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The final exam questions for math 222 - analytic geometry and calculus iii held in summer 2007. The exam covers various topics including parametric equations, lagrange multipliers, gradient vector field, critical points, green's theorem, and surface integrals.
Typology: Exams
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TO RECEIVE CREDIT YOU MUST SHOW YOUR WORK.
No. 1. (25 points) An object is moving in 3-space according to the parametric equations
x = t, y = t^2 , z = 3t,
where t is the time. Find, as functions of t,
a) position vector r =
b) velocity vector v =
c) acceleration vector a =
d) speed =
e) tangential component of acceleration aT =
f ) curvature =
g) normal component of acceleration aN =
No. 2. (15 points) Use method of Lagrange multipliers to find the largest value and the smallest value for f (x, y) = x^2 y subject to the constraint x^2 + 2y^2 = 6.
No. 4. (20 points) Let f (x, y) = 2x^3 + xy^2 + 5x^2 + y^2.
Find all the critical points for f (x, y) and then apply the second partials test to each critical point to determine its nature.
No. 5. (15 points) Use Green’s Theorem to evaluate the line integral
∫
C
ex^ + x^2 y
dx +
ey^ − xy^2
dy
where C is the circle x^2 + y^2 = 25 oriented clockwise.
No. 7. (20 points) Find the work done by the force field
F(x, y) = sin xi + cos yj
on a particle that moves along the top half of the circle x^2 + y^2 = 1 from (1, 0) to (− 1 , 0) and then the line segment from (− 1 , 0) to (− 2 , 3).
No. 8. (15 points) Show that the force field
F = (2xz + y^2 )i + 2xyj + (x^2 + 3z^2 )k
is conservative by finding a potential function for it. Now use this potential function to calculate the work done by F as it acts on an object which moves along a curve given by
C : x = t^2 , y = t + 1, z = 2t − 1 , 0 ≤ t ≤ 1.
No. 10. (15 points) Let F = 3xy^2 i + xez^ j + z^3 k. Use the divergence theorem to calculate the outward flux of F across the surface of the solid bounded by the cylinder y^2 + z^2 = 1 and the planes x = −1 and x = 2.
No. 11. (20 points) Let C be the boundary of the part of the plane 2x + y + 2z = 2 in the first octant, oriented counterclockwise as viewed from above. Let
F = e−xi + exj + ez^ k.
a). curl F =
b). Use Stokes’ theorem to calculate
C F^ ·^ dr.
