MATH 228 Calculus III (FALL 2016)
FINAL EXAM
NAME :
NOTE : There are 8 problems on this final (total of 9 pages). Use of calculators will NOT be permitted.
In order to receive full credit for any problem, you must show work leading to your answer. You have 150
minutes to complete this test.
Problem Possible
points
1
20
2
20
3
20
4
20
5
30
6
30
7
30
8
30
Total
200
Score
(This page purposely left blank and may be used for scratch work.)
MATH 228 – FINAL
Problem 1. (20pts) Let A(2, 4, 5), B(1, 5, 7) and C(−1, 6, 8).
(a) Find the equation of the plane that contains all three points.
(b) Find the area of the triangle determined by these points.
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MATH 228 – FINAL
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Problem 2. (20pts) Find the absolute maxima and minima of the function T (x, y) = x2 + xy + y 2 − 6x + 2
on the rectangular plate 0 ≤ x ≤ 5, −3 ≤ y ≤ 0.
MATH 228 – FINAL
Problem 3. (20pts) Find the maximum value of s = xy + yz + xz where x + y + z = 6.
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MATH 228 – FINAL
Page 6
Problem 4. (20pts) Find the centroid of the region bounded by the line y = 1 and the parabola y = x2 .
MATH 228 – FINAL
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Problem 5. (30pts)
(a) Let f (x, y, z) = xy 2 + z 3 and F(x, y, z) = xzi + yk. Compute
∇f =
∇·F=
∇×F=
(b) Let
xi
yj
b
r= p
+p
x2 + y 2
x2 + y 2
xj
b = p −yi
and θ
+p
.
x2 + y 2
x2 + y 2
IDENTIFY the true statement(s) among the following.
(A) ∇ × b
r=0
(B) ∇ · b
r=0
b=0
(C) ∇ × θ
b=0
(D) ∇ · θ
(c) Let C is the circle of radius 5 centered at the origin oriented clockwise. Compute
˛
b · dr =
θ
C
b=0
(E) b
r·θ
MATH 228 – FINAL
Problem 6. (30pts) Let F = 3x2 i +
Page 8
z2
j + 2z ln y k.
y
(a) Given that ∇ × F = 0, can you conclude that F is conservative? Why or why not?
(b) Let C is the path from (1, 1, 1) to (1, 2, 3) formed by two straight segments that meet at the common
endpoint (1, 2, 1). Compute
´
C
F · dr =
MATH 228 – FINAL
Page 9
Problem 7. (30pts)
(a) Give an example of an irrotational vector field that is not conservative.
(b) Explain how you know this vector field is not conservative.
(c) IDENTIFY the true statement(s) among the following.
¨
˛
(i)
F · dσ =
(∇ × F) · dr
(ii)
S
∂S
‹
˚
(iii)
(∇ · F) dσ =
|F| dV
(iv)
∂E‹
˚E
(v)
F · dσ =
(∇ · F) dV
(vi)
∂E
E
‹
˚
(∇F) dσ =
¨
S
F dV
˛
∂E
E
(∇ · F) dσ =
F · dr
˛
¨∂S
F · dr =
(∇ × F) · dσ
∂S
S
MATH 228 – FINAL
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Problem 8. (30pts) Find the outward flux of F = x2 i + xzj + 3zk across the boundary of the sphere
x2 + y 2 + z 2 ≤ 4.