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
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 system
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 Ma
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
x y z
,
,
,
where u, v
6 Cross Product of Partials.
At x(u, v), y(u, v), z(u, v) , S R u R v (u, v)
7 Surface Area.
SA =
R u R v dA .
D
Cartesian coordinates,
SA =
q
D
f x2 + f y2 + 1 dA
In
Series converges only at x = a.
Series converges absolutely x R.
Series converges for all x in an interval centered at a.
3 Interval of Convergence. Set of values x such
that the power series is co
C
19 Law of Conservation of Energy.
1
mv i2 p i
2
1
mv 2f p f =
2
20 Greens Theorem. If first partials of L, M cont on
B R2 , C piecewise-smoothH simple closed curve in
B , R region bounded by C (
Tangent Planes to Level Surfaces
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines-Diliman
19 January 2017
Math 55 (G.A.M. Velasco)
19 Jan. 2016 Tangent Planes
D
Relative Extrema of Functions of Two Variables
Math 55 - Elementary Analysis III
Gino Angelo Velasco
Institute of Mathematics
University of the Philippines-Diliman
24 January 2017
Math 55 (G.A.M. Vela
Directional Derivatives and Gradients
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines-Diliman
17 January 2017
Math 55 (G.A.M. Velasco)
17 Jan. 2017 Dir. Der. &
Power Series
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 19
Power Series
Definition
A series of the form
X
cn (x a)n = c0 + c1 (x a) + c2
Taylor and Maclaurin Series
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 14
Recall
A power series centered at a is of the form
X
cn (x a)n
Convergence Tests for Series with Negative and
Positive Terms
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 19
Recall
R
Divergence Test
If l
Sequences
Mathematics 55 - Elementary Analysis 3
Institute of Mathematics
University of the Philippines-Diliman
1 / 19
Sequences
A sequence can be thought of as an ordered list of numbers:
a 1 , a2 ,
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
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 ch
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 ch
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 equ
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
mult
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 num
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 so
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 m
Physics 71
Elementary Physics 1
Section: TWHFW
Lecture: TWTh (LP102)
Recit: F
Time: 01:15 02:15PM
Corequisite: Math 53, Math 63, or Math 100
Teacher
Name: Francesca Isabel N. de Vera
Faculty room: A10
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
ang
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 fin
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
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 P
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
Gravimetric Determination of Moisture in Fertilizer Samples
E.M.M. Medrano
Medrano, J.M. Pasco
Department of Mining, Metallurgical and Materials Engineering, College of Engineering
University of the P
SUMMARY (Coverage of First Long Exam)
Directional Derivatives
Let f be a function of x and y and ~
u = hu1 , u2 i a unit vector.
The gradient of f , denoted by f , is the vector function
f (x, y) = h
SUMMARY (Coverage of Second Long Exam)
Z
TRIPLE INTEGRALS
~ dR
~ = f (R(b) f (R(a)
F
C
Given a solid G with density function (x, y, z),
Z Z Z
1. The volume of G is V =
dV .
where R(a) and R(b) are th
1: 1:11
M:111111:11111111'III111 5112':
Se eiondIlAmgg13.111111
September I. ZLZII II I
Show 21111 necessaty Solutions
]. Let f (31,31, zI=I111 )11'w2. 1,1 I g 'r:. . Detem'nine the IfeIIowing: (SI po
rrlwt.\.-_'r,iq-
L6 Decembqr 2w
MRrHruertcs 55 Ftnsr Exena
Direction xWrite all answers clearly and legibty in blue or black ink. You have 90 minutes to finish
this exam.
L. Let f (*,a)
:
u2e2'* cos(r