Plot[E^x, {x, -3, 3}]

Series[E^x, {x, 0, 7}]

Series[Sin[x], {x, 0, 7}]

Series[Cos[x], {x, 0, 7}]

expseries = Normal[Series[E^x, {x, 2, 3}]]

?Global`*

Plot[{expseries, E^x}, {x, -3, 3}, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}]

sinseries = Normal[Series[Sin[x], {x, 0, 15}]]

Plot[{sinseries, Sin[x]}, {x, -7, 7}, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}]

t = 5

x[t] = -0.5*9.8*t^2 + 6*t

x[6]

x[5]

x[t]

?Global`*

Remove["Global`*"]

?Global`*

x[t_] = -0.5*9.8*t^2 + 6*t

x[0]

x[5]

Plot[x[t], {t, -5, 5}]

?Global`*

Remove["Global`*"]

x[t_] = Cos[2*Pi*t]

y[t_] = Sin[2*Pi*t]

Series[x[t], {t, 0, 3}]

Series[y[t], {t, 0, 3}]

vx[t_] = Dt[x[t], t]

vy[t_] = Dt[y[t], t]

Plot[vx[t], {t, 0, 0.25}]

Speed[t_] = Simplify[Sqrt[vx[t]^2 + vy[t]^2]]

Plot[Speed[t], {t, 0, 1}]