A  worksheet illustrating some of Mathematica's ability to manipulate functions.

Mathematica is capable of performing many of the standard operations of calculus.

g[x_] = x^2*Sin[x]

For instance, it can differentiate and integrate simple functions.

D[g[x], x]

Integrate[g[x], x]

Given a polynomial, Mathematica can also factor it or pick off coefficients of various terms.

p[x_] = 3 - 4 x + 4 (x^2) - 4 (x^3) + (x^4)

Factor[p[x]]

Coefficient[p[x], x^2]

It is also very useful for finding Taylor series approximations to functions.

Series[g[x], {x, 0, 8}]

We can expand around points other than zero, of course.

Series[g[x], {x, Pi, 8}]

If you don't want the coefficients in the expansion to be expressed as combinations of Pi, you can express them as decimals with the "N" (for numerical) command.

N[Series[g[x], {x, Pi, 8}]]