Evaluate the line integral

F · dr,

where C is given by the vector function r(t).

F(x, y, z) = sin(x) i + cos(y) j + xz k

r(t) = t⁵ i − t⁴ j + t k, 0 ≤ t ≤ 1

find the general solution to the second order linear non-homogeneous differential equation
(y"/2)-y' +y = cos x

Given: f(x) = x 2−36 x 2−7x+6 Find the following

a. V.A.

b. Domain

c. H.A.

d. X-intercept

e. Y-intercept

f. Graph

Find the tangential and normal components of the acceleration vector. ? (?) = ? ? ?̂+ √2 ? ?̂+ ? −??̂

-Let O (0,0,0) be the origin and A(a+1,b,c) where a,b and c are the last three digits of your student ID. Find, describe and sketch the set of points P such that OP is perpendicular to AP

- Let u=<a , b+1 , c+1> find and describe all vectors v such that |u x v|=|u|

a=1 , b=1 , c=9

For f(x)=x^2+x-2/x^2-4, determine the equation for any vertical asymptotes, the equation for any horizontal asymptotes, and the x-coordinates of any holes

csc beta = 2/3. What is the beta to the nearest degree?

Determine an example of a vector field that would yield a positive value for a line integral around a circle that is traversed once clockwise for any nonzero radius and explain how you know your vector field is correct.

Express the following systems of equations in matrix form as A x = b. What is the rank of A in each case (hint: You can find the rank of a matrix A by using rank = qr(A)$rank in R. For each problem, find the solution set for x using R.

a. X1+5X2+2X3=5

4X1-X2+3X3=-8

6X1-2X2+X3=0

b. 3X1-5X2+6X3+X4=7

4X1+2X3-3X4=5

X2-3X3+7X4=0

X2+3X4=5

. Let Π be a finite incidence geometry. Prove that, if every line in Π has exactly n points and every point in Π lies on exactly n + 1 lines, then Π is an affine plane. Come up with a similar criterion for finite geometries satisfying (EP) (those geometries are called projective planes).

why are the domain and range of square root function restricted to [0,∞) ?