MACM 101 Final: MACM101_Final-Review-Grimaldi-Notes

Evaluatethe limit by first recognizing the sum as a Riemann sum for afunction defined on [0,9].

lim9/n(sqrt(9/n)+sqrt(18/n)+sqrt(27/n)+...+(sqrt(9n/n))

n-->infinity

so far I have ...

lim n--> infinity (9-0)/n n { i=1 sqrt(9i/n) = 9/0 sqrt(x)dx

then I am lost...

Let F be a field, and p(x), q(x), r(x) be different irreducible polynomials of F[x] . How many maximal and prime ideals are there for factor ring F[x]/<p(x)q(x)> and F[x]/<p(x)q(x)r(x)>? In addition, for positive integer m,n, how many maximal and prime ideals are there for factor ring F[x]/<(p(x))^(m)q(x)^(n)>?

For the following vectors u andv, express u as the sum u= p +n where p is parallel to v and n isothogonal to v.

u = <4,3,0>

v = <1,1,1>