MATH 141
CALCULUS I
Textbook: Thomas’ CALCULUS, Eleventh Edition, Addison Wesley, 2005
References:
1) Thomas’ CALCULUS, 11-N Editions, (N = 1,2,…10)
2) Apostol T.M. Calculus and Linear Algebra, Wiley, 1967
3) Marsden J, Weinstein A. Calculus Unlimited, Benjamin,1981
4) Dunham W. The Calculus Gallery: Masterpieces from Newton to Lebesgue, 2004
5) Newton I. Philisophiae Naturalis Principia Mathematica (The Mathematical Principles of
Natural Philosophy), London, 1729
6) Silverman R.A. Essential Calculus and Analytic Geometry, Dover, 2003
7) Leibnitz G.W. Oeuvres Mathematiques, Paris, A.Franck, 1853
Syllabus:
Chapter 1: Preliminaries
Chapter 2: Limits and Continuity
Chapter 3: Differentiation
Chapter 4: Applications of Derivatives
Chapter 5: Integration
Chapter 6: Applications of Definite Integrals
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MATH 141
CALCULUS I
Detailed Syllabus
Chapter 1 PRELIMINARIES
Week 1 – Week 2

Real Numbers and the Real Line (Section 1.1)
1- Real Numbers. Real Line
2- Intervals. Absolute Value

Lines, Circles and Parabolas (Section 1.2)
1- Increments and Straight Lines
2- Slope of a Line
3- Distance Between Points. Circle and Parabola.

Functions and Their Graphs (Section 1.3)
1- Functions. Domain and Range
2- Graphs of Functions
3- Piecewise Defined Functions

Identifying Functions (Section 1.4)
1- Linear functions. Power Functions.
2- Polynomial Functions. Rational Functions
3- Even and Odd Functions

Combining Functions (Section 1.5)
1- Composition of Functions
2- Shifting and Scaling a Graph

Trigonometric Functions (Section 1.6)
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Radian Measure
Values of Trigonometric Functions.
Periodicity.
Even and Odd Trigonometric Functions
Graphs of Trigonometric Functions
Trigonometric Identities.
MATH 141
CALCULUS I
Chapter 2 LIMITS AND CONTINUITY
Week 3 – Week 4 – Week 5

Rates of Change and Limits (Section 2.1)
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Average of Instantaneous Speed
Average Rates of Change and Secant Lines
Limits of Functions
Informal Definition of Limit

Calculating Limits. Definition of a Limit. (Sections 2.2 – 2.3)
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The Limit Laws
Limits of Polynomials and Rational Functions.
Eliminating Zero Denominators Algebraically.
The Sandwich Theorem.
Precise Definition of Limit.

One- sided Limits and Limits at Infinity. (Sections 2.4 - 2.5)
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One –sided Limits
Precise Definition of One-sided Limits.
Limits Involving Sin x / x
Finite Limits as x -->±∞
Limits at Infinity of Rational Functions
Horizontal and Vertical Asymptotes
The Sandwich Theorem Revisited
Infinite Limits. Precise Definition. Dominant Terms.

Continuity (Sections 2.6)
1- Continuity at a Point. Definition.
2- Continuity Test. Discontinuities.
3- Continuous Functions. Composites. Continuous Extension to a Point
4- Intermediate Value Theorem for Continuous Functions.

Tangents and Derivatives. (Sections 2.7)
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Tangent to a Curve
Tangent to the Graph of a Function.
Slope of the Curve.
Rates of Change. Derivative at a Point.
MATH 141
CALCULUS I
Chapter 3 DIFFERENTIATION
Week 6 - Week 7 - Week 8 - Week 9

The Derivative as a Function (Sections 3.1, 3.2)
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Definition of Derivative
Differentiation
Graphing the Derivative.
One-sided Derivatives
Continuity of Differentiable Functions.
Differentiation Rules.
Higher-Order Derivatives.

The Derivative as a Rate of Change (Sections 3.3, 3.4)
1- Instantaneous Rates of Changes.
2- Instantaneous Velocity
3- Derivatives of Trigonometric Functions.

The Chain Rule (Section 3.5)
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
Implicit Differentiation. Related Rates (Sections 3.6, 3.7)
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
Derivative of a Composite Function.
The Chain Rule.
Parametric Curve
Slopes of Parametrized Curves.
Implicitly Defined Functions
Implicit Differentiation.
Derivatives of Higher Order.
Applications. Rates Equations
Linearization and Differentials (Section 3.8)
1- Linear Approximation.
2- Differantial.
3- Estimating with Differentials.
MATH 141
CALCULUS I
Chapter 4 APPLICATIONS OF DERIVATIVES
Week 10 – Week 11

Extreme Values of Functions (Section 4.1)
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Absolute Maximum. Absolute Minimum.
The Extreme Value Theorem
Local Maximum. Local Minimum.
Critical Points.

The Mean Value Theorem (Section 4.2)
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Rolle’s Theorem.
The Mean Value Theorem.
Monotonic Functions and the First Derivative Test.
Concavity and Curve Sketching.

Monotonic Functions. First Derivative Test (Section 4.3)
1- Increasing, Decreasing Functions
2- First Derivative Test for Monotonic Functions.
3- First Derivative Test for Local Extrema.

Concavity and Curve Sketching (Section 4.4)
1- Concavity. Points of Inflection
2- Second Derivative Test for Local Extrema.
3- Strategy for Graphing. Learning about Functions from Derivatives.

Applied Optimization Problems (Section 4.5)
1- Examples from Business, Industry and Economics.
2- Examples from Mathematics and Physics.

Indeterminate Forms and L’Hopital’s Rule (Section 4.6)
1- Indeterminate Forms 0/0, ∞/∞, ∞ - ∞, 0 ∞. L’Hopital’s Rule.

Newton’s Method (Section 4.7)
MATH 141
CALCULUS I
Chapter 5 INTEGRATION
Week 12 – Week 13

Antiderivatives (Section 4.8)
1- Antiderivative Definition
2- Antiderivative Formulas. Linearity Rules
3- Initial Value Problems and Differential Equations
4- Indefinite Integral. Definition

Finite Sums (Sections 5.1, 5.2)
1- Estimating with Finite Sums
2- Sigma Notations. Algebra Rules for Finite Sums.
3- Limits of Finite Sums
4- Riemann Sums

The Definite Integral (Section 5.3)
1- Limits of Riemann Sums
2- Definite Integral Definition
3- Integrable and Nonintegrable Functions
4- Properties of Definite Integrals
5- Area Under a Curve as Definite Integral
6- Average Value of Continuous Function

The Fundamental Theorem of Calculus (Section 5.4)
1- Mean Value Theorem for Definite Integrals
2- Fundamental Theorem of Calculus
3- Finding Area Using Antiderivatives

Indefinite Integrals. Substitution Rule (Sections 5.5, 5.6)
1- Substitution Rule
2- Integrals of sin^2 x, cos^2 x
3- Areas Between Curves
MATH 141
CALCULUS I
Chapter 6 APPLICATIONS OF DEFINITE INTEGRALS
Week 14
6.1 Volumes by Slicing and Rotation About an Axis
1. Solids of Revolution: The Disk Method
2. Solids of Revolution: The Washer Method
6.2 Volumes by Cylindrical Shells
6.3 Length of Plane Curves
1. Length of Parametric Curve
6.5 Areas of Surfaces of Revolution
1. Revolution about the x-Axis. Revolution about the y-Axis.
2. Revolution of Parametrized Curves