Unit 2952: Calculus Foundations and Applications

1. Introduction to Limits

1.1 Concept of Limits

• Intuitive understanding of limits
• Formal (ε-δ) definition (brief overview)

1.2 Methods for Finding Limits

• Algebraic methods (factoring, rationalizing, substitution)
• Graphical interpretation of limits
• Numerical approach using tables of values

1.3 Special Limits and Limit Laws

• Limits involving infinity
• One-sided limits
• Limit laws and properties

2. Continuity

2.1 Definition of Continuity at a Point

• Continuity vs. discontinuity
• Types of discontinuities: removable, jump, infinite

2.2 Continuity on an Interval

• Continuous functions on closed and open intervals
• Intermediate Value Theorem

3. Differentiation

3.1 Introduction to Derivatives

• Concept of the derivative as a rate of change and slope of tangent
• Definition of the derivative using limits

3.2 Differentiation Rules

• Power rule
• Sum and difference rules
• Product and quotient rules
• Chain rule

3.3 Derivatives of Common Functions

• Polynomials, exponentials, logarithms, trigonometric functions

4. Applications of Derivatives

4.1 Rates of Change

• Velocity and acceleration examples

4.2 Optimization Problems

• Local maxima and minima
• Critical points and second derivative test

4.3 Related Rates

• Solving problems involving related quantities changing with time

4.4 Curve Sketching

• Using first and second derivatives to analyze function behavior
• Identifying intervals of increase/decrease and concavity

5. Integration

5.1 Concept of Integration

• Antiderivatives and indefinite integrals
• Notation and basic properties

5.2 Definite Integrals

• Area under a curve
• Properties of definite integrals

6. Fundamental Theorem of Calculus

6.1 Statement and Explanation

• Connection between differentiation and integration

6.2 Practical Applications

• Calculating definite integrals using antiderivatives

7. Techniques of Integration

7.1 Integration by Substitution

• Method and examples

7.2 Integration by Parts

• Formula and applications

7.3 Trigonometric Integrals

• Integrals involving sine, cosine, and other trig functions

8. Applications of Integrals

8.1 Area Between Curves

• Setting up and computing area

8.2 Volumes of Revolution

• Disk and shell methods

8.3 Work Done by a Variable Force

• Calculating work using integrals

8.4 Other Real-World Applications

• Examples from physics, engineering, and economics

9. Differential Equations

9.1 Introduction to Differential Equations

• Definition and terminology

9.2 Solving First-Order Ordinary Differential Equations

• Separation of variables
• Initial value problems

10. Taylor Series

10.1 Infinite Series Representation of Functions

• Concept of approximating functions by polynomials

10.2 Maclaurin Series

• Special case of Taylor series at zero

10.3 Applications of Taylor Series

• Approximations, error estimation