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
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
Indeterminate Forms and LHpitals Rule
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Indeterminate Forms and LHpitals Rule
Mathematics 53
1 / 33
For today
0
and
0
1
Indeterminate Forms of Type
2
LHpitals Rule
3
Derivatives of Logarithmic and Exponential Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Derivatives of Logarithmic and Exponential Functions
Mathematics 53
1 / 35
For today
1
Exponential and Logarith
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.
Integrals Yielding Logarithmic and Exponential
Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Integrals Yielding Logarithmic and Exponential Functions
Mathematics 53
1 / 35
For today
1
and of the other
Second Long Exam Review
Mathematics 53
Institute of Mathematics
Math 53 (Institute of Mathematics)
Second Long Exam Review
1 / 14
Find
dy
dx .
No need to simplify.
Math 53 (Institute of Mathematics)
Second Long Exam Review
2 / 14
Find
1
dy
dx .
y=
No need
9(x + 2)2
9x(x + 2)
18(x2 3)
, f (x) =
, and f (x) =
.
3
4
(x + 3)
(x + 3)
(x + 3)5
3 3
3+ 3
Note: f ( 3) =
, f ( 3) =
, 3 1.732.
4
4
I. Given f (x) =
1. Identify the domain, and the x- and y-intercept/s of f .
2. Using limits, nd the equations of the li
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
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
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
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
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
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
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Unit 4: Integration and Its Applications
Math 53 (Unit 4)
Integr
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
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
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
Exercises on the Mean Value Theorem, Relative Extrema, Increasing and
Decreasing Functions, Points of Inection, Concavity, Absolute Extrema
IV.
2. Show that any polynomial function of degree three has exactly one point of
inection.
I. Determine whether ea
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
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M53_LE1_002
Mathematics 53
First Long Examination
Elementary Analysis I
First Semester, AY 2012-2013
I.
TRUE or FALSE. Write True if the statement is correct. Otherwise, write False.
(1 point each)
1. If the functions f and g are both discontinuous at x
_
M53_LE3_002
Mathematics 53
Third Long Examination
I.
Elementary Analysis I
First Semester, AY 2012-2013
Write TRUE if the statement is always true. Otherwise, write FALSE. (1 pt. each)
1. Let g be a function whose g(x) -7 x [0,3] and such that it crosse
Hello all.
Some reminders:
1. Exam coverage is from Review on Functions up to IVT, Squeeze and Limits of Trigonometric Functions.
2. Make sure to be on time. I won't be giving additional time for those who came in late. (Sorry.)
3. For WFX14, bring a blu
_
M53_LE5_001
Mathematics 53
Fifth Long Examination
I.
Find dy/dx. There is no need to simplify.
Elementary Analysis I
First Semester, AY 2012-2013
1.
2.
(4 points)
(Use logarithmic differentiation.)
3.
II.
III.
(5 points)
Evaluate the following limits.
1
_
M53_LE4_002
Mathematics 53
Fourth Long Examination
Elementary Analysis I
First Semester, AY 2012-2013
I.
Write TRUE if the statement is correct. Otherwise, write FALSE.
[1 point each]
1. All differentiable functions are integrable functions.
2. As the n
Math 53 Exam 4
1.
Find the following anti-derivatives
3 cos( x)
+ 8 5 x dx
(1 sin( x)1 / 2
(a)
cfw_
(b)
2.
(5 pts. each)
2 x + x 4 sec 2 ( x 3 + 15)
dx
7x2
The region R1 is bounded in the second quadrant by the curve
y = x 2 + 1 , the y-axis and the x-axi
Impertinent Premises Examples
Ad Baculum(Appeal to Force)
27
Example: We must believe that the Blessed Virgin, the Mother of God, was
immaculate in her conception because if we dont we are heretics and heretics
will be burned to the stakes.
Reasoning: We
Volume by Slicing
The Mean Value Theorem for Integrals
Mathematics 53
Institute of Mathematics (UP Diliman)
For today
1
Volumes of Solids by Slicing
2
Mean Value Theorem for Integrals
For today
1
Volumes of Solids by Slicing
2
Mean Value Theorem for Integ
Chapter 10: Dynamics of Rotational Motion
Lecture 26: Torque and Rotational Dynamics I
Objective
1. Relate torque to force and angular acceleration of
rigid body.
2. State the consequences of Newtons 2nd law for
rotation under various conditions
Force and
Lecture 37: Archimedes Principle
Lecture Objectives
1. Apply the concept of buoyancy and Archimedes
principle to various systems involving fluids and objects
in fluids and continuity equation.
Physical quantities on fluid statics
=
density
=
pressure
(abs
Collisions & Conservation of momentum
Lecture 23 Objectives
1. Explain the conditions for conservation of linear
momentum.
2. Compare and contrast elastic and inelastic collisions.
3. Solve problems involving systems where linear
momentum is conserved.
La