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Math 222 - Analytic Geometry and Calculus III Final Exam Summer 2007, Exams of Analytical Geometry and Calculus

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

2010/2011

Uploaded on 05/31/2011

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ANALYTIC GEOMETRY AND CALCULUS III - MATH 222
SUMMER 2007 - FINAL EXAM
Name : ..........................................
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 =t2, z = 3t,
where tis 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=
pf3
pf4
pf5
pf8
pf9
pfa

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ANALYTIC GEOMETRY AND CALCULUS III - MATH 222

SUMMER 2007 - FINAL EXAM

Name : ..........................................

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.