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Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice—evident in the narrative, the figures, and the questions interspersed in the narrative—is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope. This book is an expanded version of**Calculus: Early Transcendentals**by the same authors, with an entire chapter devoted to differential equations, additional sections on other topics, and additional exercises in most sections.

## Table of Contents

**1. Functions **

1.1 Review of functions

1.2 Representing functions

1.3 Inverse, exponential, and logarithmic functions

1.4 Trigonometric functions and their inverses

**2. Limits**

2.1 The idea of limits

2.2 Definitions of limits

2.3 Techniques for computing limits

2.4 Infinite limits

2.5 Limits at infinity

2.6 Continuity

2.7 Precise definitions of limits

**3. Derivatives**

3.1 Introducing the derivative

3.2 Rules of differentiation

3.3 The product and quotient rules

3.4 Derivatives of trigonometric functions

3.5 Derivatives as rates of change

3.6 The Chain Rule

3.7 Implicit differentiation

3.8 Derivatives of logarithmic and exponential functions

3.9 Derivatives of inverse trigonometric functions

3.10 Related rates

**4. Applications of the Derivative**

4.1 Maxima and minima

4.2 What derivatives tell us

4.3 Graphing functions

4.4 Optimization problems

4.5 Linear approximation and differentials

4.6 Mean Value Theorem

4.7 L’Hôpital’s Rule

4.8 Newton’s Method

4.9 Antiderivatives

**5. Integration**

5.1 Approximating areas under curves

5.2 Definite integrals

5.3 Fundamental Theorem of Calculus

5.4 Working with integrals

5.5 Substitution rule

**6. Applications of Integration**

6.1 Velocity and net change

6.2 Regions between curves

6.3 Volume by slicing

6.4 Volume by shells

6.5 Length of curves

6.6 Surface area

6.7 Physical applications

6.8 Logarithmic and exponential functions revisited

6.9 Exponential models

6.10 Hyperbolic functions

**7. Integration Techniques**

7.1 Integration Strategies

7.2 Integration by parts

7.3 Trigonometric integrals

7.4 Trigonometric substitutions

7.5 Partial fractions

7.6 Other integration strategies

7.7 Numerical integration

7.8 Improper integrals

**8. Differential Equations **

8.1 Basic ideas

8.2 Direction fields and Euler’s method

8.3 Separable differential equations

8.4 Special first-order differential equations

8.5 Modeling with differential equations

**9. Sequences and Infinite Series**

9.1 An overview

9.2 Sequences

9.3 Infinite series

9.4 The Divergence and Integral Tests

9.5 The Ratio, Root, and Comparison Tests

9.6 Alternating series

**10. Power Series**

10.1 Approximating functions with polynomials

10.2 Properties of Power series

10.3 Taylor series

10.4 Working with Taylor series

**11. Parametric and Polar Curves **

11.1 Parametric equations

11.2 Polar coordinates

11.3 Calculus in polar coordinates

11.4 Conic sections

**12. Vectors and Vector-Valued Functions**

12.1 Vectors in the plane

12.2 Vectors in three dimensions

12.3 Dot products

12.4 Cross products

12.5 Lines and curves in space

12.6 Calculus of vector-valued functions

12.7 Motion in space

12.8 Length of curves

12.9 Curvature and normal vectors

**13. Functions of Several Variables**

13.1 Planes and surfaces

13.2 Graphs and level curves

13.3 Limits and continuity

13.4 Partial derivatives

13.5 The Chain Rule

13.6 Directional derivatives and the gradient

13.7 Tangent planes and linear approximation

13.8 Maximum/minimum problems

13.9 Lagrange multipliers

**14. Multiple Integration**

14.1 Double integrals over rectangular regions

14.2 Double integrals over general regions

14.3 Double integrals in polar coordinates

14.4 Triple integrals

14.5 Triple integrals in cylindrical and spherical coordinates

14.6 Integrals for mass calculations

14.7 Change of variables in multiple integrals

**15. Vector Calculus**

15.1 Vector fields

15.2 Line integrals

15.3 Conservative vector fields

15.4 Green’s theorem

15.5 Divergence and curl

15.6 Surface integrals

15.6 Stokes’ theorem

15.8 Divergence theorem

* *

Appendix A. Algebra Review

Appendix B. Proofs of Selected Theorems
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