2.) (12 pts.) Show that F = ( xy/(1+x^2y^2) + 1 + arctan(xy))i+
(x^2/(1+x^2y^2-1)j is a...
2.) (12 pts.) Show that F = ( xy/(1+x^2y^2) + 1 + arctan(xy))i+
(x^2/(1+x^2y^2-1)j is a conservative vector field. Then use the
Fundamental Theorem for Line Integrals to find the Work done by F
from point (0,0) to point (2, 1/2).
Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 14z) k
C is the line segment from (3, 0, −1) to (6, 4, 2)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.
Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 14z) k
C is the line segment from (3, 0, −3) to (5, 5, 1)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.
The Vector Field f(x, y) = (2x + 2y^2)i + (4xy - 6y^2)j has
exactly one potential function f (x, y) that satisfies f(0, 0).
Find this potential function , then find the value of this
potential function at the point (1, 1).
Find the directional derivative of f(x,y)=arctan(xy) at the point (-2,5) in the direction of maximum decrease. What is the Domain and Range of f(x,y)=arctan(xy)?
5. Consider the differential equation
xy^5/2 +1+x^2y^3/2dy/dx =0
(a) Show that this differential equation is not exact.
(b) Find a value for the constant a such that, when you multiply
the d.e. through by xa, it becomes exact. Show your working. Do NOT
solve the resulting differential equation.
6. Consider the differential equation
(D − 3)(D − 4)y = 0.
(a) Solve this d.e., showing brief working.
(b) How many solutions does this d.e. have? Justify your
answer.
(c) How...
Let f(x, y) = xy3 − x 2 + 2y − 1. (a) Find the gradient vector
of f(x, y) at the point (2, 1).
(b) Find the directional derivative of f(x, y) at the point (2,
1) in the direction of ~u = 1 √ 10 (3i + j).
(c) Find the directional derivative of f(x, y) at point (2, 1)
in the direction of ~v = 3i + 2j.
a.) Show that the DE is exact and find a general solution
2y - y^2sec^2(xy^2)+[2x-2xysec^2(xy^2)]y' = 0
b.) Verify that the equation is not exact. Multiply by
integrating factor u(x, y) = x and show that resulting equation is
exact, then find a general solution.
(3xy+y^2) + (x^2 + xy)dy/dx = 0
c.) Verify that the equation is not exact. Multiply by
integrating factor u(x, y) = xy and show that resulting equation is
exact, then find a general solution....
. Consider a Bessel ODE of the form:
x^2y"+xy'+(x^2-1)y=0
What is the general solution to this ODE assuming that the
domain of the problem includes x=0? Why?
Consider the function f(x, y) = 3+xy−x−2y. Let D be the closed
triangular region with vertices (1, 4), (5, 0), and (1, 0). Find
the absolute maximum and the absolute minimum of f on D.