Calculus for Scientists and Engineers

1. Functions 1.1 Review of functions 1.2 Representing functions 1.3 Trigonometric functions and their inverses Review 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 Review 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 inverse trigonometric functions 3.9 Related rates Review 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'Hopital's Rule 4.8 Newton's method 4.9 Antiderivatives Review 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 Review 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 Hyperbolic functions Review 7. Logarithmic and Exponential Functions 7.1 Inverse functions 7.2 The natural logarithm and exponential functions 7.3 Logarithmic and exponential functions with general bases 7.4 Exponential models 7.5 Inverse trigonometric functions 7.6 L'Hopital's rule and growth rates of functions Review 8. Integration Techniques 8.1 Basic approaches 8.2 Integration by parts 8.3 Trigonometric integrals 8.4 Trigonometric substitutions 8.5 Partial fractions 8.6 Other integration strategies 8.7 Numerical integration 8.8 Improper integrals Review 9. Differential Equations 9.1 Basic ideas 9.2 Direction fields and Euler's method 9.3 Separable differential equations 9.4 Special first-order differential equations 9.5 Modeling with differential equations Review 10. Sequences and Infinite Series 10.1 An overview 10.2 Sequences 10.3 Infinite series 10.4 The Divergence and Integral Tests 10.5 The Ratio, Root, and Comparison Tests 10.6 Alternating series Review 11. Power Series 11.1 Approximating functions with polynomials 11.2 Properties of power series 11.3 Taylor series 11.4 Working with Taylor series Review 12. Parametric and Polar Curves 12.1 Parametric equations 12.2 Polar coordinates 12.3 Calculus in polar coordinates 12.4 Conic sections Review 13. Vectors and Vector-Valued Functions 13.1 Vectors in the plane 13.2 Vectors in three dimensions 13.3 Dot products 13.4 Cross products 13.5 Lines and curves in space 13.6 Calculus of vector-valued functions 13.7 Motion in space 13.8 Length of curves 13.9 Curvature and normal vectors Review 14. Functions of Several Variables 14.1 Planes and surfaces 14.2 Graphs and level curves 14.3 Limits and continuity 14.4 Partial derivatives 14.5 The Chain Rule 14.6 Directional derivatives and the gradient 14.7 Tangent planes and linear approximation 14.8 Maximum/minimum problems 14.9 Lagrange multipliers Review 15. Multiple Integration 15.1 Double integrals over rectangular regions 15.2 Double integrals over general regions 15.3 Double integrals in polar coordinates 15.4 Triple integrals 15.5 Triple integrals in cylindrical and spherical coordinates 15.6 Integrals for mass calculations 15.7 Change of variables in multiple integrals Review 16. Vector Calculus 16.1 Vector fields 16.2 Line integrals 16.3 Conservative vector fields 16.4 Green's theorem 16.5. Divergence and curl 16.6 Surface integrals 16.6 Stokes' theorem 16.8 Divergence theorem Review