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Calculus Concepts: An Informal Approach to the Mathematics of Change, 5th Edition
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Overview
Designed for a one or two-semester Applied Calculus course, this innovative text features a graphing calculator approach, incorporating real-life applications and such technology as graphing utilities and Excel® spreadsheets to help students learn mathematical skills that they will use in their lives and careers. The texts overall goal is to improve learning of basic calculus concepts by involving students with new material in a way that is different from traditional practice. The development of conceptual understanding coupled with a commitment to make calculus meaningful to the student are guiding forces. The material involves many applications of real situations through its data-driven, technology-based modeling approach. The ability to correctly interpret the mathematics of real-life situations is considered of equal importance to the understanding of the concepts of calculus.
CALCULUS CONCEPTS, Fifth Edition, presents concepts in a variety of forms, including algebraic, graphical, numeric, and verbal. Targeted toward students majoring in liberal arts, economics, business, management, and the life and social sciences, the text's focus on technology along with its use of real data and situations make it a sound choice to help students develop an intuitive, practical understanding of concepts.
- Many of the book's examples and activities are new. In addition, many data sets have been revised to incorporate more recent data.
- The concept of limits is introduced early in Chapter 1 and used throughout the discussion of models in the remainder of that chapter. The concept is also used to help students understand differentiation and integration.
- Formerly presented in a self-contained chapter, coverage of sine models has been incorporated throughout the text in optional sections and activities.
- Differential equations and slope fields are introduced in a pair of optional sections located at the end of the integration chapters.
- The text has been carefully rewritten so that narrative sections are as clear and concise as possible.
- While a real-world context is still used as the platform for most of the discussion, some of the less critical details of these contextual descriptions are now presented to the side of the primary narrative in Notes, allowing students to focus on key ideas without potentially getting distracted.
- Abundant use of data underscores the data-driven nature of this book.
- Each chapter opens with a real-life situation and several questions about the situation that can be answered using the concepts and skills to be covered.
- Each section incorporates a brief concept development narrative, interspersed with Quick Examples that highlight specific skills as well as formal examples that illustrate the application of the skills and concepts in a real-world setting.
- A NAVG (numeric, algebraic, verbal, and graphical) compass icon indicates places in the text where a concept is demonstrated through multiple representations. This feature helps students recognize connections between different representations, and is particularly helpful for students who use alternative learning styles.
- A Concept Inventory listed at the end of each section gives students a brief summary of the major ideas developed in the section.
- A Concept Review activity section at the end of each chapter provides practice with techniques and concepts. Complete answers to the Concept Review activities are included in the answer key located at the back of the text.
- Concept Check is an end-of-chapter checklist that describes the main concepts and skills taught in the chapter and identifies sample odd-numbered activities corresponding to each item. Students can complete these representative activities to help them assess their understanding of the chapter content and identify the areas on which they need to focus their study.
- A Chapter Summary reviews and connects major chapter topics and concepts, and further emphasizes their practical importance.
- Activities, such as online Projects and Writing Across the Curriculum, reinforce the authors' innovative approach. Spreadsheet and Graphing Calculator Activities give students the opportunity to use technology as they learn difficult calculus concepts.
- Section Activities at the end of each section reinforce concepts and allow students to explore topics using, for the most part, actual data in a variety of real-world settings. These activities, designed to encourage students to communicate in written form, include questions and interpretations pertinent to the data. The activities do not mimic the examples in the chapter discussion, requiring students to think more independently. Possible answers to odd-numbered activities are given at the end of the book.
Each chapter concludes with a summary, a concept check, and review activities
1. INGREDIENTS OF CHANGE: FUNCTIONS AND LIMITS.
Functions--Four Representations. Function Behavior and End Behavior Limits.
Limits and Continuity. Linear Functions and Models. Exponential Functions and Models.
Models in Finance. Constructed Functions. Logarithmic Functions and Models. Quadratic Functions and Models. Logistic Functions and Models. Cubic Functions and Models. Cyclic Functions and Models. Representations of a Sine Function. Characteristics of Sine Functions.
2. DESCRIBING CHANGE: RATES.
Measures of Change over an Interval. Measures of Change at a Point. Rates of Change--Notation and Interpretation. Rates of Change--Numerical Limits and Non-existence. Rates of Change Defined over Intervals. Sketching Rate-of-Change Graphs.
3. DETERMINING CHANGE: DERIVATIVES.
Simple Rate-of-Change Formulas. Exponential, Logarithmic, and Cyclic Rate-of-Change Formulas. Rates of Change for Functions That Can Be Composed. Rates of Change of Composite Functions. Rates of Change for Functions That Can Be Multiplied. Rates of Change for Product Functions. Limits of Quotients and L'Hôpital's Rule.
4. ANALYZING CHANGE: APPLICATIONS OF DERIVATIVES.
Linearization. Relative Extreme Points. Relative Extreme Points. Inflection Points and Second Derivatives. Marginal Analysis. Optimization of Constructed Functions. Related Rates.
5. ACCUMULATING CHANGE: LIMITS OF SUMS AND THE DEFINITE INTEGRAL.
An Introduction to Results of Change. Limit of Sums and the Definite Integral.
Accumulation Functions. The Fundamental Theorem. Antiderivative Formulas for Exponential, Natural Log, and Sine Functions. The Definite Integral--Algebraically.
Differences of Accumulated Change. Average Value and Average Rate of Change. Integration of Product or Composite Functions.
6. ANALYZING ACCUMULATED CHANGE: INTEGRALS AND ACTION.
Perpetual Accumulation and Improper Integrals. Streams in Business and Biology. Calculus in Economics--Demand and Elasticity. Calculus in Economics--Supply and Equilibrium. Calculus in Probability--Part 1. Calculus in Probability--Part 2. Differential Equations--Slope Fields and Solutions. Differential Equations--Proportionality and Common Forms.
7. INGREDIENTS OF MULTIVARIABLE CHANGE: FUNCTIONS AND RATES.
Multivariable Functions and Contour Graphs. Cross-Sectional Models and Rates of Change. Partial Rates of Change. Compensating for Change.
8. ANALYZING MULTIVARIABLE CHANGE: OPTIMIZATION.
Extreme Points and Saddle Points. Multivariable Optimization. Optimization Under Constraints. Least-Squares Optimization.
Answers to Odd Activities.
Index of Applications.
Subject Index.
1. INGREDIENTS OF CHANGE: FUNCTIONS AND LIMITS.
Functions--Four Representations. Function Behavior and End Behavior Limits.
Limits and Continuity. Linear Functions and Models. Exponential Functions and Models.
Models in Finance. Constructed Functions. Logarithmic Functions and Models. Quadratic Functions and Models. Logistic Functions and Models. Cubic Functions and Models. Cyclic Functions and Models. Representations of a Sine Function. Characteristics of Sine Functions.
2. DESCRIBING CHANGE: RATES.
Measures of Change over an Interval. Measures of Change at a Point. Rates of Change--Notation and Interpretation. Rates of Change--Numerical Limits and Non-existence. Rates of Change Defined over Intervals. Sketching Rate-of-Change Graphs.
3. DETERMINING CHANGE: DERIVATIVES.
Simple Rate-of-Change Formulas. Exponential, Logarithmic, and Cyclic Rate-of-Change Formulas. Rates of Change for Functions That Can Be Composed. Rates of Change of Composite Functions. Rates of Change for Functions That Can Be Multiplied. Rates of Change for Product Functions. Limits of Quotients and L'Hôpital's Rule.
4. ANALYZING CHANGE: APPLICATIONS OF DERIVATIVES.
Linearization. Relative Extreme Points. Relative Extreme Points. Inflection Points and Second Derivatives. Marginal Analysis. Optimization of Constructed Functions. Related Rates.
5. ACCUMULATING CHANGE: LIMITS OF SUMS AND THE DEFINITE INTEGRAL.
An Introduction to Results of Change. Limit of Sums and the Definite Integral.
Accumulation Functions. The Fundamental Theorem. Antiderivative Formulas for Exponential, Natural Log, and Sine Functions. The Definite Integral--Algebraically.
Differences of Accumulated Change. Average Value and Average Rate of Change. Integration of Product or Composite Functions.
6. ANALYZING ACCUMULATED CHANGE: INTEGRALS AND ACTION.
Perpetual Accumulation and Improper Integrals. Streams in Business and Biology. Calculus in Economics--Demand and Elasticity. Calculus in Economics--Supply and Equilibrium. Calculus in Probability--Part 1. Calculus in Probability--Part 2. Differential Equations--Slope Fields and Solutions. Differential Equations--Proportionality and Common Forms.
7. INGREDIENTS OF MULTIVARIABLE CHANGE: FUNCTIONS AND RATES.
Multivariable Functions and Contour Graphs. Cross-Sectional Models and Rates of Change. Partial Rates of Change. Compensating for Change.
8. ANALYZING MULTIVARIABLE CHANGE: OPTIMIZATION.
Extreme Points and Saddle Points. Multivariable Optimization. Optimization Under Constraints. Least-Squares Optimization.
Answers to Odd Activities.
Index of Applications.
Subject Index.
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