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.
2.
3.
4.
II. Given: (12 points)
Discuss the

Indeterminate Forms and LHpitals Rule
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Indeterminate Forms and LHpitals Rule
Mathematics 53
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For today
0
and
0
1
Indeterminate Forms of Type
2
LHpitals Rule
3

Derivatives of and Integrals Yielding Inverse Circular
Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Derivatives of and Integrals Yielding Inverse Circular Functions
Mathematics 53
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For today
1

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
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For today
1
and of the other

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
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For today
1
Exponential and Logarith

Related Rates, Local Linear Approximation and
Dierentials
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman) Related Rates, Local Linear Approximation and Dierentials
Mathematics 53
1 / 37
For today
1
Related Rates

The Chain Rule, Implicit Dierentiation,
Higher Order Derivatives
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman) Chain Rule, Implicit Di., Higher Order
Mathematics 53
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For today
1
The Chain Rule
2
Implicit

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
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For today
1
Rates of Change
2
Rectilinear Motion
Institute of Ma

Slopes and the Derivative
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Slopes and the Derivative
Mathematics 53
1 / 43
For today
1
The Tangent Line
2
Denition of the Derivative
3
Techniques of Dierentiation
4

Applications of the Denite Integral
Volumes of Solids of Revolution
Mathematics 53
Institute of Mathematics - UP Diliman
7 September 2012
Mathematics 53 (Part 4)
Arc Length and Volumes by Slicing
7 September 2012
1 / 64
For today
1
Solids of Revolution: D

Applications of the Denite Integral
Arc Length, Volumes by Slicing,
Solids of Revolution: Disk or Washer Method
Mathematics 53
Institute of Mathematics - UP Diliman
5 September 2012
Mathematics 53 (Part 4)
Arc Length and Volumes by Slicing
5 September 201

Applications of the Denite Integral
The Area of a Plane Region
Mathematics 53
Institute of Mathematics - UP Diliman
1 September 2012
Mathematics 53 (Part 4)
Area of a Plane Region
1 September 2012
1 / 59
Review of Parabolas and Circles
Parabolas opening u

The Denite Integral and
the Fundamental Theorem of Calculus
Mathematics 53
Part 4
Institute of Mathematics
Mathematics 53 (Part 4)
The Denite Integral
Institute of Mathematics
1 / 51
Outline
1
The Denite Integral
2
Mean-Value Theorem for Integrals
3
The F

Integration by Substitution
Introduction to the Area of a Plane Region
Mathematics 53
Institute of Mathematics - UP Diliman
24 August 2012
Math 53 (Part 4)
Substitution and Area
24 August 2012
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Outline
1
Integration by Substitution
2
Introduction th

Antidierentiation and the Indenite Integral
Part 4: Integration and Its Applications
Mathematics 53
Institute of Mathematics - UP Diliman
22 August 2012
Math 53 (Part 4)
The Indenite Integral
22 August 2012
1 / 32
Outline
1
Antiderivatives or Indenite Int

Applications of Absolute Extrema: Optimization
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.5
Optimization Problems
Optimization Problems
maximizing/minimizing a function f on a closed and nite interval
Optimization Problems
maximizing/m

Absolute Extrema of a Function
Mathematics 53
Institute of Mathematics (UP Diliman)
Lecture 3.4
For today
1
Denitions and Examples
2
Absolute Extrema On Finite Closed Intervals
3
Absolute Extrema Of Functions with One Relative Extremum
4
On Innite or Open

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
Asymptotes
For today
1
Concavity and Points of Inection
2
The Second Deri

Techniques of Dierentiation
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Techniques of Dierentiation
Mathematics 53
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For today
1
Dierentiation Rules
2
The Chain Rule
Institute of Mathematics (UP Diliman)

Limits at Innity and
Continuity at a Point
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
Limits at Innity and Continuity at a Point
Mathematics 53
1 / 60
For today
1
Limits at Innity
2
Continuity of Functions
I

One-sided limits and
Innite Limits
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman)
One-sided limits and Innite Limits
Mathematics 53
1 / 49
For today
1
One-Sided Limits (continuation)
2
Innite Limits
Institute of