Any smooth well-behaved scalar function of one variable f(x) can be approximated in the vicinity of x0 by:

f(x) = f(x0) + f'(x0)(x-x0) + (1/2) f"(x0) (x-x0)^2 + ...

A similar formula works for vector functions of many variables F(X):

F(X) = F(X₀) + J(X₀)*(X-X₀) + ...

where J(X) is called the Jacobian matrix:

J_kl = dF_k / dX_l

J_kl is the k,l element of the Jacobian matrix, F_k is the kth element of the vector function F, and X_l is the lth element of the vector variable X.

Some Applications:

0. Numerical Derivatives  

given y(t), what is y'(t)?

y(t) y(t_a) + y'(t_a) (t-t_a) so

y'(t_a) [y(t)-y(t_a)]/(t-t_a)

1. Root-Finding

2. Numerical Integration

3. Solving ODE IVP's

4. Extrapolating to the Exact Result