Essential Calculus: With Applications


Calculus is an extremely powerful tool for solving a host of practical problems in fields as diverse as physics, biology, and economics, to mention just a few. In this rigorous but accessible text, a noted mathematician introduces undergraduate-level students to the problem-solving techniques that make a working knowledge of calculus indispensable for any mathematician.
The author first applies the necessary mathematical background, including sets, inequalities, absolute value, ...
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Essential Calculus with Applications

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Calculus is an extremely powerful tool for solving a host of practical problems in fields as diverse as physics, biology, and economics, to mention just a few. In this rigorous but accessible text, a noted mathematician introduces undergraduate-level students to the problem-solving techniques that make a working knowledge of calculus indispensable for any mathematician.
The author first applies the necessary mathematical background, including sets, inequalities, absolute value, mathematical induction, and other "precalculus" material. Chapter Two begins the actual study of differential calculus with a discussion of the key concept of function, and a thorough treatment of derivatives and limits. In Chapter Three differentiation is used as a tool; among the topics covered here are velocity, continuous and differentiable functions, the indefinite integral, local extrema, and concrete optimization problems. Chapter Four treats integral calculus, employing the standard definition of the Riemann integral, and deals with the mean value theorem for integrals, the main techniques of integration, and improper integrals. Chapter Five offers a brief introduction to differential equations and their applications, including problems of growth, decay, and motion. The final chapter is devoted to the differential calculus of functions of several variables.
Numerous problems and answers, and a newly added section of "Supplementary Hints and Answers," enable the student to test his grasp of the material before going on. Concise and well written, this text is ideal as a primary text or as a refresher for anyone wishing to review the fundamentals of this crucial discipline.
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Product Details

  • ISBN-13: 9780486660974
  • Publisher: Dover Publications
  • Publication date: 8/1/1989
  • Series: Dover Books on Mathematics Series
  • Edition description: Enlarged
  • Pages: 320
  • Sales rank: 783,319
  • Product dimensions: 6.13 (w) x 9.19 (h) x 0.63 (d)

Table of Contents

To the Instructor; To the Student
Chapter 1. Mathematical Background
1.1 Introductory Remarks
1.2 Sets
1.3 Numbers
1.4 Inequalities
1.5 The Absolute Value
1.6 Intervals and Neighborhoods
1.7 Rectangular Coordinates
1.8 Straight Lines
1.9 More about Straight Lines
Chapter 2. Differential Calculus
2.1 Functions
2.2 More about Functions
2.3 Graphs
2.4 Derivatives and Limits
2.5 More about Derivatives
2.6 More about Limits
2.7 Differentiation Technique
2.8 Further Differentiation Technique
2.9 Other Kinds of Limits
Chapter 3. Differentiation as a Tool
3.1 Velocity and Acceleration
3.2 Related Rates and Business Applications
3.3 Properties of Continuous Functions
3.4 Properties of Differentiable Functions
3.5 Applications of the Mean Value Theorem
3.6 Local Extrema
3.7 Concavity and Inflection Points
3.8 Optimization Problems
Chapter 4. Integral Calculus
4.1 The Definite Integral
4.2 Properties of Definite Integrals
4.3 The Logarithm
4.4 The Exponential
4.5 More about the Logarithm and Exponential
4.6 Integration Technique
4.7 Improper Integrals
Chapter 5. Integration as a Tool
5.1 Elementary Differential Equations
5.2 Problems of Growth and Decay
5.3 Problems of Motion
Chapter 6. Functions of Several Variables
6.1 From Two to n Dimensions
6.2 Limits and Differentiation
6.3 The Chain Rule
6.4 Extrema in n Dimensions
Tables; Selected Hints and Answers; Supplementary Hints and Answers; Index
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  • Posted June 12, 2009

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    Useful as a review of those aspects of calculus that it covers.

    Richard Silverman's text provides a careful, largely self-contained, treatment of the aspects of differential and integral calculus that it covers. The exposition is clear. Most of the theorems are proved. The examples are illuminating. The problems, which range from routine calculations to problems that require insight and ingenuity to solve, are interesting. Hints or answers to virtually every problem in the book are provided in two answer keys, the second of which was adapted for this edition from an instructor's manual for an earlier edition of the text. Thus, the text is suitable for self-study. However, it is dense. Consequently, a student who is new to calculus might benefit from using another text with a more leisurely treatment of the material in order to be exposed to more examples and to obtain more practice on the techniques of differentiation and integration introduced here before being confronted with the difficult problems at the end of the exercise sets. Using the text to review calculus is also problematic since important topics are omitted from the text. The text does not cover the calculus of trigonometric functions. The method of integration by partial fractions is omitted. The applications of integration do not include the calculation of volumes, surface areas, or arc lengths. There is minimal coverage of sequences and the treatment of series is limited to geometric series. The text begins with a review of the foundations you need to understand in order to learn differential and integral calculus. Those foundations include sets, number systems, proofs by mathematical induction, inequalities, absolute value, intervals, lines, functions, and graphs. Limits are introduced via the derivative. Silverman emphasizes the geometric meaning of the derivative as the slope of the tangent line at the point where the derivative is taken by using the Delta notation. He continues to use this notation throughout the text, which sometimes makes the algebraic proofs harder to understand. After discussing some properties of limits, continuity, and the derivative, Silverman discusses differentiation techniques, including the sum, difference, product, and quotient rules and the Chain Rule. He discusses velocity and acceleration in terms of the derivative and demonstrates how to solve related rate problems. After a discussion of the properties of continuous and differentiable functions, Silverman applies differential calculus to finding extrema, determining concavity, and solving optimization problems. He discusses the properties of the antiderivative and the definite integral before introducing the natural logarithm of x, where x > 0, as the area under the curve 1/x between 1 and x. He derives the properties of the logarithm from the integral and introduces the exponential function with base e as the inverse of the logarithm. His treatment of exponential and logarithmic functions is a particular strength of the text. Silverman demonstrates the techniques of integration by substitution and integration by parts before showing how to solve simple differential equations and apply them to growth and decay problems and motion problems. The text concludes with a brief treatment of topics in multi-variable calculus, including limits, the Chain Rule, and extrema problems. When used in conjunction with other texts, this text is useful as a review of the topics it covers and as a source of interesting problems.

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