A. Diagnostic Test: Algebra
B. Diagnostic Test: Analytic Geometry
C. Diagnostic Test: Functions
D. Diagnostic Test: Trigonometry
1. Functions and Limits
1.1. Functions and Their Representations
1.1.1. Representations of Functions
1.1.2. Piecewise Defined Functions
1.1.3. Symmetry
1.1.4. Increasing and Decreasing Functions
1.1. Exercises
1.2. A Catalog of Essential Functions
1.2.1. Mathematical Modeling
1.2.2. Transformations of Functions
1.2.3. Combinations of Functions
1.2. Exercises
1.3. The Limit of a Function
1.3.1. Intuitive Definition of a Limit
1.3.2. One-Sided Limits
1.3.3. Precise Definition of a Limit
1.3. Exercises
1.4. Calculating Limits
1.4. Exercises
1.5. Continuity
1.5. Exercises
1.6. Limits Involving Infinity
1.6.1. Infinite Limits
1.6.2. Limits at Infinity
1.6.3. Infinite Limits at Infinity
1.6.4. Precise Definitions
1.6. Exercises
Chapter 1. Review
2. Derivatives
2.1. Derivatives and Rates of Change
2.1.1. The Tangent Problem
2.1.2. The Velocity Problem
2.1.3. Derivatives
2.1.4. Rates of Change
2.1. Exercises
2.2. The Derivative as a Function
2.2.1. Other Notations
2.2.2. Differentiable Functions
2.2.3. How Can a Function Fail to Be Differentiable?
2.2.4. Higher Derivatives
2.2. Exercises
2.3. Basic Differentiation Formulas
2.3.1. Power Functions
2.3.2. New Derivatives from Old
2.3.3. The Sine and Cosine Functions
2.3.4. Applications to Rates of Change
2.3. Exercises
2.4. The Product and Quotient Rules
2.4.1. The Product Rule
2.4.2. The Quotient Rule
2.4.3. Trigonometric Functions
2.4. Exercises
2.5. The Chain Rule
2.5.1. How to Prove the Chain Rule
2.5. Exercises
2.6. Implicit Differentiation
2.6. Exercises
2.7. Related Rates
2.7. Exercises
2.8. Linear Approximations and Differentials
2.8.1. Applications to Physics
2.8.2. Differentials
2.8. Exercises
Chapter 2. Review
3. Applications of Differentiation
3.1. Maximum and Minimum Values
3.1. Exercises
3.2. The Mean Value Theorem
3.2. Exercises
3.3. Derivatives and the Shapes of Graphs
3.3.1. What Does f ′ Say about f?
3.3.2. What Does f ″ Say about f?
3.3. Exercises
3.4. Curve Sketching
3.4.1. Guidelines for Sketching a Curve
3.4.2. Graphing with Technology
3.4. Exercises
3.5. Optimization Problems
3.5.1. Applications to Business and Economics
3.5. Exercises
3.6. Newton’s Method
3.6. Exercises
3.7. Antiderivatives
3.7.1. Rectilinear Motion
3.7. Exercises
Chapter 3. Review
4. Integrals
4.1. Areas and Distances
4.1.1. The Area Problem
4.1.2. The Distance Problem
4.1. Exercises
4.2. The Definite Integral
4.2.1. Evaluating Integrals
4.2.2. The Midpoint Rule
4.2.3. Properties of the Definite Integral
4.2. Exercises
4.3. Evaluating Definite Integrals
4.3.1. Indefinite Integrals
4.3.2. Applications
4.3. Exercises
4.4. The Fundamental Theorem of Calculus
4.4.1. Differentiation and Integration as Inverse Processes
4.4.2. Average Value of a Function
4.4. Exercises
4.5. The Substitution Rule
4.5.1. Definite Integrals
4.5.2. Symmetry
4.5. Exercises
Chapter 4. Review
5. Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions
5.1. Inverse Functions
5.1.1. The Calculus of Inverse Functions
5.1. Exercises
5.2. The Natural Logarithmic Function
5.2.1. Logarithmic Differentiation
5.2. Exercises
5.3. The Natural Exponential Function
5.3.1. Differentiation
5.3.2. Integration
5.3. Exercises
5.4. General Logarithmic and Exponential Functions
5.4.1. General Exponential Functions
5.4.2. Exponential Graphs
5.4.3. Exponential Integrals
5.4.4. The Power Rule Revisited
5.4.5. General Logarithmic Functions
5.4.6. The Number e as a Limit
5.4. Exercises
5.5. Exponential Growth and Decay
5.5.1. Population Growth
5.5.2. Radioactive Decay
5.5.3. Newton’s Law of Cooling
5.5.4. Continuously Compounded Interest
5.5. Exercises
5.6. Inverse Trigonometric Functions
5.6. Exercises
5.7. Hyperbolic Functions
5.7.1. Inverse Hyperbolic Functions
5.7. Exercises
5.8. Indeterminate Forms and l’Hospital’s Rule
5.8.1. Indeterminate Products
5.8.2. Indeterminate Differences
5.8.3. Indeterminate Powers
5.8. Exercises
Chapter 5. Review
6. Techniques of Integration
6.1. Integration by Parts
6.1. Exercises
6.2. Trigonometric Integrals and Substitutions
6.2.1. Trigonometric Integrals
6.2.2. Trigonometric Substitutions
6.2. Exercises
6.3. Partial Fractions
6.3. Exercises
6.4. Integration with Tables and Computer Algebra Systems
6.4.1. Tables of Integrals
6.4.2. Computer Algebra Systems
6.4.3. Can We Integrate All Continuous Functions?
6.4. Exercises
6.5. Approximate Integration
6.5.1. Simpson’s Rule
6.5. Exercises
6.6. Improper Integrals
6.6.1. Type 1: Infinite Intervals
6.6.2. Type 2: Discontinuous Integrands
6.6.3. A Comparison Test for Improper Integrals
6.6. Exercises
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