Math 227-103 (CRN 10968)
Carter
Fall 2015
Final Exam Study Model
General Instructions. Write your name on only the outside of your blue books. Do not
write on this test sheet, do all of your work inside your blue books. Write neat complete
solutions to each of the problems in the blue book. Please put your test sheet inside the blue
book as you leave.
1. (20 points) Be able to determine the optimal values and the points at
which the optima occur for a function of the form
f (x, y) = ±(x − 1)2 ± (y − 2)2 + C
over a triangular region such as that bounded by y = 0, x = 0 and
x + y = 4. Observe that I may, or may not, tell you a set of values of f
at particular points.
2. (20 points) Use Lagrange multipliers to find the maximum and minimum
values of the function f (x, y) = xy, subject to the constraint 4x2 + 9y 2 =
36.
3. (20 points) Evaluate the integral
Z (2,2,2)
y
x
1
1
dx +
− 2 dy − 2 dz.
z y
z
(1,1,1) y
4. (20 points) Use Green’s Theorem to evaluate
I
y 2 dx + x2 dy
C
where C is the unit square {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
1
5. Three more problems at about 20 points each selected from:
• §15.2 p. 844-845 #2,4,21,27
• §15.3 p. 856-857 # 9,13,17,26
• §15.4 p. 867-869 # 1,7,21,25
• §15.5 p. 878-879 # 23,31,42
• §15.7 p. 898-900 # 2,4,13
• §15.8 p. 909-910 # 3,17,18,25
6. (20 points) Compute the line integral
Z
1
(y dx − x dy)
2 ~γ
when ~γ is the straight line segment that starts at (a, b) and ends at (c, d).
7. Consider the vector field
x~ı + y~ + z~k
~
F =
ρ3
p
where ρ = x2 + y 2 + z 2 . Compare this problem to Example 5,
§15.8, p. 905-906 and the subsequent discussion on p. 906.
(a) (5 points) Compute
∂ρ
∂x .
(b) (5 points) Use symmetry to give the values for
∂ρ
∂y
and
∂ρ
∂z .
∂
(c) (5 points) Use the results above to compute ∂x
(xρ−3 ), and again
∂
∂
symmetrically compute ∂y
(yρ−3 ) and ∂z
(zρ−3 ).
(d) (5 points) Use the divergence theorem to compute
Z Z
F~ · ~n dσ
F
2
where F is the boundary of the domain D = {(x, y, z) : a2 ≤ x2 +
y 2 + z 2 ≤ b2 } ~n is the outward pointing normal, and dσ denotes the
surface area form.
(e) (5 points) Now compute the outward flux of F~ across the surface of
a ball of radius a that is centered at the origin. Specifically, compute
Z Z
F~ · ~n dσ
S
where
S = {(x, y, z) : x2 + y 2 + z 2 = a2 }
whose outward pointing unit normal is ~n =
3
x~ı+y~+z~k
.
a