MATH261 FINAL EXAM FALL 2014
NAME:
CSU ID:
Problem Points
1
20
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit. Recall:
cos2 (a) =
1 + cos(2a)
2
and
sin2 (a) =
1 − cos(2a)
2
GOOD LUCK !!!
2
18
3
20
4
24
5
26
6
26
7
18
8
26
9
22
Total
200
Score
1. (20 pts) (Reminder: Two equations intersect if there is at least one common point.)
Consider the planes
P1 :
2x − y + 2z = 3
P2 :
x+y+z =3
P3 :
4x − 2y + 4z = 6
P4 :
2x − y + 2z = 4
and the lines
r1 (t) = h3t, −3 + 2t, −2ti
L1 :
L2 :
r2 (s) = h1 + 3s, −3 + 2s, −2si
For each problem below, circle either True or False.
(a) True or False: P1 intersects P2
(b) True or False: P1 intersects P3
(c) True or False: P1 intersects P4
(d) True or False: L1 intersects P1
(d) True or False: L2 intersects P1
2. (18 pts) Given a(t) = (2t − 1)i −
1
π
sin(πt)j + et−1 k complete the following.
(a) (10 pts) Find v(t) with condition v(1) = (−1/3)i − j + 2k
(b) (8 pts) Find r(t) with condition r(1) = h0, 0, 0i.
3. (20 pts) Given a(0) = h0, 1, 1i and v(0) = h3, −4, 5i find the following.
(a) (8 pts) T(0) and aT (0).
(b) (8 pts) |a(0)| and aN (0).
(c) (4 pts) N(0). Do Not Simplify
4. (24 pts) Consider the function f (x, y) = xy 2 − x2 y + x2 − xy. Classify each of the
critical points given below as a local max, local min, saddle point or a point where the
second derivative test fails. Fill in the table.
Critical Point Max Min Saddle Fails
(0,0)
(0,1)
(1/3,2/3)
(1,1)
5. (a) (16 pts) Find the linearization L(x, y) of f (x, y) = (x−1)2 −(x−1)3 (y+2)+3(y+2)
at the point (1, −2). The final form of the linearization must be L(x, y) = A +
Bx + Cy.
(b) (10 pts) Find an upper bound on the magnitude of the error in the approximation
f (x, y) ≈ L(x, y) for the rectangular prism |x − 1| ≤ 0.1 and |y + 2| ≤ 0.2.
6. (26 pts) Consider a mass of constant density δ = 1 over the region bounded by the
x-axis, y = 3, x = y 2 + 1 and x = −y 2 − 1.
(a) (4 pts) Graph the region.
(b) (6 pts) Find the mass M .
(c) (6 pts) Find Mx .
(d) (6 pts) Find My .
(e) (4 pts) Find the center of mass.
7. (18 pts) Rewrite the integral for the function f in spherical coordinates.
Z π Z 3 Z √9−r2
0
0
f dz rdr dθ =
0
8. (26 pts) Use Stokes’ Theorem to find the work done by F = h−y 2 , z, xi progressing
around the triangle C with vertices (0,0,0), (2,0,0) and (0,2,2), oriented counterclockwise as viewed from above. Steps to be performed are:
(a) (10 pts) Define the plane in parametrized form using the variables x and z.
(b) (8 pts) Graph the projection of the plane onto the xz plane.
(c) (8 pts) Evaluate the appropriate integral.
9. (22 pts) Let the surface S be the boundary of the volume enclosed by z = 0, x2 +y 2 = 4,
and y + z = 5. Let F = xzi + yzj + xyk.
(a) (10 pts) Graph the volume.
(b) (12 pts) Using the Divergence Theorem, set up
S F · n dσ (where n is the
outward-pointing surface normal) as a volume integral. Do Not Evaluate.
RR