MULTIVARAIBLE CALCULUS 1. Compute the work required to move a particle through t
ID: 2882290 • Letter: M
Question
MULTIVARAIBLE CALCULUS
1. Compute the work required to move a particle through the force field F = hx, y, 1i from (1, 0, 0) to (1, 2, 2) along the curve K parametrized by r(t) = hcos(t), tsin(0.25t), ti.
2. Let G = y 2 , 2xy, 2z and C be the circle in the xy-plane centered at(0, 0, 0) with radius 3.
(a) Compute curl (G):
b) Compute the work required to move a particle through G along one complete revolution of C
3. Let f(x, y) = 3xy + 1 x ln(y).
(a) Find the critical points of f(x, y) in the first quadrant and classify each critical point as a local maximum, a local minimum or a saddle point.
(b) Find the direction in which f(x, y) is increasing most rapidly at (1, e).
(c) Compute the directional derivative of f(x, y) at (1, e) in the direction h6, 5i.
(d) Find an equation of the plane tangent to z = f(x, y) at (1, e, 3e)
4. Let F = harctan(y), ln(1 + xz), xz + yi and B be the solid in the first octant bounded on the sides by the planes x = 1 and y = 3x and above by the plane z = 2x + 3y + 1.
(a) Compute div (F):
(b) Compute the flux of F across the (closed) boundary of B.
(c) Compute the surface area of the top face of B.
5. Evaluate: Z 4 0 Z 2 y cos(1 + x 3 ) dx dy
6. Evaluate I T 5y dx + 3x dy if triangle T has vertices (0, 0), (1, 0) and (1, 1) and clockwise orientation
1. Compute the work required to move a particle through the force field F (r, -y, -1) from (1,0,0 2. Let G (y2,2ry, 22) and C be the circle in the ary-plane centered at (0,0,0) with radius 3. (a) Compute curl (G) (b) Compute the work required to move a particle through G along one complete revolution of C. 3. Let f(ac, y) -3.ry ln(y) (a) Find the critical points of f(z,y) in the first quadrant and classify each critical point as a loca maximum, a local minimum or a saddle point (b) Find the direction in which f(r,y) is increasing most rapidly at (1,e). (c) Compute the directional derivative of f(z,y) at (1, e) in the direction (6,-5) (d) Find an equation of the plane tangent to 2 f(r, y) at (1,e, 3e). 4. Let F (arctan(y),ln(1 az), az y) and B be the solid in the first octant bounded on the side by the planes z 1 and y 33r and above by the plane z (a) Compute div (F) (b) Compute the flux of F across the (closed) boundary of B. (c) Compute the surface area of the top face of B. cos(1 ara)dr dy 5. Evaluate: 6. Evaluate 5y dar 3r dy if triana T has vertices (0,0), (1, 0) and (1,1) and clockwise orientation. gleExplanation / Answer
1) given F=<x,-y ,-1>
r(t)=<cos(t),tsin(0.25t),t> ,0<=t<=2
r'(t)=<-sin(t),sin(0.25t) +0.25tcos(0.25t),1>
F(r(t))=<cos(t),-tsin(0.25t) ,-1>
work done =[0 to 2]F(r(t)).r'(t) dt
work done =[0 to 2]<cos(t),-tsin(0.25t) ,-1>.<-sin(t),sin(0.25t) +0.25tcos(0.25t),1> dt
work done =[0 to 2][(-cos(t)sin(t))+(-tsin(0.25t)(sin(0.25t) +0.25tcos(0.25t))) -1] dt
work done =[0 to 2][-(1/2)(cos(t))2- (1/2)(tsin(0.25t))2 -t]
work done =[-(1/2)(cos(2))2- (1/2)(2sin(0.25*2))2 -2] -[-(1/2)(cos0)2- (1/2)(0*sin(0))2 -0]
work done =[-(1/2)- (1/2)(2)2 -2] -[-(1/2)- 0 -0]
work done =-22 -2
work done =-2(+1)
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