greenhart

We've all run across those 'fundamental' theorems.

Fundamental theorem of arithmetic: That every natural number has unique prime factorization.

Fundamental theorem of algebra: That every polynomial can be factored into linear factors over the Complex numbers.

Fundamental Theorem of Calculus: That differentiation and integration are almost inverses.

Fundamental Theorem of finitely generated abelian groups: Complete classification up to isomorphism of all these types of groups.

What I am interested in is what 'fundamental' means. What significance do these theorems have in contrast to other theorems?

As a scholarship job position at my college, I am going to help my professor to grade Calculus Papers.

I've never done this before, but from anyone that has any previous experience:

Do you have any suggestions/ hints/ pointers?

My concern is the balance of grading fairly and in a way that will help the students learn?

Does anyone know what to expect?

Thank you!

Suppose you had a non-polynomial function:

For example,

f(x) = -1-x+3sin(x)

but you are not necessarily interested in the exact value of the roots,

but you want to know how many there are;

Is there a general method for doing so?

The purpose of this would be to guess how many roots there are before using Newton's method to approximate their values. (Without using a calculator)