MATHEMATICS 53
FIRST LONG EXAM
First Semester AY 2012-2013
July 5, 2012
This exam is good for 75 minutes. Show all necessary solutions. Box all final answers.
I. Evaluate the following limits. (5 points each)
1.
1
1
lim + ( 2 1 (+1)2 )
1
14 2
2 2 3+1|+|2
Applications of the Denite Integral
Volumes of Solids of Revolutions (Part 2)
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.6
For today
1
Volumes of Solids of Revolution: Cylindrical Shell Method
2
Other Examples: Volumes of Solids of Rev
Applications of Integration:
Volumes by Slicing
Volumes of Solids of Revolution (Disks and Washers)
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.5
For today
1
Volumes of Solids by Slicing
2
Volumes of Solids of Revolution: Disk or Washer
Particular Antiderivatives
Introduction to the Area of a Plane Region
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.1
For today
1
Particular Antiderivatives
2
Application to Rectilinear Motion
3
Introduction the Area of a Plane Region
For
The Denite Integral
The Fundamental Theorem of Calculus
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.3
For today
1
The Denite Integral
2
Mean-Value Theorem for Integrals
3
The First Fundamental Theorem of Calculus
4
The Second Fundamenta
Antiderivaties and Antidierentiation
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.1
For today
1
Antiderivatives and Antidierentiation
2
Integration by Substitution
For today
1
Antiderivatives and Antidierentiation
2
Integration by Substi
Unit 4: Integration and Its Applications
Introduction
Mathematics 53
Institute of Mathematics - UP Diliman
7 February 2013
Math 53 (Unit 4)
Integration and Applications
7 February 2013
1 / 7
Unit 4: Integration and Its Applications
Math 53 (Unit 4)
Integr
Absolute Extrema and Optimization
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.4
For today
For today
Absolute Extrema
Denition
A function f is said to have an absolute maximum value on an
interval I at x0
Absolute Extrema
Denition
A func
The Mean Value Theorem and Relative Extrema
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.1
For today
1
The Mean Value Theorem
2
Critical Numbers
3
Increasing/Decreasing Functions
4
The First Derivative Test For Relative Extrema
For today
Concavity and The Second Derivative Test
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.2
For today
1
Concavity and Points of Inection
2
The Second Derivative Test
3
Graphing Polynomials
For today
1
Concavity and Points of Inection
2
The S
2nd LONG EXAMINATION
SLOPES AND THE DERIVATIVE
To test differentiability of a function,
o
See if continuous 'yung function in the
point/interval
o
Show f and the value of
Know the definition of derivative.
ANG LOCAL LINEAR APPROXIMATION AY ISANG
MAARTENG
1ST LONG EXAMIINATION
REVIEW OF FUNCTIONS
Be comfortable in functions and its properties
(domain, range, zeroes, etc.)
Know if your infinite limit is + or - . 0 as
denominator doesnt mean your limit is infinity.
It would help if you know the graphs
Use co
E12: Carboxylic Acid
and Acid Derivatives
Background
Acidity
EWG stabilize carboxylate ion = enhance acidity
Carboxylic Acid Derivatives
Expected Results
Test
Solubility of
carboxylic acid in 5%
NaHCO3
CH3COOH
C6H5COOH
Test for acetic acid
Test for benzo
Mathematics 53
2nd Semester, A.Y. 2014-2015
Q3, R3, W8, X8
Exercises 21 - Derivatives and Integrals
of Other Classes of Functions
I. Evaluate the following indenite integrals. (Hints included.)
1.
6 ln x
dx
x(1 + ln4 x)
14.
(ex 7x = (7e)x )
(tan1 u)
2.
se
Applications of Denite Integrals:
Area of a Plane Region and Arc Length
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 4.4
For today
1
Area of a Plane Region
2
Arc Length
Review of Parabolas and Circles
Parabolas opening upwards or downwards
Integrals of Hyperbolic Functions
Inverse Hyperbolic Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Integrals of Hyp Fcns, Inverse Hyp Fcns
Mathematics 53
1 / 37
For today
1
Integrals of Hyperbolic Fun
Mathematics 53
2nd Semester, A.Y. 2014-2015
Q2, R2, X14
Sample Exam - First Long Exam (2nd Semester 2013-2014)
General Directions: This exam is for 80 minutes only. Use ONLY blue or black ink. Show clear, complete and
concise solutions to earn full points
MATHEMATICS 53 Second Semester, A.Y. 2013 - 2014
Fourth Long Exam . 28 February 2014
GENERAL INSTRUCTIONS: This exam is for 80 minutes only. Write your answers in a
b'luebook using black or blue pen. Show all necessary solutions and box your ﬁnal answers.
The Mean Value Theorem and Relative Extrema
Mathematics 53
Institute of Mathematics (UP Diliman)
March 5, 2015
Institute of Mathematics (UP Diliman)
MVT and Relative Extrema
March 5, 2015
1 / 51
For today
1
The Mean Value Theorem
2
Increasing/Decreasing F
Rates of Change and Rectilinear Motion
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Rates of Change and Rectilinear Motion
Mathematics 53
1 / 30
For today
1
Rates of Change
2
Rectilinear Motion
Institute of Ma
The Intermediate Value Theorem
The Squeeze Theorem
Limits and Continuity of Trigonometric Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
IVT, Squeeze, Trigonometric Limits
Mathematics 53
1 / 42
On Inco