Examples, Section 2.4
Calculus I
Andy Long

#7, p. 146

y[t_]:=c*Cos[t]+t^2*Sin[t]
y'[t]
(* to graph it we need to set the parameter c to some particular value. I just chose 2 at random: *)
c=2
Plot[{y[t],y'[t]},{t,-2Pi,2Pi}]

(* Now let's see that in action: c is a parameter -- a scalar, that we set -- but we're going to let it change and see how that affects the shape of the derivative. *)
Clear[c]
Manipulate[
Module[
{},
Plot[{c*Cos[t]+t^2*Sin[t],-c*Sin[t]+2t*Sin[t]+t^2Cos[t]},{t,-2Pi,2Pi}]
]
,{c,-10,10,.01}
]

#26, p. 146:

f[x_]:=3 x + 6 Cos[x];
Together[f'[x]]
(* Observe how I put the tangent line into the plot: in point-slope form: *)
xp=Pi/3;
width=1;
Simplify[f'[xp](x-xp)+f[xp]]
Plot[{f[x],f'[xp](x-xp)+f[xp]},{x,xp-width,xp+width}]
Plot[{f[x],f'[x]},{x,xp-width,xp+width}]

#35, p. 147:

f[x_]:=8Sin[x];
f'[x]
f''[x]
Plot[{f[x],f'[x],f''[x]},{x,0,Pi}]
f[2Pi/3]
f'[2Pi/3]
f''[2Pi/3]

#49 and 50, p. 147:

D[Sin[x],{x,99}]
D[x Sin[x],{x,35}]