Math 19AB: Topics

The text for Math 19AB, Calculus for Science. Mathematics, and Engineering, is Calculus, Early Transcendentals, second edition, by Jon Rogawski.
Chapters 2–4, on limits and differentiation are covered in Math 19A; chapters 5–10, on integration and infinite series, are covered in Math 19B.

Chapter 2: Limits

2.1 Limits, Rates of Change, and Tangent Lines
2.2 Limits: A Numerical and Graphical Approach
2.3 Basic Limit Laws
2.4 Limits and Continuity
2.5 Evaluating Limits Algebraically
2.6 Trigonometric Limits
2.7 Limits at Infinity
2.8 Intermediate Value Theorem

Chapter 3: Differentiation

3.1 Definition of the Derivative
3.2 The Derivative as a Function
3.3 Product and Quotient Rules
3.4 Rates of Change
3.5 Higher Derivatives
3.6 Trigonometric Functions
3.7 The Chain Rule
3.8 Derivatives of Inverse Functions
3.9 Derivatives of General Exponential and Logarithmic Functions
3.10 Implicit Differentiation
3.11 Related Rates

Chapter 4: Applications of the Derivative

4.1 Linear Approximation and Applications
4.2 Extreme Values
4.3 The Mean Value Theorem and Monotonicity
4.4 The Shape of a Graph
4.5 L’Hopital’s Rule
4.6 Graph Sketching and Asymptotes
4.7 Applied Optimization
4.8 Newton’s Method

Chapter 5: The Integral

5.1 Approximating and Computing Area
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus, Part I
5.4 The Fundamental Theorem of Calculus, Part II
5.5 Net Change as the Integral of a Rate
5.6 Substitution Method
5.7 Further Transcendental Functions
5.8 Exponential Growth and Decay

Chapter 6: Applications of the Integral

6.1 Area Between Two Curves
6.2 Setting Up Integrals: Volume, Density, Average Value
6.3 Volumes of Revolution
6.4 The Method of Cylindrical Shells
6.5 Work and Energy

Chapter 7: Techniques of Integration

7.1 Integration by Parts
7.2 Trigonometric Integral
7.3 Trigonometric Substitution
7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions
7.5 The Method of Partial Fractions
7.6 Improper Integrals
7.7 Probability and Integration
7.8 Numerical Integration

Chapter 8: Further Applications of the Integral and Taylor Polynomials

8.1 Arc Length and Surface Area
8.2 Fluid Pressure and Force
8.3 Center of Mass
8.4 Taylor Polynomials

Chapter 10: Infinite Series

10.1 Sequences
10.2 Summing an Infinite Series
10.3 Convergence of Series with Positive Terms
10.4 Absolute and Conditional Convergence
10.5 The Ratio and Root Tests
10.6 Power Series
10.7 Taylor Series

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