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出版者:Thomson Learning

作者:Kaufmann, Jerome E./ Schwitters, Karen L.

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页数:720

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出版时间:2006-1

价格:$ 192.04

装帧:HRD

isbn号码:9780495105527

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• 代数
• 中级代数
• 数学
• 高等教育
• 教材
• 学习
• 方程
• 函数
• 多项式
• 因式分解

Calculus: A Comprehensive Introduction A Thorough Exploration of Fundamental Concepts and Advanced Applications This text is designed to provide a rigorous yet accessible foundation in calculus, suitable for students embarking on their first serious encounter with the subject. It moves beyond rote memorization of formulas, focusing instead on the conceptual underpinnings that drive mathematical analysis and its application across the sciences and engineering. The book is structured into three major parts: Differential Calculus, Integral Calculus, and Infinite Series, with an introductory section dedicated to essential pre-calculus review to ensure a solid starting point for all readers. --- Part I: Foundations and Differential Calculus Chapter 1: Prerequisites and the Concept of a Limit This chapter sets the stage by reviewing the necessary algebraic manipulations, functions, and graphical analysis skills. The central theme, however, is the introduction of the limit. We meticulously define the formal $(epsilon, delta)$ definition of a limit, demonstrating its power in precisely describing the behavior of functions near specific points. Continuity is defined through the language of limits, exploring both intuitive and formal interpretations. The concept of one-sided limits and limits involving infinity are handled with precision, preparing the ground for the derivative. Chapter 2: The Derivative The derivative is introduced first through the geometric problem of finding tangent lines and then through the physical interpretation of instantaneous rate of change. We build the fundamental differentiation rules systematically—the constant rule, power rule, constant multiple rule, and sum/difference rule—proving each one from the limit definition. Subsequent sections delve into the product rule and quotient rule. A dedicated section covers the Chain Rule, emphasizing its importance as a composition tool. We explore implicit differentiation, laying the groundwork for related rates problems, a cornerstone application in physics and engineering. Chapter 3: Applications of the Derivative This section moves from computation to application. We explore how the first derivative reveals information about a function's increasing/decreasing behavior and locates local extrema (Maxima and Minima). The Second Derivative Test is introduced to classify these extrema and analyze concavity. The chapter culminates in a comprehensive treatment of curve sketching, incorporating all discovered information to produce accurate graphical representations of complex functions. Optimization problems, ranging from simple geometric maximization to complex constraint-based scenarios, are solved using differential techniques. Finally, L'Hôpital's Rule is introduced as a powerful tool for resolving indeterminate forms arising from limits, directly connecting back to the core concept of the derivative. Chapter 4: Transcendental Functions Exponential and logarithmic functions are treated not merely as rules to be memorized, but as functions defined precisely through their unique properties (e.g., the exponential function as the unique solution to $dy/dx = y$ with $y(0)=1$). Differentiation and integration rules for these functions are derived. Inverse trigonometric functions are explored, paying careful attention to their restricted domains and ranges. Hyperbolic functions ($sinh, cosh$, etc.) are introduced as elegant alternatives derived from exponential functions, showcasing their utility in modeling physical phenomena like hanging cables. --- Part II: Integral Calculus Chapter 5: The Indefinite Integral and Antiderivatives This chapter reverses the process of differentiation. We define the antiderivative and introduce the notation for the indefinite integral. Basic integration rules are established, often mirroring the differentiation rules. Techniques for integrating powers of $x$, exponential functions, and trigonometric functions are covered. A crucial section focuses on substitution (u-substitution), framed as the reverse of the Chain Rule, which is essential for tackling more complex integrals. Chapter 6: Definite Integrals and the Fundamental Theorem of Calculus The Riemann sum is introduced as the formal definition of the area under a curve, providing the necessary intuition for integration. We build upon this concept to define the definite integral. The heart of integral calculus, the Fundamental Theorem of Calculus (FTC), is presented in its two parts. The first part establishes the relationship between the derivative and the definite integral, while the second part provides the practical method for evaluating definite integrals using antiderivatives. Applications include finding total change from a rate, area between curves, and average value of a function. Chapter 7: Techniques of Integration Mastering integration requires a diverse toolkit. This chapter systematically develops these techniques: 1. Integration by Parts: Derived directly from the product rule, this is essential for integrals involving products of dissimilar functions (e.g., $x e^x$). 2. Trigonometric Integrals: Methods for integrating powers of sine, cosine, tangent, and secant, often involving half-angle identities and strategic substitutions. 3. Trigonometric Substitution: A powerful geometric technique used to rationalize expressions involving $sqrt{a^2 - x^2}$, $sqrt{a^2 + x^2}$, and $sqrt{x^2 - a^2}$. 4. Partial Fraction Decomposition: A systematic algebraic method for integrating rational functions by breaking them down into simpler, integrable components. 5. Improper Integrals: Extending the concept of the definite integral to functions with discontinuities or to limits extending to infinity. Chapter 8: Applications of Integration This chapter demonstrates the broad utility of the definite integral beyond simple area calculations. We calculate volumes of solids of revolution using the Disk, Washer, and Shell methods, focusing on choosing the most appropriate method based on the axis of rotation. Other essential applications include finding the arc length of a curve and calculating the surface area generated by revolving a curve about an axis. Work done by a variable force is introduced as an application in physics. --- Part III: Infinite Series Chapter 9: Sequences and Series This section transitions from finite sums to infinite processes. A sequence is formally defined, and concepts like convergence and divergence are introduced, often analyzed using limit theorems. The concept of a series is defined as the limit of its partial sums. We investigate geometric series and the harmonic series, providing concrete examples of convergence and divergence based on the ratio test. The necessary condition for convergence (the $n$-th term test) is established early on. Chapter 10: Tests for Convergence and Power Series This is the most analytically challenging part of the introductory sequence. We develop rigorous tests to determine the convergence of complex series: the integral test, comparison tests (direct and limit), the ratio test, and the root test. These tests are presented with clear procedural guidelines and examples illustrating when and why each test is most effective. The concept of a Power Series—a series whose coefficients are functions of $x$—is introduced. We find the Interval of Convergence and the Radius of Convergence for these series, a critical step toward Taylor and Maclaurin series. Chapter 11: Taylor and Maclaurin Series Building on the power series framework, we derive Taylor series representations for elementary functions, viewing these functions as being infinitely differentiable polynomials centered at a point $a$. Maclaurin series (Taylor series centered at $a=0$) are presented as the most common form. The chapter concludes with the estimation of error using Taylor's Remainder Theorem, allowing students to quantify the accuracy of polynomial approximations of transcendental functions. --- Appendices The text includes comprehensive appendices covering advanced algebraic identities, trigonometric identities necessary for integration techniques, and tables of common derivatives and integrals for quick reference. Each chapter is supplemented with a wide array of problems, categorized into conceptual checks, computational drills, and challenging applications that integrate concepts across different sections.

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