University Calculus

Table of Contents

1. Functions
   • 1.1 Functions and Their Graphs
   • 1.2 Combining Functions; Shifting and Scaling Graphs
   • 1.3 Trigonometric Functions
   • 1.4 Graphing with Software
   • 1.5 Exponential Functions
   • 1.6 Inverse Functions and Logarithms
2. Limits and Continuity
   • 2.1 Rates of Change and Tangent Lines to Curves
   • 2.2 Limit of a Function and Limit Laws
   • 2.3 The Precise Definition of a Limit
   • 2.4 One-Sided Limits
   • 2.5 Continuity
   • 2.6 Limits Involving Infinity; Asymptotes of Graphs
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
3. Derivatives
   • 3.1 Tangent Lines and the Derivative at a Point
   • 3.2 The Derivative as a Function
   • 3.3 Differentiation Rules
   • 3.4 The Derivative as a Rate of Change
   • 3.5 Derivatives of Trigonometric Functions
   • 3.6 The Chain Rule
   • 3.7 Implicit Differentiation
   • 3.8 Derivatives of Inverse Functions and Logarithms
   • 3.9 Inverse Trigonometric Functions
   • 3.10 Related Rates
   • 3.11 Linearization and Differentials
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
4. Applications of Derivatives
   • 4.1 Extreme Values of Functions on Closed Intervals
   • 4.2 The Mean Value Theorem
   • 4.3 Monotonic Functions and the First Derivative Test
   • 4.4 Concavity and Curve Sketching
   • 4.5 Indeterminate Forms and L'Hopital's Rule
   • 4.6 Applied Optimization
   • 4.7 Newton's Method
   • 4.8 Antiderivatives
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
5. Integrals
   • 5.1 Area and Estimating with Finite Sums
   • 5.2 Sigma Notation and Limits of Finite Sums
   • 5.3 The Definite Integral
   • 5.4 The Fundamental Theorem of Calculus
   • 5.5 Indefinite Integrals and the Substitution Method
   • 5.6 Definite Integral Substitutions and the Area Between Curves
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
6. Applications of Definite Integrals
   • 6.1 Volumes Using Cross-Sections
   • 6.2 Volumes Using Cylindrical Shells
   • 6.3 Arc Length
   • 6.4 Areas of Surfaces of Revolution
   • 6.5 Work
   • 6.6 Moments and Centers of Mass
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
7. Integrals and Transcendental Functions
   • 7.1 The Logarithm Defined as an Integral
   • 7.2 Exponential Change and Separable Differential Equations
   • 7.3 Hyperbolic Functions
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
8. Techniques of Integration
   • 8.1 Integration by Parts
   • 8.2 Trigonometric Integrals
   • 8.3 Trigonometric Substitutions
   • 8.4 Integration of Rational Functions by Partial Fractions
   • 8.5 Integral Tables and Computer Algebra Systems
   • 8.6 Numerical Integration
   • 8.7 Improper Integrals
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
9. Infinite Sequences and Series
   • 9.1 Sequences
   • 9.2 Infinite Series
   • 9.3 The Integral Test
   • 9.4 Comparison Tests
   • 9.5 Absolute Convergence; The Ratio and Root Tests
   • 9.6 Alternating Series and Conditional Convergence
   • 9.7 Power Series
   • 9.8 Taylor and Maclaurin Series
   • 9.9 Convergence of Taylor Series
   • 9.10 Applications of Taylor Series
   • Questions to Guide Your Review
   • Practice Exercises
   • Additional and Advanced Exercises
10. Parametric Equations and Polar Coordinates
    • 10.1 Parametrizations of Plane Curves
    • 10.2 Calculus with Parametric Curves
    • 10.3 Polar Coordinates
    • 10.4 Graphing Polar Coordinate Equations
    • 10.5 Areas and Lengths in Polar Coordinates
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
11. Vectors and the Geometry of Space
    • 11.1 Three-Dimensional Coordinate Systems
    • 11.2 Vectors
    • 11.3 The Dot Product
    • 11.4 The Cross Product
    • 11.5 Lines and Planes in Space
    • 11.6 Cylinders and Quadric Surfaces
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
12. Vector-Valued Functions and Motion in Space
    • 12.1 Curves in Space and Their Tangents
    • 12.2 Integrals of Vector Functions; Projectile Motion
    • 12.3 Arc Length in Space
    • 12.4 Curvature and Normal Vectors of a Curve
    • 12.5 Tangential and Normal Components of Acceleration
    • 12.6 Velocity and Acceleration in Polar Coordinates
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
13. Partial Derivatives
    • 13.1 Functions of Several Variables
    • 13.2 Limits and Continuity in Higher Dimensions
    • 13.3 Partial Derivatives
    • 13.4 The Chain Rule
    • 13.5 Directional Derivatives and Gradient Vectors
    • 13.6 Tangent Planes and Differentials
    • 13.7 Extreme Values and Saddle Points
    • 13.8 Lagrange Multiplier
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
14. Multiple Integrals
    • 14.1 Double and Iterated Integrals over Rectangles
    • 14.2 Double Integrals over General Regions
    • 14.3 Area by Double Integration
    • 14.4 Double Integrals in Polar Form
    • 14.5 Triple Integrals in Rectangular Coordinates
    • 14.6 Applications
    • 14.7 Triple Integrals in Cylindrical and Spherical Coordinates
    • 14.8 Substitutions in Multiple Integrals
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
15. Integrals and Vector Fields
    • 15.1 Line Integrals of Scalar Functions
    • 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
    • 15.3 Path Independence, Conservative Fields, and Potential Functions
    • 15.4 Green's Theorem in the Plane
    • 15.5 Surfaces and Area
    • 15.6 Surface Integrals
    • 15.7 Stokes' Theorem
    • 15.8 The Divergence Theorem and a Unified Theory
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
16. First-Order Differential Equations (online at bit.ly/2pzYlEq)
    • 16.1 Solutions, Slope Fields, and Euler's Method
    • 16.2 First-Order Linear Equations
    • 16.3 Applications
    • 16.4 Graphical Solutions of Autonomous Equations
    • 16.5 Systems of Equations and Phase Planes
17. Second-Order Differential Equations (online at bit.ly/2IHCJyE)
    • 17.1 Second-Order Linear Equations
    • 17.2 Non-homogeneous Linear Equations
    • 17.3 Applications
    • 17.4 Euler Equations
    • 17.5 Power-Series Solutions

Appendix

• A.1 Real Numbers and the Real Line
• A.2 Mathematical Induction
• A.3 Lines and Circles
• A.4 Conic Sections
• A.5 Proofs of Limit Theorems
• A.6 Commonly Occurring Limits
• A.7 Theory of the Real Numbers
• A.8 Complex Numbers
• A.9 The Distributive Law for Vector Cross Products
• A.10 The Mixed Derivative Theorem and the increment Theorem

Additional Topics (online at bit.ly/2IDDl8w)

• B.1 Relative Rates of Growth
• B.2 Probability
• B.3 Conics in Polar Coordinates
• B.4 Taylor's Formula for Two Variables
• B.5 Partial Derivatives with Constrained Variables

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