Math 54 First Examination
2 September 2014
I. Solve the following indenite integrals. (5; 6; 6; 5 points)
. . 411:2 ~ 3x + 7
1. n. 5 2. d . d
/51 r(cos(m)+31n "0) a 3 /(m2)(4m2+1) x
(1:1: _ _
2. fmT/Z 4. /sm(2.r:) CCOSI d3:
II. Evaluate the following im
27
Magnetic Field and
Magnetic Forces
1
27
Fundamental nature
of Magnetism
Magnetism
Magnetic Field
Magnetic Field Lines and Magnetic Flux
Motion of Charged Particles in Magnetic Field
Magnetic Force on a CurrentCarrying Conductor
Force and Torque on a
University of the Philippines Chemical Engineering Society, Inc. (UP KEM)
Math 54 3rd Long Exam Reviewer
Cartesian coordinate system in three dimensions
Point represented by an ordered triple of real numbers
(x, y, z)

v
A vector
can be represented by an
University of the Philippines Chemical Engineering Society, Inc. (UP KEM)
Math 54 2nd Long Exam Reviewer
Conic section figure formed when a right circular cone is
cut by a plane
1.
2.
Circle
Parabola all points whose distance from the focus is
equal to it
Vector equation
UP
nd a parametrization of the given equation of the
curve, e.g. for a line passing through Po (xo , yo , zo )
and parallel to vector v = a, b, c :
K E M
r(t) = xo + at, yo + bt, zo + ct
Operations on vector functions
if F and G are realv
M ATHEMATICS 54
First Semester AY 2010  2011
15 October 2010
Final Exam Sample 1
Write all necessary solutions on your bluebooks and box all final answers. You have two hours to finish this
exam. Good luck!
I. True or False. Write TRUE if the statement i
Partial Fractions Exercises
Partial Fractions
Mathematics 54Elementary Analysis 2
Institute of Mathematics
University of the PhilippinesDiliman
1 / 17
Partial Fractions Exercises
Case 1 Case 2 Case 3 Case 4
Introduction
Recall.
A function h is a rational
MATHEMATICS 54 First Semester AY 20152016
EXAMINATION 5 26 November 2014
black or blue pcn only. Write legibly and show complete solutions. Box the nal numerical
answers.
Note: Any form of cheating in. naminations or any act of dishonesty in relation to
MATHEMATICS 54 lst Semester AY 20152016
EXAM 4 REVIEWER
XX
2 .
I. Find the domain of ROE) = <ln(t2 l), t 4 et>.
t2
'AI;
2 2 . 4 . .
_ 75? t3 31+talg t2 6)j+ln(4t2)k ,iftag
II. Let R(t) = 5 75 * t 3. Determine whether or
E225'+2inzii ,ift=
not the funct
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
Exercises 14  Surfaces
I. Read each problem carefully and solve them systematically.
1. Consider the surface S : x2 + y 2 + 4z 4 = 0.
(a) Identify the type of quadric and the traces of S on the coor
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
Exercises 11  Polar Curves III
I. Write the capital letter pertaining to your choice of the best answer.
1. Which of the following is not an ellipse?
12
(A) the set of points whose distances from (C
Mathematics 54
1st Semester, A.Y. 20152016
Exercises 8  Parametric Equations and Parametric Curves
Q, W16, X16
Do as indicated.
d2 y
1. Given the parametric equations x = ln t and y = t3 + 1 where t > 0, nd
without
dx2
eliminating the parameter.
2. Give
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
3
Exercises 12  R , Vectors
I. Write TRUE if the statement is always true, otherwise write FALSE.
1. For any vectors A and B, A B = B A.
2. If A, B and C are nonzero vectors, then A (B C) = (C
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
Exercises 15  VectorValued Functions
I. Domain and Parametrization of Space Curves
1. Find the domain of the following vectorvalued functions.
ln(1 t)
t2 4
+
t2 + 1
2t + 4
1
t2 4, sin t,
(b) R(t)
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
Exercises 18  Motions in Space
I. Problems about Motions in Space
1. A particle is moving in space with velocity V (t) = 2 2, cos 2t, sin 2t such that at t = 0,
it is at (1, 0, 1). Find:
(a) the pos
Mathematics 54
1st Semester, A.Y. 20152016
Q, W16, X16
Exercises 16  Moving Trihedral, Curvature
I. Do as indicated.
1. Let R be a vector function such that T (t) =
3
1 3
cos 2t, ,
sin 2t
2
2 2
and R(0) =
1, 0, 1 . Find:
(a) the unit tangent, unit norma
Mathematics 54
Second Semester, AY 201213
Fourth Long Exam
March 7, 2013
This exam is for one hour and twenty minutes only. Use only black or blue ink. Show all necessary
solutions and box all final answers. Calculators are not allowed.
ln(1 t) t2 4
~
1
I.
1. dom f = (x, y) R2 1 xy 0 and x + 2y 6= 0
f (0, y) f (0, 0)
fy (0, 0) = lim
= lim
y0
y0
y0
2.
0
y0
= lim 1 = 1
y0
f (x + x, y + y) f (x, y) fx x fy y
p
(x,y)(0,0)
(x)2 + (y)2
lim
=
=
k = 2:(x 1)2 = y + 2
k = 0:(x 1)2 = y
2.
k = 2:(x 1)2 = y 2
(par
III. 1. Let C1 : x = 0, y = t. Then
00
= 0.
t0 t12
f (x, y) = lim
lim
(x,y)(0,0)
along C1
On the other hand, if C2 : x = t2 , y = t, then
t12
1
.
=
t0 (t4 + t4 )3
8
f (x, y) = lim
lim
(x,y)(0,0)
along C2
Since limits along two smooth paths through (0, 0)
Mathematics 54
30 September 2014
Second Long Examination
General Instructions: Show all necessary solutions and box your final answers. Use only black or blue ballpens.
I. Multiple Choice. Write the capital letter of the correct answer. [1 point
each]
5
5
MATHEMATICS 54
1st Semester 20082009
5th LONG EXAM
October 8, 2008
Provide a neat, complete and logical solution for each problem. Express numerical answers in DECIMAL FORM. No solution, no credit.
p
I. Given the function g(x, y) = 9 9x2 + 9y 2 :
1. Find
MATHEMATICS 54
1st Semester 20132014
Fifth Long Exam
Follow all directions. Use black or blue pen only and avoid erasures. This exam is good for
80 minutes.
I. TRUE OR FALSE. Write the TRUE if the statement is always true, otherwise write
FALSE.
(1 pt. e
Mathematics 54 Third Long Exam
General Direction: Use black or blue bullpen. Show neat. and complete solution to obtain full points.
I. Do as indicated.
1. Find the distance between the plane 61 + 6y + 32 = 2 and the point (%, , 1).
m
l Consider the vecto
Mathematics 54
Fourth Long Exam Answer Key
Second Semester, AY 201213
March 7, 2013
1. (a) (, 2) (2, +)
6
e6
e
, 0, 4
i + 4k or
(b)
17
17
~
2. Let z = t. Then x = et and y = 2t2 3t2 = t2 . So R(t)
= het , t2 , ti.
1
~
~
3. The only possible point of di
M ATHEMATICS 54
S AMPLE F IFTH E XAM
1 ST S EMESTER AY 20152016
1. Determine and sketch the domain of f (x, y) =
2. Sketch the contour plot of f (x, y) =
ln(x2 + y 2 2x)
1 x2
1 x2
with k = 1, 0, 1.
(y + 1)2
2y 2 x2/3
does not exist.
(x,y)(0,0) x2 + y 3
(
MATHEMATICS 54
CONIC SECTIONS
1st Semester 20162017
WFW (LEYSON)
I. Sketch the following parabolas. Label the vertex, focus and endpoints of the latus rectum
with their corresponding coordinates and give the equation of the directrix.
1. (x + 2)2 = 10(y
MATHEMATICS 54
TRIGONOMETRIC INTEGRALS
Evaluate the following integrals.
Z
1.
sin3 4x(sin 4x + cos5 4x) dx
Z
2.
Z
3.
Z
4.
Z
5.
Z
6.
Z
7.
Z
8.
sin5 x
dx
3
cos x
1st Semester 20162017
WFW (LEYSON)
Z
sec x tan4 x dx
9.
cot3 x
dx
csc2 x
Z
10.
Z
Z
4
csc x
dx