The Calculus Tutoring Book

Description:

This book fills an educational void by adapting unique classroom-tested techniques that students find most congenial...that strip the shroud of mystery from an esoteric subject...that prepare students for applications of calculus in later courses.

 

Table of Contents:

Preface.

FUNCTIONS.

Introduction.

The Graph of a Function.

The Trigonometric Functions.

Inverse Functions and the Inverse Trigonometric Functions.

Exponential and Logarithm Functions.

Solving Inequalities Involving Elementary Functions.

Graphs of Translations, Reflections, Expansions and Sums.

Review of Problems for Chapter 1.

LIMITS.

Introduction.

Finding Limits of Combinations of Functions.

Indeterminate Limits.

Review Problems for Chapter 2.

THE DERIVATIVE PART I.

Preview.

Definition and Some Applications of the Derivative.

Derivatives of the Basic Functions.

Nondifferentiable Functions.

Derivatives of Constant Multiples, Sums, Products and Quotients.

The Derivative of a Composition.

Implicit Differentiation and Logarithmic Differentiation.

Antidifferentiation.

Review Problems for Chapter 3.

THE DERIVATIVE PART II.

Relative Maxima and Minima.

Absolute Maxima and Minima.

L'Hôpital's Rule and Orders of Magnitude.

Indeterminate Products, Differences and Exponential Forms.

Drawing Graphs of Functions.

Related Rates.

Newton's Method.

Differentials.

Separable Differential Equations.

Review Problems for Chapter 4.

THE INTEGRAL PART I.

Preview.

Definition and Some Applications of the Integral.

The Fundamental Theorem of Calculus.

Numerical Integration.

Nonintegrable Functions.

Improper Integals.

Review Problems for Chapter 5.

THE INTEGRAL PART II.

Further Applications of the Integral.

The Centroid of a Solid Hemisphere.

Area and Arc Length.

The Surface Area of a Cone and a Sphere.

Integrals with a Variable Upper Limit.

Review Problems for Chapter 6.

ANTIDIFFERENTIATION.

Introduction.

Substitution.

Pre-Table Algebra I.

Pre-Table Algebra II: Partial Faction Decomposition.

Integration by Parts.

Recursion Formulas.

Trigonometric Substitution.

Choosing a Method.

Combining Techniques of Antidifferentiation with the Fundamental Theorem.

Review Problems for Chapter 7.

SERIES.

Introduction.

Geometric Series.

Convergence Tests for Positive Series I.

Convergence Tests for Positive Series II.

Alternating Series.

Power Series Functions.

Power Series Representations for Elementary Functions I.

Power Series Representations for Elementary Functions II (Maclaurin Series).

The Taylor Reminder Formula and an Estimate for the Number e.

Power Series in Powers of x - b (Taylor Series).

Review Problems for Chapter 8.

VECTORS.

Introduction.

Vector Addition, Subtraction, Scalar Multiplication and Norms.

The Dot Product.

The Cross Product.

The Scalar Triple Product.

The Velocity Vector.

The Acceleration Vector.

Review Problems for Chapter 9.

TOPICS IN THREE-DIMENSIONAL ANALYTIC GEOMETRY.

Spheres.

Planes.

Lines.

Cylindrical and Quadric Surfaces.

Cylindrical and Spherical Coordinates.

Review Problems for Chapter 10.

PARTIAL DERIVATIVES.

Graphs and Level Sets.

Partial Derivatives.

Chain Rules for First-Order Partial Derivatives.

Chain Rules for Second-Order Partial Derivatives.

Maxima and Minima.

The Gradient.

Differentials and Exact Differential Equations.

Review Problems for Chapter 11.

MULTIPLE INTEGRALS.

Definition and Some Applications of the Double Integral.

Computing Double Integrals.

Double Integration in Polar Coordinates.

Area and Volume.

Further Applications of the Double Integral.

Triple Integrals.

Triple Integration in Spherical Coordinates.

Center of Mass.

Review Problems for Chapter 12.

APPENDIX.

A1: Distance and Slope.

A2: Equations of Lines.

A3: Circles, Ellipses, Hyperbolas and Parabolas.

A4: The Binomial Theorem.

A5: Determinants.

A6: Polar Coordinates.

Solutions to the Problems.

Abbreviations Used in the Solutions.

List of Symbols.

Index.

Authors' Biographies.

描述：

這本書填補了教育上的空白，採用了經過班級測試的獨特技巧，學生們覺得最親切...這些技巧揭開了一門神秘的學科的面紗...為學生們在後續課程中應用微積分做好準備。

 

目錄：

前言。

函數。

介紹。

函數的圖形。

三角函數。

反函數和反三角函數。

指數和對數函數。

解涉及基本函數的不等式。

平移、反射、擴展和總和的圖形。

第一章問題回顧。

極限。

介紹。

找出組合函數的極限。

不確定極限。

第二章問題回顧。

導數第一部分。

預覽。

導數的定義和一些應用。

基本函數的導數。

不可微分的函數。

常數倍數、總和、乘積和商的導數。

復合函數的導數。

隱式微分和對數微分。

反微分。

第三章問題回顧。

導數第二部分。

相對極大值和極小值。

絕對極大值和極小值。

洛必達法則和量級。

不確定乘積、差異和指數形式。

繪製函數圖形。

相關速率。

牛頓法。

微分。

可分離微分方程。

第四章問題回顧。

積分第一部分。

預覽。

積分的定義和一些應用。

微積分基本定理。

數值積分。

不可積分的函數。

不定積分。

第五章問題回顧。

積分第二部分。

積分的進一步應用。

固體半球的重心。

面積和弧長。