Ex. 6: Mobile Emissions
Inventory
Introduction
In 2008, the Environmental Management Bureau (EMB) reported the following air pollutant
emissions from vehicles (in tons/yr) broken down into emissions from cars, UVs, trucks,
buses and motorcycles/tricycles:
Ex. 5: Understanding Your Electricity
Bill: Metering Information
Introduction
Electricity is what powers most of our daily activities. The understanding of electricity for
most people is through consumption. This is concretely expressed by the devices and
ES 12-K
Group 8
November 27, 2013
Jason Chang
Michelle Chong
Aldwyn Tan
Luigi Tanbio
Exercise 2: No Place Like Home (Up Close)
I. Introduction:
With the recent rise of technology, people have become so consumed with electronics
and social media that we ha
COURSE SYLLABUS
Course Number: ES-12
Title: Introduction to Environmental Science, Laboratory
Department: Environmental Science
School: School of Science & Engineering
Semester: 2nd semester
School Year: 2013-2014
Instructor: Ms. Deborah M. Villa
A. COURS
Ex. 1: Plant Care Project
INTRODUCTION
This special project is designed to introduce the students to the proper care of the environment using a plant
of their choice as a means to achieve this objective.
OBJECTIVES
At the end of this exercise, it is aimed
ALDWYN TAN
February 26, 2014
ES 12- K
Group 8
Leader:
Members:
Jason Chang
Michelle Chong
Kristina Deacon
Aldwyn Tan
Luigi Tanbio
Ex. 1: Plant Care Project
Introduction
The calamondin, Citrofrutunella microcarpa, also widely known as calamansi, originated
Ex 3: Killing Me Softly
Effects of Acid Rain on Plant Growth
INTRODUCTION
Rainwater is normally slightly acid because it absorbs carbon dioxide as it passes through the
astmosphere, forming a weak acid known as carbonic acid. Other gases in the atmosphere
1
Directional Derivatives and Gradients
The directional derivative of a function f is a more general derivative compared with the partial
derivatives of the function. It gives the slope of the tangent line to the surface along a certain
direction from a p
I. Directional Derivatives and Gradients
~ is the
1. (a) Use the definition of the directional derivative to find DU~ f (x, y) if f (x, y) = 12 x2 4y 2 and U
unit vector in the direction /6.
(b) Find DU~ f (2, 1) and give its geometric interpretation.
(c)
Triple Integrals
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Triple Integrals
Triple Integral over a Box
Suppose f (x, y, z) is defined on a rectangular box
B = cfw_(x, y, z) : a x b, c y d, r z s.
By dividing the i
Math 55
Problem Set 1
THQ1 / THR1
2nd Semester, AY 15-16
General Instructions: Show complete solutions. You can work in groups of at most 3 members.
Deadline is on February 9 (Tuesday).
1. Given the function
g(x, y) = 4xy 2 + 2x2 y x2 x.
Use the Second De
Triple Integrals in Cylindrical
& Spherical Coordinates
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Cylindrical & Spherical Coordinates
Cylindrical Coordinates
Consider a point P in space. Let Q be the projection of
Line Integrals of Scalar Fields
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Line Integrals of Scalar Fields
Curtain Area Problem
Problem
Let C be a smooth curve defined by a vector function
~
R(t)
= x(t) + y(t)
, wi
Vector Fields
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Vector Fields
Introduction
The following shows air velocity vectors that indicate the wind speed
and direction at points above the surface elevation in Phili
The Fundamental Theorem of Line Integrals
and Greens Theorem
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
FTLI and Greens Theorem
Recall: FTOC
Recall from Math 53:
Theorem
Let f (x) be a function that is continuous o
Directional Derivatives and the Gradient Vector
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Directional Derivatives & Gradient
Recall: Partial Derivatives
Let z = f (x, y).
z
x
z
y
f (x + h, y) f (x, y)
h0
h
f (x, y
Lagrange Multiplier
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Lagrange Multiplier
Recall
R
R
R
(a, b) is a critical point of f (x, y) if f (a, b) = ~0
(fx (a, b) = fy (a, b) = 0) or either one of fx (a, b) or fy (
Relative and Absolute Extrema
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Relative and Absolute Extrema
Recall
R
R
R
For a function f (x)
a is a critical number of f if either f 0 (a) = 0 or f 0 (a) does
not exist.
Line Integrals of Vector Fields
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Line Integrals of Vector Fields
Work Along a Curve Problem
We know that the work W done by a force F~ in moving an
object is given by
W = F
Double Integrals in Polar Coordinates
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals in Polar Coordinates
Recall
If f is continuous on a Type I region D such that
D = cfw_(x, y) : a x b, g1 (x) y g2 (x
Double Integrals over Rectangles
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals over Rectangles
Recall
If f (x) is defined for a x b, we subdivide the interval [a, b]
into subintervals of length x. Cho
Double Integrals over General Regions
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals over General Regions
Recall
If f (x, y) is integrable over the rectangular region
R = [a, b] [c, d], then
b
d
f (x,
Applications of Double Integrals
Lucky Galvez
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Applications of Double Integrals
Volume of a Solid
Let f (x, y) 0 for all (x, y) in a closed and bounded region
R R2 .
The volume V of the
Math 20.3A section: _
Student Name: _
ATENEO DE MANILA UNIVERSITY
MATHEMATICS DEPARTMENT
APPLIED CALCULUS FOR LIFE SCIENCES I (MATH 20.3A)
LECTURER: MR. ROBERT C. PANGILINAN
PRACTICE LONG EXAM # 3
February 12, 2011
Instructions:
1.
Score: _
Letter
Grade:
Math 20.3A section: _
Student Name: _
ATENEO DE MANILA UNIVERSITY
MATHEMATICS DEPARTMENT
APPLIED CALCULUS FOR LIFE SCIENCES I (MATH 20.3A)
LECTURER: MR. ROBERT C. PANGILINAN
PRACTICE LONG EXAM # 1
December 7, 2010
Instructions:
1.
Score: _
1. Time allotte
Math 20.3A section: _
Student Name: _
ATENEO DE MANILA UNIVERSITY
MATHEMATICS DEPARTMENT
APPLIED CALCULUS FOR LIFE SCIENCES I (MATH 20.3A)
LECTURER: MR. ROBERT C. PANGILINAN
PRACTICE LONG EXAM # 2
January 13, 2011
Instructions:
1.
Find
2.
If
Letter
Grade:
PARAMETER AND STATISTIC
! A
measure computed on the basis of data obtained
from a sample is termed a statistic.
! Any quantity obtained from a sample for the
purpose of estimating a population parameter is
called a sample statistic, or briefly statistic.