Calculus: Concepts and Connections (Paperback)

Description

This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.

Table of Contents

 

0 Preliminaries

1 Limits and Continuity

2 Differentiation

3 Applications of Differentiation

4 Integration

5 Applications of the Definite Integral

6 Differential Equations

7 Infinite Series

8 Vectors and the Geometry of Space

9 Vector-Valued Functions

10 Functions of Several Variables and Differentiation

11 Multiple Integrals

12 Vector Calculus

Appendix A Additional Polar Graphs

Appendix B Formal Definition of the Limit

Appendix C Derivation of Sine Function

Appendix D The Natural Logarithm as an Integral

Appendix E Conic Sections in Polar Coordinates

Appendix F Proofs of Selected Theorems

Appendix G Answers to Odd-Numbered Exercises

描述

這本現代微積分教科書非常強調培養學生的概念理解能力，並建立微積分主題與現實世界的關聯。它是為平均學生而寫的，這些學生對這個主題大多不熟悉，需要相當的動力。它按照相對標準的順序進行介紹，早期涵蓋了超越函數，並在整本書中融入了引人思考的應用、例子和練習。該教材還提供了關於技術在解決問題中適當角色的平衡指導，包括其好處和潛在的陷阱。在實際情況下，概念是從圖形、數值、代數和語言的角度發展的（“四規則”），以使學生對微積分有全面的理解。

目錄

0 前言
1 極限與連續性
2 微分
3 微分應用
4 積分
5 定積分應用
6 微分方程
7 無窮級數
8 向量與空間幾何
9 向量值函數
10 多變數函數與微分
11 多重積分
12 向量微積分
附錄 A 額外的極坐標圖形
附錄 B 極限的正式定義
附錄 C 正弦函數的推導
附錄 D 自然對數作為積分
附錄 E 極坐標中的圓錐曲線
附錄 F 選定定理的證明
附錄 G 奇數練習題的答案