Math 213 Final Exam December 15, 2006
Read each problem carefully. Please show all your work for each problem! Use only those
methods discussed thus far in class. Always simplify when possible. No calculators!
1. (10 points) Find a parametric equation of the line of intersection of the planes x+y= 4
and 2x−y−z= 2.
2. (10 points) Find the velocity and the position vector of a particle whose acceleration
vector is a(t) = cos 2ti+ sin 2tj+kand the initial velocity and position are v(0) = i
and r(0) = j, respectively.
3. (10 points) Use differentials to estimate the maximum error in the calculated volume
of a rectangular box, if its dimensions are measured to be 10 cm, 20 cm, and 30
cm, respectively, each within an error of at most 0.1 cm. Write the answer as the
calculated volume ±the estimated error.
4. (10 points) Find and classify all the critical points of the function
f(x, y) = x2+ 3xy −y2+ 16x−2y+ 5.
5. (10 points) Use Lagrange Multipliers to find the minimum value of the function
f(x, y, z) = x2+y2+z2sub ject to the constraint x+y−z= 5.
6. (a) (10 points) Use the Fundamental Theorem of line integrals to compute
Z(3,2,1)
(1,2,3)
yz dx +xz dy +xy dz.
(b) (10 points) Verify the answer in part (a) by explicitly computing the integral
over a straight line segment connecting (1,2,3) with (3,2,1).
7. (10 points) Evaluate the surface integral
ZZS
y dS,
where Sis the surface z=x+y2, 0 ≤x≤1, 0 ≤y≤2.
8. (10 points) Use Stoke’s Theorem to evaluate
ZZS
curl F·dS,
where F(x, y, z) = x2ey zi+y2ezx j+z2exyk, and Sis the hemisphere x2+y2+z2= 4,
z≥0, oriented upward.
9. (10 points) Use the Divergence Theorem to calculate the flux of the vector field F
across the surface S, that is, RRSF·ndS, where F(x, y, z) = x2yi+xy2j+ 2xyz k,
Sis the surface of the tetrahedron bounded by the planes x= 0, y= 0, z= 0, and
x+ 2y+z= 2, and nis the outward normal to S.