Examples from section 2.3
Calculus I
Andy Long

#13, p. 136

f[p_]:=Sqrt[p]-p
f'[p]

#22, p. 136

g[x_]:=Sqrt[x]+1/(x^(1/3))
f[x_]:=(g[x])^2
f'[x]
Expand[%]
2*g'[x]*g[x]
Expand[%]

#48, p. 137

(*  Notice the symmetry. f is odd, so its derivative is even. *)
f[x_]:=x+1/x;
Plot[f[x],{x,-2,2},PlotRange->{-10,10}]
Plot[{f[x],f'[x]},{x,-2,2}]

#58: p. 137

f[x_]:=Sqrt[x]/(x+1);
Together[f'[x]]
f'[4]
(* Observe how I put the tangent line into the plot: in point-slope form: *)
T[x_]:=f'[4](x-4)+f[4]
normal[x_]:=-1/f'[4]*(x-4)+f[4]
Plot[{f[x],T[x],normal[x]},{x,0,6},PlotRange->{-1.5,1.5}]
Plot[{f[x],f'[x]},{x,0,6}]

#64: p. 137

s[t_]:=t^4-2t^3+t^2-t;
s'[t]
s''[t]
s''[1]
Plot[{s[t],s'[t],s''[t]},{t,0,1}]
{s[1],s'[1],s''[1]}
Plot[{s[t],s'[t],s''[t]},{t,0,6}]
{s[6],s'[6],s''[6]}