Calculus for Scientists and Engineers

William Briggs has been on the mathematics faculty at the University of Colorado at Denver for twenty-three years. He received his BA in mathematics from the University of Colorado and his MS and PhD in applied mathematics from Harvard University. He teaches undergraduate and graduate courses throughout the mathematics curriculum with a special interest in mathematical modeling and differential equations as it applies to problems in the biosciences. He has written a quantitative reasoning textbook, Using and Understanding Mathematics; an undergraduate problem solving book, Ants, Bikes, and Clocks; and two tutorial monographs, The Multigrid Tutorial and The DFT: An Owner’s Manual for the Discrete Fourier Transform. He is the Society for Industrial and Applied Mathematics (SIAM) Vice President for Education, a University of Colorado President’s Teaching Scholar, a recipient of the Outstanding Teacher Award of the Rocky Mountain Section of the Mathematical Association of America (MAA), and the recipient of a Fulbright Fellowship to Ireland.  Lyle Cochran is a professor of mathematics at Whitworth University in Spokane, Washington. He holds BS degrees in mathematics and mathematics education from Oregon State University and a MS and PhD in mathematics from Washington State University. He has taught a wide variety of undergraduate mathematics courses at Washington State University, Fresno Pacific University, and, since 1995, at Whitworth University. His expertise is in mathematical analysis, and he has a special interest in the integration of technology and mathematics education. He has written technology materials for leading calculus and linear algebra textbooks including the Instructor’s Mathematica Manual for Linear Algebra and Its Applications by David C. Lay and the Mathematica Technology Resource Manual for Thomas’ Calculus. He is a member of the MAA and a former chair of the Department of Mathematics and Computer Science at Whitworth University.  Bernard Gillett is a Senior Instructor at the University of Colorado at Boulder; his primary focus is undergraduate education. He has taught a wide variety of mathematics courses over a twenty-year career, receiving five teaching awards in that time. Bernard authored a software package for algebra, trigonometry, and precalculus; the Student’s Guide and Solutions Manual and the Instructor’s Guide and Solutions Manual for Using and Understanding Mathematics by Briggs and Bennett; and the Instructor’s Resource Guide and Test Bank for Calculus and Calculus: Early Transcendentals by Briggs, Cochran, and Gillett. Bernard is also an avid rock climber and has published four climbing guides for the mountains in and surrounding Rocky Mountain National Park.

• 1. Functions 1.1 Review of functions 1.2 Representing functions1.3 Inverse, exponential, and logarithmic functions1.4 Trigonometric functions and their inverses  2. Limits2.1 The idea of limits2.2 Definitions of limits2.3 Techniques for computing limits 2.4 Infinite limits 2.5 Limits at infinity 2.6 Continuity2.7 Precise definitions of limits  3. Derivatives3.1 Introducing the derivative3.2 Rules of differentiation3.3 The product and quotient rules3.4 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 functions3.9 Derivatives of inverse trigonometric functions3.10 Related rates  4. Applications of the Derivative4.1 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 Theorem4.7 L’Hôpital’s Rule 4.8 Newton’s Method 4.9 Antiderivatives  5. Integration5.1 Approximating areas under curves5.2 Definite integrals 5.3 Fundamental Theorem of Calculus 5.4 Working with integrals 5.5 Substitution rule  6. Applications of Integration6.1 Velocity and net change6.2 Regions between curves6.3 Volume by slicing6.4 Volume by shells6.5 Length of curves6.6 Surface area6.7 Physical applications6.8 Logarithmic and exponential functions revisited6.9 Exponential models6.10 Hyperbolic functions  7. Integration Techniques7.1 Integration Strategies7.2 Integration by parts 7.3 Trigonometric integrals 7.4 Trigonometric substitutions7.5 Partial fractions 7.6 Other integration strategies7.7 Numerical integration7.8 Improper integrals  8. Differential Equations 8.1 Basic ideas 8.2 Direction fields and Euler’s method8.3 Separable differential equations 8.4 Special first-order differential equations8.5 Modeling with differential equations  9. Sequences and Infinite Series9.1 An overview 9.2 Sequences9.3 Infinite series 9.4 The Divergence and Integral Tests9.5 The Ratio, Root, and Comparison Tests9.6 Alternating series  10. Power Series10.1 Approximating functions with polynomials10.2 Properties of Power series10.3 Taylor series10.4 Working with Taylor series  11. Parametric and Polar Curves 11.1 Parametric equations11.2 Polar coordinates 11.3 Calculus in polar coordinates 11.4 Conic sections  12. Vectors and Vector-Valued Functions12.1 Vectors in the plane12.2 Vectors in three dimensions12.3 Dot products12.4 Cross products12.5 Lines and curves in space 12.6 Calculus of vector-valued functions 12.7 Motion in space12.8 Length of curves12.9 Curvature and normal vectors  13. Functions of Several Variables13.1 Planes and surfaces13.2 Graphs and level curves13.3 Limits and continuity13.4 Partial derivatives13.5 The Chain Rule 13.6 Directional derivatives and the gradient13.7 Tangent planes and linear approximation13.8 Maximum/minimum problems13.9 Lagrange multipliers  14. Multiple Integration14.1 Double integrals over rectangular regions14.2 Double integrals over general regions14.3 Double integrals in polar coordinates14.4 Triple integrals14.5 Triple integrals in cylindrical and spherical coordinates14.6 Integrals for mass calculations14.7 Change of variables in multiple integrals  15. Vector Calculus15.1 Vector fields15.2 Line integrals15.3 Conservative vector fields15.4 Green’s theorem15.5 Divergence and curl15.6 Surface integrals15.6 Stokes’ theorem15.8 Divergence theorem  Appendix A. Algebra ReviewAppendix B. Proofs of Selected Theorems