Summary of Tests for Series
Test
Condition(s)
of Convergence
Series
Condition(s)
of Divergence
Comment
lim 0
This test cannot be used
to show convergence.
nth-Term
Geometric Series
| 1
Telescoping Series
lim
p-Series
Alternating Series
Integral
(f
Lecture 18 Objectives:
Work and Kinetic Energy theorem II
1. Solve problems using work-kinetic energy theorem
2. Relate work, power and kinetic energy.
Additional for today:
- Submit Bluebook #1 and Problem set
- Announce LE1 scores
Sample Problem 1: Moti
Convergence Tests
(for Series with Positive and Negative Terms)
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Convergence Tests II
Recall
R
Divergence Test
If lim an 6= 0, then the series
n
R
P
an is divergent.
n=1
Th
Slope of position-time : velocity
Slope of velocity-time : acceleration
Tangent lines: instantaneous velocity and
acceleration
Acceleration of
rising ball is the
same as of the
falling ball!
Acceleration due to
gravity is constant!
1
Review Question:
Two
Review Problem:
A 10.-N box is being pushed on top of a table with
force magnitude of 5.0N. If the coefficient of kinetic
friction between the box and the table is 0.25, what is
(a) the acceleration of the box and (b) the net force
on the box?
Given: Wbox
REVIEW: relative velocities are quantified based on a reference
Reference frame is where
velocity is measured.
Inertial reference frame is
non-accelerating reference
frame.
Things to remember
Identify each reference frame in the problem
Identify target va
Review:
1
Chapter 5: APPLYING NEWTONS LAWS
Lecture 11: Objectives
Apply Newtons laws to obtain quantitative and
qualitative conclusions about the contact and noncontact forces acting on a body in equilibrium.
2
Solving Problems using equations of motion
Lecture 22: Equilibrium, Energy Plots and Center of Mass
Objectives
1. Define the concept of equilibrium and its relationship with the
potential energy function.
2. Determine the stable, unstable, and neutral equilibrium points
given the potential energy
Series
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Series
Recall
A sequence cfw_an = cfw_a1 , a2 , . . . is convergent if
lim an = L
n
exists for some real number L. Otherwise, it is divergent.
In many applications
Lecture 27 Objectives:
Rotational kinematics II & Moment of Inertia
1. Apply the rotational kinematic relations in rotating objects.
2. Calculate the moment of inertia about a given axis of given
multiple objects
Recall:
Relationship between linear and ro
Lecture 10 Objectives:
Newtons 2nd and 3rd Laws of Motion
1. Draw correct free-body diagrams for a given body.
2. Identify action-reaction pairs.
1
Yesterday: A body in equilibrium that remains in equilibrium!
Newtons 1st Law of Motion: body in equilibriu
Lecture 28:
Parallel axis theorem and conservation of energy in
rotational motion
Objectives
1. Apply parallel axis theorem
2. Solve problems using the law of conservation of energy
to rotating systems
Moment of inertia
2
Moment of inertia is proportion
Fundamental Theorem for Line Integrals
Math 55 - Elementary Analysis III
Gino Angelo Velasco
Institute of Mathematics
University of the Philippines-Diliman
14 March 2017
Math 55 (G.A.M. Velasco)
14 Mar. 2017 Fund. Thm. Line Integrals
Theorem (The Fundamen
Surface Integrals
Math 55 - Elementary Analysis III
Gino Angelo Velasco
Institute of Mathematics
University of the Philippines-Diliman
23/28 March 2017
Math 55 (G.A.M. Velasco)
23/28 Mar. 2017 Surface Integrals
Given: smooth parametric surface with vector
Review: Parallel and perpendicular components of
acceleration
Parallel
Acceleration
Velocity
Perpendicular
Parallel
Parallel to , path
Same direction
Change in particles
speed (magnitude)
No change in direction
Perpendicular
Perpendicular to ,
No change i
Name:_
_
Recit Teacher:_
Date: _
Physics 71 TWHFW 2nd semester 2016-17: Recitation Quiz 6
INSTRUCTIONS: Answer the following questions clearly and legibly. Show your complete solutions and box you final answer.
Any act of dishonesty such as cheating shall
Lecture 13 Objectives
1. Differentiate the properties of static friction and
kinetic friction.
2. Compare the magnitude of sought after quantities
such as frictional force, normal force, threshold
angles for sliding, acceleration etc.
3. Solve equilibrium
2. Vector Fields
1. Triple Integrals
Fubinis Theorem: Let f be cont on a box B = [a, b]
[c, d] [r, s], then
ZZZ
Z bZ dZ s
f (x, y, z)dV =
f (x, y, z)dydxdz
a
B
c
dZ
Z
r
sZ
b
=
f (x, y, z)dxdzdy = . . .
c
r
For general solids: Type 1
ZZZ
ZZ Z
f (x, y, z
UP Engineering Society
Math 55: Final Examination
Building Bridges, Breaking Barriers
This exam is good for two hours. Use only black of blue ink. Show all
necessary solutions and box all final answers.
I. Let , = 3 + 2 3 + 2
1. Find the directional deriv
Mathematics 55
LE2 Thirdie Tips
TRIPLE INTEGRALS
I. Triple Integrals in Rectangular Coordinates
1. Given a solid G with density function f (x, y, z):
ZZZ
(a) Volume of G:
dV
Note. For Triple Integrals in Rectangular and Cylindrical Coordinates, be sure to
Results and Discussion
When heat flows from one region to another per unit time, the condition for heat current is used,
H=
kA (T H T C )
,(1)
L
where A is the cross-sectional area of the rod, L is the length of the rod, T H and TC are the temperature of
MATHEMATICS 55
PROBLEM SET I (Part 2)
1st Semester AY 2015-2016
Due: 11 September 2015
Work independently. Do not consult anyone except your instructor. Write your
complete solutions on your PS1 bluebook using non-erasable black or blue pen. Any
form of e
Fourth Long Exam Review
Mathematics 53
Institute of Mathematics
Math 53 (Institute of Mathematics)
Fourth Long Exam Review
1 / 13
Z
1. Find the general antiderivative of:
Math 53 (Institute of Mathematics)
(x + 3)
Fourth Long Exam Review
p
5
x2 6x + 9 dx
CS 135 Algorithm Design and Analysis
Department of Computer Science, UP Diliman
Asst. Prof. Philip Christian Zuniga
Asst. Prof. Rose Ann Zuniga
Rules:
February 6, 2017
MUHW, MUHW-1, MUHX,
MXTW, MXTW-1
Problem Portfolio 3
Write your solution to the followi
Names/Student numbers:
_
_
Recit Teacher/Section: _
Date: _
Physics 71TWHFW 2nd semester 2016-17 Recitation Quiz 1
INSTRUCTIONS: Answer the following questions clearly and legibly. Show your complete solutions and box you
final answer. Any act of dishones
Action and Reaction Pairs?
- force exerted by the box on the table AND normal
force of the table on the box
- weight of the box AND normal force of the table on
the box
- force exerted by the table on the earth AND weight
of the table
1
- friction AND pus
Names/Student numbers:
_
Recit Teacher/Section: _
Date: _
Physics 71TWHFW 2nd semester 2016-17 Recitation Quiz 4
INSTRUCTIONS: Answer the following questions clearly and legibly. Show your complete solutions and box you final answer.
Any act of dishonesty
Names/Student numbers:
_
_
Recit Teacher/Section: _
Date: _
Physics 71TWHFW 2nd semester 2016-17 Recitation Quiz 2
INSTRUCTIONS: Answer the following questions clearly and legibly. Show your complete solutions and box
you final answer. Any act of dishones
Review Problem: Above a plane inclined 15o is a cart with weight
connects to a bucket with weight w2. Assume that there is no friction
on the plane and the mass of the rope is negligible. The system moves
with constant acceleration of 1.0m/s2.
a
a
cart
bu
Convergence Tests for Series with Positive
Terms
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 38
Recall
R
R
The series
P
an converges if its sequence of partial sums
n=1
cfw_Sn is convergent. O
Math 55 (THQ3, THR1, THV1) Homework Due: 07 JanuaryFebruary 2017, Tuesday
Write your complete solutions on a whole sheet of yellow pad paper. Papers submitted 20 minutes
after the start of the class will be checked but the score will not be recorded. Work
Math 55 (THQ3, THR1, THV1) Homework
Due: 09 March 2017, Thursday
Write your complete solutions on a whole sheet of yellow pad paper. Papers submitted 20 minutes
after the start of the class will be checked but the score will not be recorded. Work independ
Math 55 (THQ3, THR1, THV1) Homework
Due: 27 April 2017, Thursday
Write your complete solutions on a whole sheet of yellow pad paper. Papers submitted 20 minutes
after the start of the class will be checked but the score will not be recorded. Work independ
Series
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 21
Recall
A sequence cfw_an = cfw_a1 , a2 , . . . is convergent if
lim an = L
n
exists for some real number L. Otherwise, it is divergent.
In