]},{x,-2,2}]
(* oh, that one!
:*)

#22, p. 221

(* Here's one that will do that! *)
Plot[Piecewise[{{1-(x+2),x<-2},{1-Sin[Pi*x/2],-2<x<2},{1-(x-2),x>-2}}],{x,-3,3}]

#30, p. 222

(* 
As a third degree polynomial, we know pretty well what it's shape will be (and that it will possess symmetry). 
*)
f[x_]:=36x+3x^2-2x^3
f'[x]
Solve[f'[x]==0,x]
f''[x]
Solve[f''[x]==0,x]
Plot[{f[x],f'[x],f''[x]},{x,-5,6}]

#32, p. 222

(* 
As a fourth degree polynomial, we know pretty well what it's shape will be. But notice the lack of symmetry -- quartics are the first polynomial functions that can fail to have some point or axis of symmetry. 
*)
f[x_]:=200+8x^3+x^4
f'[x]
Solve[f'[x]==0,x]
f''[x]
Solve[f''[x]==0,x]
Plot[{f[x],f'[x],f''[x]},{x,-8,2}]

#52, p. 222: graph of coffee....