MTH 222 Final

December 19, 2011

Name:

This test consists of 10 problems on 11 pages (including this cover sheet).The exam is worth 200 points. Do not separate the pages of this exam. Ifany pages do become detached, write your name on them and point themout to me. Please read the instructions for each individual exercise carefully.Show an appropriate amount of work for each exercise so that Ican see not only the answer but also how you obtained it.

This is a closed book exam.

Relax.

Problem Points Score

1 15

2 15

3 20

4 20

5 25

6 20

7 20

8 25

9 20

10 20

Total 200

Problem 1. Find an equation for the plane through the point (2,−3, 5) withnormal vector 3 i+ 2 j+ k.

Problem 2. Find an equation for the surface obtained by rotating the parabolay = x2 about the y-axis.

Problem 3. Find the curvature of the twisted cubic r(t) = 〈t, t2, t3〉 at ageneral point.

Problem 4. Find the local maxima, local minima and saddle points of thefunction

f(x, y) = 2x3 − y3 + 3x2 − 3y2.

Problem 5. Use the change of variables x = u2 − v2, y = 2uv to evaluatethe integral

∫∫Ry dA, where R is the region bounded by the x-axis and the

parabolas y2 = 4− 4x and y2 = 4 + 4x, y ≥ 0.

Problem 6. Evaluate∫Cy sin z ds, where C is the circular helix given by the

equations x = cos t, y = sin t, z =√3 t,0 ≤ t ≤ 2π.

Problem 7. Let C be the curve r(t) = cos t i+ 3 sin t j+ tk, 0 ≤ t ≤ π6. Let

F(x, y, z) = 2xy3z5 i+ 3x2y2z5 j+ 5x2y3z4 k.

Use the Fundamental Theorem of Line Integrals to evaluate∫C

F · dr.

Problem 8. Let F(x, y, z) = xyz i+xy2z3 j+x3y5z7 k. Compute curlF anddivF.

Problem 9. Evaluate∫∫

Sy dS, where S is the surface z = x+y2, 0 ≤ x ≤ 1,

0 ≤ y ≤ 2.

Problem 10. Let C be the curve r(t) = 3 sin t i+ 3 cos t j, 0 ≤ t ≤ 2π. Let

F(x, y, z) = yez i+ x(ez + 2) j+ exyz k.

Use Stokes’ Theorem to evaluate ∫C

F · dr.

Problem 11. Let E be the solid region

E = {(x, y, z) | x2 + 4y2 + 9z2 ≤ 36}

and let S be the boundary surface of E with outward orientation. Let

F(x, y, z) = (x+ sin(yez2

))i+ (y + e−z cosx2)j+ (z + exy)k.

Use The Divegence Theorem to evaluate∫∫S

F · dS.