q[t_] = {Cos[t], Sin[t]}; r = 1;
Manipulate[ParametricPlot[{q[x]}, {x,0, 2 Pi}, PlotRange -> {{1-t, 1+t},{-t,t}}, AspectRatio -> Automatic, PlotStyle -> {{Red, Thick}, {Green, Thick}, {Black}}], {{t,2.5}, .000001, 2.5}]

f[x_] = (1/300)(x^3 -3 x^2 + 1)Sin[2x / Pi] +4;
Manipulate[
Plot[{f[x+6], m( x - x0) + y0}, {x, 0, 12}, 
PlotRange -> {0, 12},
AspectRatio -> Automatic,
Ticks -> {Range[-10,10], Range[-10,10]},
PlotStyle -> {{Red, Thick}, {Blue, Thick}}], {{m,1/2}, -10, 10}, {{x0,0}, -10, 10},{{y0,0}, -10, 10}]

g[x_] = (1/6)(x^4); r = 1;
Manipulate[Plot[{ g[r] + g'[r](x-r), g[r] + (g'[r] + .05)(x-r), g[x]}, {x, r-t, r+t}, PlotRange -> {g[r] - t, g[r] + t}, AspectRatio -> Automatic, PlotStyle -> {{Darker[Green], Thick}, {Blue, Thick}, {Red, Thick}}], {{t,3}, .000001, 3}]

h[x_] = (x-1)^2- 3/2;
Manipulate[Plot[{h[x], h[t] + h'[t](x - t)}, {x, -4, 4}, PlotRange -> {-2, 6}, AspectRatio -> Automatic, PlotStyle -> {{Blue, Thick}, {Red, Thick}}], {{t,-2}, -3, 3}]

g[x_] = Abs[x]; r = 0;
Manipulate[Plot[{g[x]}, {x, r-t, r+t}, PlotRange -> {g[r] - t, g[r] + t}, AspectRatio -> Automatic, PlotStyle -> { {Blue, Thick}, {Red, Thick}}], {{t,3}, .0000001, 3}]

k[x_] = (1/8) (x+.5)^6 + .5; z = .75;
Manipulate[Plot[{k[x], k[z] + k'[z](x-z), k[z] + (k'[z]+h)(x-z)}, {x, -.5, 1.5}, PlotRange -> {-1, 6}, Ticks -> {{},{}},
PlotStyle -> {{Black, Thick}, {Red, Thick}, {Darker[Green], Thick}}], {{h,5}, 0, 5}]