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Math 20C – Calculus III Final Exam Review – UCSD Supplemental Instruction Practice Questions with Full Solutions

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This document is a supplemental final exam review for Math 20C (Calculus III for Science and Engineering). It includes six fully solved practice problems covering core multivariable calculus topics: finding the area of a triangle formed by coordinate plane intersections, determining parallel planes through given points, evaluating double integrals using multiple methods, optimization with constraints and global extrema, partial derivatives with parametrization, and computing arc length of a vector-valued curve. The provided solutions show step-by-step methods to reinforce understanding and prepare effectively for exams. Topics in this document are not guaranteed to be on the final exam, but have a far higher chance to show up on exam day.

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math 200 final review


I the 6x 47 12 intersects X-axis , y-axis,
the
plane
=
.
+
3y +
and z-axis in points P, Q ,
and R respectively find the area of the
ofPAR
triangle


1 .




5) find the plane that is parallel to the plane 4x-5y + z =
2

which contains (4 , 0 ,
-1) .



2 evaluate
SSp(Fxy) &A R 20 , 2] x(0 , 13
=
.




sevaluate So Y
dx
dy

4let Xty2
f(x ,y) =

,
find maxtmin values w/ the
constraint 2x2+ ya = 2

b)x find global maxt min over the
region
D
G(x y)/2x2 y2124
=

, +




5 let .
f(x , y) In)1 + X + 2y)
=
·
find of (u , 1) /2 21
a =
,

where X =
1 + 1 12 + and
y
=
cos(20-12)




6 .
let Fit =< t3-1) 1-t32t> , ,
find
length
traced over the bounded
0xt = 1
curre
by

, 1 the 6x 47 12 intersects the X-axis , y-axis,
plane
=
.
+
3y +
and z-axis in points P, Q ,
and R respectively find the area of the
ofPAR PR
triangle A 11 BX ACI
=




6X =
12
T5 = -
AB 2
,
4 , 07
X= 2 -


AR (2 , 0 %
=
,
AC =
c
-
2 0
, ,
33


12
5122 + 62 + 82
3y =

11 AB X All
y 4 =

=H
BX (0 , 4 , 0
t22
=



=

z 3 =
-


203
CR 10 , 0 , 3)
=




i(12 0) j) - - -

b -


0) k(0 8)+ +




12i +
6j + 8k
< 12
,
6 , 87

find the plane that is parallel to the plane(X-5y +z =
2

which contains (4 , 0 ,
-1) . normal rector
- of the plane
= 5 , 1)
-
is coefficent


4(x 4) 5(y 0) + 1(z + 1)
- - - = 0

4X 16 - -




5y + 0 + z +1 = 0


4x
5y + z 15
-

=




.
2 evaluate
SSp(#xy) &A R 20 , 2] x(0 , 13
=




Solo (ixydy) dy u =
1 + xy

du = dy


Sils
*
+
2
-
ul
Sol-x
18 x -

In (1 + x)
2- 1n3

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August 18, 2025
Number of pages
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Written in
2024/2025
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