Calculus I with Precalculus: A One-Year Course Book

Note: Each

Chapter concludes with Problem Solving. P. Prerequisites P.1 Solving Equations Section Project: Projectile Motion P.2 Solving Inequalities P.3 Graphical Representation of Data P.4 Graphs of Equations P.5 Linear Equations in Two Variables
1.Functions and Their Graphs
1.1 Functions
1.2 Analyzing Graphs of Functions
1.3 Shifting, Reflecting, and Stretching Graphs
1.4 Combinations of Functions
1.5 Inverse Functions
1.6 Mathematical Modeling Section Project: Hooke's Law
2.Polynomial and Rational Functions
2.1 Quadratic Functions
2.2 Polynomial Functions of Higher Degree
2.3 Polynomial and Synthetic Division
2.4 Complex Numbers Section Project: The Mandelbrot Set
2.5 The Fundamental Theorem of Algebra
2.6 Rational Functions Section Project: Rational Functions
3.Limits and Their Properties
3.1 A Preview of Calculus
3.2 Finding Limits Graphically and Numerically
3.3 Evaluating Limits Analytically
3.4 Continuity and One-Sided Limits
3.5 Infinite Limits Section Project: Graphs and Limits of Functions Progressive Summary 1: Flowchart of Calculus
4.Differentiation
4.1 The Derivative and the Tangent Line Problem
4.2 Basic Differentiation Rules and Rates of Change
4.3 The Product and Quotient Rules and Higher-Order Derivatives
4.4 The Chain Rule
4.5 Implicit Differentiation Section Project: Optical Illusions
4.6 Related Rates
5.Applications of Differentiation
5.1 Extrema on an Interval
5.2 Rolle's Theorem and the Mean Value Theorem
5.3 Increasing and Decreasing Functions and the First Derivative Test
5.4 Concavity and the Second DerivativeTest
5.5 Limits at Infinity
5.6 A Summary of Curve Sketching
5.7 Optimization Problems Section Project: Connecticut River
5.8 Differentials
6.Integration
6.1 Antiderivatives and Indefinite Integration
6.2 Area
6.3 Riemann Sums and Definite Integrals
6.4 The Fundamental Theorem of Calculus Section Project: Demonstrating the Fundamental Theorem
6.5 Integration by Substitution
6.6 Numerical Integration Progressive Summary 2: Flowchart of Calculus
7.Exponential and Logarithmic Functions
7.1 Exponential Functions and Their Graphs
7.2 Logarithmic Functions and Their Graphs
7.3 Using Properties of Logarithms
7.4 Exponential and Logarithmic Equations
7.5 Exponential and Logarithmic Models Section Project: Comparing Models
8.Exponential and Logarithmic Functions and Calculus
8.1 Exponential Functions: Differentiation and Integration
8.2 Logarithmic Functions and Differentiation Section Project: An Alternate Definition of ln x
8.3 Logarithmic Functions and Integration
8.4 Differential Equations: Growth and Decay Progressive Summary 3: Flowchart of Calculus
9.Trigonometric Functions
9.1 Radian and Degree Measure
9.2 Trigonometric Functions: The Unit Circle
9.3 Right Triangle Trigonometry
9.4 Trigonometric Functions of Any Angle
9.5 Graphs of Sine and Cosine Functions Section Project: Approximating Sine and Cosine Functions
9.6 Graphs of Other Trigonometric Functions
9.7 Inverse Trigonometric Functions
9.8 Applications and Models
10.Analytic Trigonometry
10.1 Using Fundamental Trigonometric Identities
10.2 Verifying Trigonometric Identities
10.3 Solving Trigonometric Equations Section Project: Modeling a Sound Wave
10.4 Sum and Difference Formulas
10.5 Multiple-Angle and Product-to-Sum Formulas
11.Trigonometric Functions and Calculus
11.1 Limits of Trigonometric Functions Section Project: Graphs and Limits of Trigonometric Functions
11.2 Trigonometric Functions: Differentiation
11.3 Trigonometric Functions: Integration
11.4 Inverse Trigonometric Functions: Differentiation
11.5 Inverse Trigonometric Functions: Integration
11.6 Hyperbolic Functions Section Project: St. Louis Arch Progressive Summary 4: Flowchart of Calculus
12.Topics in Analytic Geometry
12.1 Introduction to Conics: Parabolas
12.2 Ellipses and Implicit Differentiation
12.3 Hyperbolas and Implicit Differentiation
12.4 Parametric Equations and Calculus
12.5 Polar Coordinates and Calculus
12.6 Graphs of Polar Equations
12.7 Polar Equations of Conics Section Project: Polar Equations of Planetary Orbits Progressive Summary 5: Flowchart of Calculus
13.Additional Topics in Trigonometry
13.1 Law of Sines
13.2 Law of Cosines
13.3 Vectors in the Plane Section Project: Adding Vectors Graphically
13.4 Vectors and Dot Products
13.5 Trigonometric Form of a Complex Number
14.Systems of Equations and Matrices
14.1 Systems of Linear Equations in Two Variables
14.2 Multivariable Linear Systems
14.3 Systems of Inequalities Section Project: Area Bounded by Concentric Circles
14.4 Matrices and Systems of Equations
14.5 Operations with Matrices
14.6 The Inverse of a Square Matrix
14.7 The Determinant of a Square Matrix Section Project: Cramer's Rule Appendices A. Proofs of Selected Theorems B. Applications of Integration C. Study Capsules
1.Graphing Algebraic Functions
2.Limits of Algebraic Functions
3.Differentiation of Algebraic Functions
4.Calculus of Algebraic Functions
5.Calculus of Exponential and Logarithmic Functions
6.Trigonometric Functions
7.Calculus of Trigonometric and Inverse Trigonometric Functions