MATHEMATICS 54
LE4 SAMPLEX 1
et + t + 3 + ln(3 t) k , if t 6= 2,
~
I. Let F (t) =
t2
e2 , 5, 1 ,
if t = 2.
1. Find the domain of F~ .
2. Determine if F~ is continuous at t = 2.
II. Give a vector equation for the curve of intersection of the elliptic p
MATHEMATICS 54 (Leyson)
PARTIAL DERIVATIVES
Definition. Let f be a function of two variables x and y and (x, y) domf .
The partial derivative of f with respect to x is the function fx (x, y) defined by
f (x + x, y) f (x, y)
,
x0
x
fx (x, y) := lim
provid
MATHEMATICS 54 (Leyson)
Differentiability and Approximation
Definition. A function f of two variables x and y is said to be differentiable at (x0 , y0 ) if
fx (x0 , y0 ) and fy (x0 , y0 ) both exist and
f fx (x0 , y0 )x fy (x0 , y0 )y
p
= 0.
(x,y)(0,0)
(x
MATHEMATICS 54 (Leyson)
I. Exercises from the slides.
Z 1
1.
x3 e3x dx
Z0 e
ln x dx
2.
INTEGRATION BY PARTS
Z
x3 ln x dx
3.
Z
(ln x)2 dx
4.
1
II. Evaluate the following integrals.
Z
1.
x5 ex dx
Z
2.
x3 cos 2x dx
Z
3.
x ln x dx
Z
4.
x(ln x)2 dx
Z
5.
e3x si
MATHEMATICS 54 (Leyson)
LIMITS AND CONTINUITY
Definition. Let f be a function of x and y defined on a set D that contains points arbitrarily
close to (a, b). Then
lim f (x, y) = L if for every > 0 there exists > 0 such that
(x,y)(a,b)
f (x, y) L < whene
MATHEMATICS 54 (Leyson)
FUNCTIONS OF SEVERAL VARIABLES
Exercises from the slides
xy
. Evaluate f (2, 1, 2) and f (5a2 , a2 , a).
1. Let f (x, y, z) =
z2
p
2. Find the domain of f (x, y) = y 2 x.
3. Find the domain of f (x, y) = ln(xy).
p
9 x2 9y 2
4. Find
MATHEMATICS 54 (Leyson)
TRIGONOMETRIC INTEGRALS
I. Exercises from the slides.
Z 1
sin2 (x) cos2 (x) dx
1.
Z
4.
cos 4x cos 3x dx
0
Z
2.
Z
3.
cos3 x
dx
sin x
2.
Z
3.
Z
4.
5.
csc4 x
dx
cot2 x
II. Evaluate the following integrals.
Z
1.
sin 4x(sin4 4x + cos 4x
MATHEMATICS 54 (Leyson)
I. Exercises from the slides.
Z 2
x 4
1.
dx
x
Z
2.
II. Evaluate the following integrals.
Z
1
1.
3 dx
(4 x2 ) 2
Z
x3
2.
dx
1 x2
Z
1
3.
dx
x2 3 x2
Z
5
4.
1 e4x 2 dx
Z
1
5.
3
dx
(1 + x2 ) 2
Z
6.
x2
dx
x2 + 6
Z
7.
x2
Z
8.
Z
9.
1
dx
4x2
Mathematics 54 WFV
Common Quiz 7
Name:
Score:
This is a takehome quiz to be submitted on April 18 to your discussion teachers. WORK
INDEPENDENTLY! Write your complete solutions on this sheet. You may use the back
page when necessary. Any form of erasure
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
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
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
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
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
MATHEMATICS 54 (Leyson)
I. Exercises from the slides.
Z
4
1.
dx
7
2 (x + 3)
Z 1
3
2.
dx
5
x
Z
1
dx
3.
1 + x2
0
Z
2
4.
xex dx
IMPROPER INTEGRALS
8
Z
6.
6
4
dx
(x 6)3
2
Z
x2 ln x dx
7.
0
2
Z
8.
0
dx
(2x 1)2/3
0
Z
3 x4
x e dx
5.
9.
4
3
+
Z
7.
2
sin x
dx
MATHEMATICS 54 (Leyson)
Evaluate the following integrals.
Z
12
1.
3
x x
Z
1
2.
3
x (x 1)
Z
2
3.
2
2
x (x + 1)
Z
4x 3
4.
dx
2
x 2x 3
Z
8x 2
dx
5.
x3 x
Z
5x + 10
6.
dx
2
2x + 11x + 12
Z
2x2 + 3
dx
7.
x(x 1)2
PARTIAL FRACTIONS
Z
17x2 + 7x 3
dx
(x 1)2 (2x + 1
MATHEMATICS 54 (Leyson)
SEPARABLE DIFF. EQUATIONS
I. Exercises from the slides.
1. Find the equation of the curve that passes through the point (2, 0) and whose slope at
any point (x, y) is xy.
2. Find a solution to the differential equation xy 0 + y = y
Second Long Examination
I. Determine whether the following statements are true or false.
Math 54
(5pts)
1. The graph of the polar equation defined by r = 5 6 cos , [0, 2] is smooth,
closed and simple.
1
is symmetric with respect to the polar axis.
2. The
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.
~
1. Let R(t)
=
e
Math 54 Fourth Examination
October 6, 2010
I. Check whether the following vectorvalued function is continuous at t = 0:
2
t + t 1 cos t ln(t + 1)
if
t=
6 0
,
,
~
R(t)
=
2t
t
sec2 t
h1, 0, 0i
if
t=0
II. Let C be the curve traced by the vector valued fun
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
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