附註:Includes bibliographical references (page 263) and index.
1. Basics -- 1.0. Introductory remarks -- 1.1. Number systems -- 1.2. Coordinates in one dimension -- 1.3. Coordinates in two dimensions -- 1.4. The slope of a line in the plane -- 1.5. The equation of a line -- 1.6. Loci in the plane -- 1.7. Trigonometry -- 1.8. Sets and functions -- 1.8.1. Examples of functions of a real variable -- 1.8.2. Graphs of functions -- 1.8.3. Plotting the graph of a function -- 1.8.4. Composition of functions -- 1.8.5. The inverse of a function -- 1.9. A few words about logarithms and exponentials -- 2. Foundations of calculus -- 2.1. Limits -- 2.1.1. One-sided limits -- 2.2. Properties of limits -- 2.3. Continuity -- 2.4. The derivative -- 2.5. Rules for calculating derivatives -- 2.5.1. The derivative of an inverse -- 2.6. The derivative as a rate of change -- 3. Applications of the derivative -- 3.1. Graphing of functions -- 3.2. Maximum/minimum problems -- 3.3. Related rates -- 3.4. Falling bodies -- 4. The integral -- 4.0. Introduction -- 4.1. Antiderivatives and indefinite integrals -- 4.1.1. The concept of antiderivative -- 4.1.2. The indefinite integral -- 4.2. Area -- 4.3. Signed area -- 4.4. The area between two curves -- 4.5. Rules of integration -- 4.5.1. Linear properties -- 4.5.2. Additivity -- 5. Indeterminate forms -- 5.1. l'Hôpital's rule -- 5.1.1. Introduction -- 5.1.2. l'Hôpital's rule -- 5.2. Other indeterminate forms -- 5.2.1. Introduction -- 5.2.2. Writing a product as a quotient -- 5.2.3. The use of the logarithm -- 5.2.4. Putting terms over a common denominator -- 5.2.5. Other algebraic manipulations -- 5.3. Improper integrals : a first look -- 5.3.1. Introduction -- 5.3.2. Integrals with infinite integrands -- 5.3.3. An application to area -- 5.4. More on improper integrals -- 5.4.1. Introduction -- 5.4.2. The integral on an infinite interval -- 5.4.3. Some applications.
6. Transcendental functions -- 6.0. Introductory remarks -- 6.1. Logarithm basics -- 6.1.1. A new approach to logarithms -- 6.1.2. The logarithm function and the derivative -- 6.2. Exponential basics -- 6.2.1. Facts about the exponential function -- 6.2.2. Calculus properties of the exponential -- 6.2.3. The number e -- 6.3. Exponentials with arbitrary bases -- 6.3.1. Arbitrary powers -- 6.3.2. Logarithms with arbitrary bases -- 6.4. Calculus with logs and exponentials to arbitrary bases -- 6.4.1. Differentiation and integration of loga x and ax -- 6.4.2. Graphing of logarithmic and exponential functions -- 6.4.3. Logarithmic differentiation -- 6.5. Exponential growth and decay -- 6.5.1. A differential equation -- 6.5.2. Bacterial growth -- 6.5.3. Radioactive decay -- 6.5.4. Compound interest -- 6.6. Inverse trigonometric functions -- 6.6.1. Introductory remarks -- 6.6.2. Inverse sine and cosine -- 6.6.3. The inverse tangent function -- 6.6.4. Integrals in which inverse trigonometric functions arise -- 6.6.5. Other inverse trigonometric functions -- 6.6.6. An example involving inverse trigonometric functions -- 7. Methods of integration -- 7.1. Integration by parts -- 7.2. Partial fractions -- 7.2.1. Introductory remarks -- 7.2.2. Products of linear factors -- 7.2.3. Quadratic factors -- 7.3. Substitution -- 7.4. Integrals of trigonometric expressions -- 8. Applications of the integral -- 8.1. Volumes by slicing -- 8.1.0. Introduction -- 8.1.1. The basic strategy -- 8.1.2. Examples -- 8.2. Volumes of solids and revolution -- 8.2.0. Introduction -- 8.2.1. The method of washers -- 8.2.2. The method of cylindrical shells -- 8.2.3. Different axes -- 8.3. Work -- 8.4. Averages -- 8.5. Arc length and surface area -- 8.5.1. Arc length -- 8.5.2. Surface area -- 8.6. Hydrostatic pressure -- 8.7. Numerical methods of integration -- 8.7.1. The trapezoid rule -- 8.7.2. Simpson's rule.
摘要:Explains how to understand calculus in a more intuitive fashion. Uses practical examples and real data. Covers both differential and integral calculus.
