MATH261 FINAL EXAM FALL 2013
NAME:
CSU ID:
Problem Points
1
26
2
30
3
20
4
24
5
20
6
30
7
20
8
30
Total
200
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. Suppose that r1 (t) and r2 (t) are parameterizations of two curves in R3 such that
r1 (2) = r2 (2) = h−2, 4, 5i.
In the following, v1 (2) is the velocity vector of the curve given by r1 at time t = 2,
v2 (2) is the velocity vector of the curve given by r2 at time t = 2, etc.
(a) (8pts) Given v1 (2) = h3, 1, 1i and v2 (2) = h4, 2, 1i, find a vector normal to the
plane “containing” v1 (2) and v2 (2). Then, write the equation of this plane in the
form ax + by + cz = d.
(b) (8pts) Find |v1 (2)| and T1 (2).
(c) (10pts) Given a1 (2) = h1, −1, 2i, find aT1 (2), aN1 (2), and N1 (2).
2. Consider the curve C given by r(t) = 21 t2 i+4t−1 j+( 12 t−t2 )k, and the surface S defined
by x2 − 4y 2 − 4z = 0.
(a) (5pts) S is a quadric surface. What type of quadric surface is it? (Identify the
type by name.)
(b) (7pts) Verify that the point (2, 2, −3) lies both on the surface S and on the curve
C.
(c) (9pts) Find a tangent vector (it need not be a unit vector) to the curve C at the
point (2, 2, −3). (Keep in mind the point.)
(d) (9pts) Find a normal vector (it need not be a unit vector) to the surface S at the
point (2, 2, −3). (Keep in mind the point.)
√
3. (10pts each part) Consider the function f (x, y) = (x + y) x − y at x = 6, y = 2.
(a) Find the linearization of f at the point (6, 2).
(b) If x = 6 + .25 and y = 2 − .5, use the linearization of part (a) to estimate
f (6.25, 1.5), simplifying your answer.
4. (8pts each part) Given that a minimum exists, we can use the Method of Lagrange
Multipliers to find the minimum value of g(x, y) = x2 + y 2 on the surface xy 3 = 1, and
we can identify the point(s) at which this minimum value occurs.
(a) Set up the system of equations one must solve in order to use this Method to find
the points at which the minimum occurs.
(b) Verify that two solutions to the system you wrote down in (a) are (3−3/8 , 31/8 ),
(−3−3/8 , −31/8 ).
(c) If you know that the two solutions in (b) are the only two solutions to the system
in (a), what is the minimum value of g on the surface?
5. (3pts each part, plus 5pts for sketch.) Consider the triple iterated integral of f (x, y, z)
Z 1 Z 1 Z 1−x
−1
z2
0
f (x, y, z)dydxdz.
Sketch the solid defined by the bounds, then, after each of the following integrals, write
down the correct order of integration from the following list: dxdydz, dxdzdy, dzdxdy,
dydzdx, dzdydx.
(a)
Z 1 Z 1−y Z √x
0
(b)
√
− x
Z 1 Z √x Z 1−x
0
(c)
0
√
− x
f (x, y, z)
f (x, y, z)
0
Z 1 Z √1−y Z 1−y
√
− 1−y
0
z2
f (x, y, z)
(d)
Z 1 Z 1−z 2 Z 1−y
−1
(e)
0
z2
Z 1 Z 1−x Z √x
0
0
√
− x
f (x, y, z)
f (x, y, z)
6. (10pts each part) Consider the curve C, which consists of the two pieces defined by
r1 (t) = ti + tj, 0 ≤ t ≤ 4, and r2 (t) = 14 (4 − t)2 i + (4 − t)j, 0 ≤ t ≤ 4, traversed
counterclockwise.
(a) Sketch the path, with arrows indicating the direction of travel. (It must be large
enough to easily see.) (Hint: Note that for the second curve, the x coordinate
always equals 41 of the square of the y coordinate.)
(b) Let F = 2yi + 2xyj. Using the appropriate Green’s Theorem, find the work done
by F, traveling around C counterclockwise.
(c) Set up the double integral that Green’s theorem says computes the flux outward
across C for the vector field F of part (b). Do Not Evaluate.
7. (10pts each part) Consider the solid D which is the part of the solid cylinder x2 +z 2 ≤ 1
above the xy-plane, with 0 ≤ y ≤ 2.
(a) Sketch the solid D. How many “pieces” does the boundary of D have?
(b) Let F be the vector field xi + zj. Use the Divergence Theorem to compute the
flux of F outward across the closed surface that forms the boundary of the solid
D.
8. (10pts each part) Let S be the bottom half of the sphere defined by by x2 +y 2 +(z−1)2 =
1 for z ≤ 1, oriented using the outward unit normal n to S. Consider the force
F = z 2 i + 3xj − 2y 3 k.
(a) Parametrize the surface. It is suggested that the appropriate variables would be
r, θ.
(b) Calculate ∇ × F
(c) Set up the surface integral
RR
S
∇ × F · ndσ. Do Not Evaluate.