Math 242
December 21, 2023
Final Exam Review
The exam is closed textbook. No electronic devices are allowed during the exam. The exam will
contain a number of problems, of types similar to the problems given in this review sheet. Please prepare
for each problem type. The test will also contain one or two problems of different type (but based on
topics that we covered).
You are allowed to bring a 2-sided formula sheet, handwritten, standard size.
1. Determine whether the given lines are parallel, skew, or intersecting.
(a) The first line has parametric equations
x = 3 + t; y = 2 − t; z = 7 − 2t
and the second line has vector equation
r = ⟨2, 4, 4⟩ + t⟨−3, 4, 1⟩.
(b) The first line has symmetric equation
x−2=
3 − 2y
z+1
=
,
4
3
and the second line has parametric equations
x = 10s; y = −4 − s; z = 5 + 2s.
2. If |u| = 2, |v| = 3, and u · v = −1 calculate
answer:
√
√
answer:
15
85
√
(d) |u × v|
answer:
11
(c) |2u + 3v|
√
(b) |u − v|
answer:
(a) |u + v|
35
3. Sketch at least five level curves for the following functions
y
x2
(b) g(x, y) = x2 − y
y
(c) h(x, y) = e x
(d) p(x, y) = x2
(e) s(x, y) =
answer: 0
2(x2 −y 2 )
(x2 +y 2 )2
uxyz .
answer: gxx = − (x2 +y2 )2 , gyy =
x
(x−y)2
(b) g(x, y) = ln(x2 + y 2 ), gxx , gyy .
z + z −1
(c) u(x, y, z) = sin(xy) + arctan
,
x − x−1
y
answer: fx = − (x−y)
2 , fy =
4. Find the indicated partial derivatives
x
(a) f (x, y) =
, fx , fy .
x−y
10
1 + x2 + y 2
2(x2 −y 2 )
(a) f (x, y) =
5. Use the Second Derivative Test to find the local minima and maxima as well as the saddle points of
the functions
√
√
er: Local minimum at (0, 2/ 3), local maximum at (0, −2/ 3), saddle points at (−1, −1) and (1, −1)
(b) f (x, y) = x2 +x2 y+y 3 −4y
answer: Local minimum at (1, 1), saddle point at (0, 0)
(a) f (x, y) = 1 + x3 + y 3 − 3xy
Math 242
December 21, 2023
Final Exam Review
6. Calculate the double integral
ZZ
(a)
x dA,
D
answer:
where D is the region bounded by y = x2 and y = x + 2.
ZZ
xy dA,
(b)
9
4
D
where D is the disk x2 + y 2 ≤ 4.
answer: 0
7. Switch the order of integration, then calculate the resulting integral
Z9 Z3
(a)
xey dx dy
sin x
dx dy
x
answer: 1 − cos 1
(b)
0
− 9e)
Z1 Z1
1 9
(e
2
√
y
answer:
1
y
8. (a) Let E be the solid that is bounded by the paraboloids z = 4 − x2 − y 2 and z = x2 + y 2 − 4.
Evaluate the triple integral
ZZZ
(x2 + y 2 ) dV.
E
answer: 64π/3
(b) Let E be the part of the solid sphere of radius 4 with y ≥ x. Evaluate the triple integral
ZZZ
y dV.
E
√
answer: 32 2π
(c) Let E be the solid that is bounded by the planes y = x, y = −x, y = 2, z = 0, and z = 3.
Evaluate the triple integral
ZZZ
2yz dV.
E
answer: 48
9. (a) Let C be the path that starts at (1, 2) and moves in a straight line to (3, 5).
i. Find a parameterization for C, including bounds for t. Z
F · dr, where F = ⟨y − 2, x + y⟩.
ii. Use that parameterization to evaluate the line integral
C
answer: 39/2
(b) Let C be the path that starts at (3, 0) and moves along the ellipse
x2
y2
+
= 1 to the point
9
4
(0, 2).
i. Find a parameterization for C, including bounds for t.
Z
ii. Use that parameterization to evaluate the line integral F · dr, where F = ⟨xy, y⟩.
answer: x = 3 cos θ, y = 2 sin θ,0 ≤ θ ≤ π/2
answer: −4
C
Math 242
December 21, 2023
Final Exam Review
10. Let F = ⟨ey , xey + sin z + 3y 2 , y cos z⟩.
(a) Show that F is conservative by calculating curl F.
(b) Find a function f such that F = ∇f .
(c) Using the Fundamental Theorem of Line Integrals, calculate
0 ≤ t ≤ 1.
R
C
F·dr, where r(t) = ⟨3t, t2 −t, t3 ⟩,
answer: 3
y
+ 2z⟩.
z
(a) Show that F is conservative by calculating curl F.
11. Let F = ⟨yz, xz + ln z, xy +
(b) Find a function f such that F = ∇f .
(c) Using the Fundamental Theorem of Line Integrals, calculate
Z
F · dr ,
C
where r(t) = ⟨t2 , 2t, (t − 1)2 ⟩, 0 ≤ t ≤ 2.
answer: 16
R √
12. (a) Use Green’s Theorem to calculate C ( x3 + 1 + ey ) dx + 2xey dy, where C is the path that
starts at (0, 0), goes right to (2, 0), then goes up to (2, 4), then in a straight line back to (0, 0).
answer: (e4 − 5)/2
R
(b) Use Green’s Theorem to calculate C y 2 x dx − (x2 y + y 2 ey ) dy, where C is the path that starts
at (2, 0) and goes once counter-clockwise around the circle x2 + y 2 = 4.
answer: 0
p
13. (a) Consider the part of the cone z = x2 + y 2 that lies beneath the plane z = 1. Find a parameterization for this surface, and sketch the domain for your parameters.
RR
(b) Using your parameterization from part (a), calculate the surface integral
S F · dS, where
4
F = ⟨x, y, z ⟩. Assume that S is oriented downward.
answer: π/3
14. Consider the cylinder S
x2 + y 2 = 9 ,
0 ≤ z ≤ 5,
(a) Find a parameterization for this surface, and sketch the domain for your parameters.
RR
(b) Using your parameterization from part (a), calculate the surface integral
S F · dS, where
F = xi + yj + ez k. Assume outward orientation.
answer: 90π
15. Let E be the solid that is bounded by the paraboloids z = 4 − x2 − y 2 and z = x2 + y 2 − 4. Use the
x3
Divergence Theorem to find the flux of F = ⟨xz + , −yz, y 2 z⟩ through the boundary of E, oriented
3
outward.
answer: 64π/3
16. Let E be the solid that is bounded by the planes y = x, y = −x, y = 2, z = 0, and z = 3. Use
the Divergence Theorem to find the flux of F = ⟨y 2 + z 2 , y 2 z, x2 + y 2 ⟩ through the boundary of E,
oriented outward.
answer: 48