Shortest distance between two general curves Have to find shortest distance between
$$f(x)=-0.000024729x^7+0.0011461x^6-0.019857x^5+0.15402x^4-0.46314x^3-0,068261x^2+2.2506x-8.3436$$
and
$$g(x)=0.000016282x^8-0.00094728x^7+0.022526x^6-0.28122x^5+1.9699x^4-7.6833x^3+15.691x^2-14.557x+0.22000$$
I need all the help I can get! Thanks.

Write the Euclidean distance (squared) as $((x_1,f(x_1)) - (x_2, g(x_2)))^2$.  Now you have a straightforward optimization in two variables, where you take derivatives, set to zero, ...
Here's a graph of your two functions:

where I've added the minimizing points (and joining line)... read on...
Pick an arbitrary point $x_1$ on $f(x)$ and another point $x_2$ on $g(x)$.  Draw a line between them.  That's the distance between those two points.  Right?
This Euclidean distance is a function of the two variables.  Since you tagged your question with "calculus" you surely must know how to find the (local) minimum of a function of two variables.
Apply that to your distance function.  (Actually, do it for the square of the distance function... simpler.)
You'll get many local minima and maxima, but the graphs will guide you to finding the global minimum.  (Also, you can use second-order derivatives.)
Here's a "closeup" of the solution:

