An edition of The ABC's of Calculus
(2011)
The ABC's of Calculus
by Angelo B. Mingarelli
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Single variable differential and integral Calculus and its applications. 2011 Edition.
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Table of Contents
Preface to the e-text edition.
vii
1.
Functions and Their Properties
Page 1
1.1.
The Meaning of a Function
Page 1
1.2.
Function Values and the Box Method
Page 5
1.3.
The Absolute Value of a Function
Page 12
1.4.
A Quick Review of Inequalities
Page 21
1.4 .1.
The triangle inequalities
Page 23
1.5.
Chapter Exercises
Page 30
1.6.
Using Computer Algebra Systems (CAS),
Page 31
2.
Limits and Continuity
Page 33
2.1.
One-Sided Limits of Functions
Page 35
2.2.
Two-Sided Limits and Continuity
Page 40
2.3.
Important Theorems About Continuous Functions
Page 59
2.4.
Evaluating Limits at Infinity
Page 63
2.5.
How to Guess a Limit
Page 66
2.6.
Chapter Exercises
Page 76
3.
The Derivative of a Function
Page 79
3.1.
Motivation
Page 80
3.2.
Working with Derivatives
Page 89
3.3.
The Chain Rule
Page 95
3.4.
Implicit Functions and Their Derivatives
Page 108
3.5.
Derivatives of Trigonometric Functions
Page 113
3.6.
Important Results About Derivatives
Page 121
3.7.
Inverse Functions
Page 128
3.8.
Inverse Trigonometric Functions
Page 136
3.9.
Derivatives of Inverse Trigonometric Functions
Page 140
3.1.0.
Relating Rates of Change
Page 143
3.1.1.
Newton’s Method for Calculating Roots 151
3.1.2.
L’Hospital’s Rule
Page 161
3.1.3.
Chapter Exercises
Page 173
3.1.4.
Challenge Questions
Page 174
3.1.5.
Using Computer Algebra Systems
Page 175
4.
Exponentials and Logarithms
Page 177
4.1.
Exponential Functions and Their Logarithms
Page 178
4.2.
Euler’s Number, e = 2.718281828
Page 184
4.3.
Euler’s Exponential Function and the Natural Logarithm
Page 189
4.4.
Derivative of the Natural Logarithm
Page 193
4.5.
Differentiation Formulae for General Exponential Functions . . .
Page 196
4.6.
Differentiation Formulae for General Logarithmic Functions . . .
Page 201
4.7.
Applications
Page 204
4.8.
Chapter Exercises
Page 210
4.9.
Using Computer Algebra Systems
Page 211
5.
Curve Sketching
Page 213
5.1.
Solving Polynomial Inequalities
Page 213
5.2.
Solving Rational Function Inequalities
Page 225
5.3.
Graphing Techniques
Page 232
5.4.
Application of Derivatives to Business and Economics
Page 256
5.5.
Single variable optimization problems
Page 258
5.6.
Chapter Exercises
Page 259
6.
Integration
Page 261
6.1.
Antiderivatives and the Indefinite Integral
Page 262
6.2.
Definite Integrals
Page 277
6.3.
The Summation Convention
Page 286
6.4.
Area and the Riemann Integral
Page 291
6.5.
Chapter Exercises
Page 304
6.6.
Using Computer Algebra Systems
Page 306
7.
Techniques of Integration
Page 309
7.1.
Trigonometric Identities
Page 309
7.2.
The Substitution Rule
Page 311
7.3.
Integration by Parts
Page 323
7.3 .1.
The Product of a Polynomial and a Sine or Cosine
Page 328
7.3 .2.
The Product of a Polynomial and an Exponential
Page 331
7.3 .3.
The Product of a Polynomial and a Logarithm
Page 334
7.3 .4.
The Product of an Exponential and a Sine or Cosine . . .
Page 337
7.4.
Partial Fractions
Page 346
7.4 .1.
Review of Long Division of Polynomials
Page 347
7.4 .2.
The Integration of Partial Fractions
Page 350
7.5.
Products of Trigonometric Functions
Page 365
7.5 .1.
Products of Sines and Cosines
Page 365
7.5 .2.
Fourier Coefficients
Page 374
7.5 .3.
Products of Secants and Tangents
Page 378
7.6.
Trigonometric Substitutions
Page 386
7.6 .1.
Completing the Square in a Quadratic (Review)
Page 386
7.6 .2.
Trigonometric Substitutions
Page 390
7.7.
Numerical Integration
Page 400
7.7 .1.
The Trapezoidal Rule
Page 401
7.7 .2.
Simpson’s Rule for n Even
Page 408
7.8.
Improper Integrals
Page 414
7.9.
Rationalizing Substitutions
Page 429
7.9 .1.
Integrating rational functions of trigonometric expressions
Page 432
7.10.
Chapter Exercises
Page 437
7.11.
Using Computer Algebra Systems
Page 443
8.
Applications of the Integral
Page 445
8.1.
Motivation
Page 445
8.2.
Finding the Area Between Two Curves
Page 448
8.3.
The Volume of a Solid of Revolution
Page 464
8.4.
Measuring the length of a curve
Page 477
8.5.
Moments and Centers of Mass
Page 489
8.6.
Chapter Exercises
Page 502
8.7.
Using Computer Algebra Systems
Page 502
9.
Simple Differential Equations
Page 505
9.1.
Why Study Differential Equations?
Page 505
9.2.
First-order Separable Equations
Page 512
9.3.
Laws of Growth and Decay
Page 518
9.4.
Using Computer Algebra Systems
Page 525
10.
Multivariable Optimization Techniques
Page 527
10.1.
Functions of More Than One Variable
Page 527
10.2.
Continuity
Page 528
10.2 .1.
Discontinuity at a point
Page 529
10.3.
Partial Derivatives
Page 531
10.4.
Higher Order Partial Derivatives
Page 533
10.5.
The Chain Rule for Partial Derivatives
Page 535
10.6.
Extrema of Functions of Two Variables
Page 540
10.6 .1.
Maxima and Minima
Page 540
10.6 .2.
The method of Lagrange multipliers
Page 544
10.7.
Chapter Exercises
Page 551
11.
Advanced Topics
Page 553
11.1.
Infinite Sequences
Page 553
11.2.
Sequences with Infinite Limits
Page 560
11.3.
Limits from the Right
Page 563
11.4.
Limits from the Left
Page 569
11.5.
Summary
Page 575
11.6.
Continuity
Page 576
11.7.
Limits of Functions at Infinity
Page 578
11.8.
Infinite Limits of Functions
Page 581
11.9.
The Epsilon-Delta Method of Proof
Page 585
12.
Appendix A: Review of Exponents and Radicals
Page 597
13.
Appendix B: The Straight Line
Page 603
14.
Appendix C: A Quick Review of Trigonometry
Page 609
14.1.
The right-angled isosceles triangle (RT45)
Page 610
14.2.
The RT30 triangle
Page 610
14.3.
The basic trigonometric functions
Page 611
14.4.
Identities
Page 613
14.4.1.
The Law of Sines
Page 614
14.4.2.
The Law of Cosines
Page 615
14.4.3.
Identities for the sum and difference of angles
Page 616
15.
Appendix D: The Natural Domain of a Function
Page 623
Solutions Manual.
627
1.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 627
1.2.
Exercise Set 1 (page 10)
Page 627
1.3.
Exercise Set 2 (page 19)
Page 627
1.4.
Exercise Set 3 (page 28)
Page 628
1.5.
Chapter Exercises (page 30 )
Page 629
Solutions.
Page 631
2.1.
Exercise Set 4 (page 39)
Page 631
2.2.
Exercise Set 5 (page 45)
Page 631
2.2.
Exercise Set 6 (page 49)
Page 631
2.2.
Exercise Set 7 (page 56)
Page 632
2.2.
Exercise Set 8 (page 57)
Page 632
2.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 633
2.4.
Exercise Set 9 (page 65)
Page 633
2.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 633
2.6.
Chapter Exercises (page 76)
Page 633
Solutions.
Page 635
3.1.
Exercise Set 10 (page 88)
Page 635
3.2.
Exercise Set 11 (page 93)
Page 635
3.3.
Exercise Set 12 (page 105)
Page 636
3.4.
Exercise Set 13 (page 112)
Page 637
3.5.
Exercise Set 14 (page 119)
Page 637
3.6.
Exercise Set 15 (page 127)
Page 637
3.7.
Exercise Set 16 (page 134)
Page 639
3.8.
Exercise Set 17 (page 139)
Page 639
3.9.
Exercise Set 18 (page 142)
Page 639
3.1.0.
Special Exercise Set (page 149)
Page 640
3.1.1.
Exercise Set 19 (page 160)
Page 640
3.1.2.
Exercise Set 20 (page 172)
Page 641
3.1.3.
Chapter Exercises (page 173)
Page 641
Solutions.
Page 643
4.1.
Exercise Set 21 (page 183)
Page 643
4.2.
Exercise Set 22 (page 188)
Page 643
4.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 644
4.4.
Exercise Set 23 (page 196)
Page 644
4.5.
Exercise Set 24 (page 199)
Page 644
4.6.
Exercise Set 25 (page 203)
Page 645
4.7.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 646
4.8.
Chapter Exercises (page 210)
Page 646
Solutions.
Page 649
5.1.
Exercise Set 26 (page 216)
Page 649
5.2.
Exercise Set 27 (page 223)
Page 649
5.3.
Exercise Set 28 (page 230)
Page 650
5.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 652
5.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 652
5.6.
Single variable optimization problems (page 258)
Page 653
5.7.
Chapter Exercises: Use Plotter (page 259)
Page 654
Solutions.
Page 657
6.1.
Exercise Set 29 (page 275)
Page 657
6.2.
Exercise Set 30 (page 289)
Page 657
6.3.
Exercise Set 31 (page 284)
Page 659
6.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 660
6.5.
Chapter Exercises (page 304)
Page 660
Solutions.
665
7.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 665
7.2.
Exercise Set 32 (page 321)
Page 665
7.3.
Exercise Set 33 (page 345)
Page 666
7.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 667
7.4 .1.
Exercise Set 34 (page 349)
Page 667
7.4 .2.
Exercise Set 35 (page 364)
Page 668
7.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 670
7.5 .1.
Exercise Set 36 (page 373)
Page 670
7.5 .2.
Exercise Set 37 (page 385)
Page 671
7.6.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 672
7.6 .1.
Exercise Set 38 (page 390)
Page 672
7.6 .2.
Exercise Set 39 (page 398)
Page 673
7.7.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 675
7.7 .1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 675
7.7 .2.
Exercise Set 40 (page 411)
Page 675
7.8.
Exercise Set 41 (page 426)
Page 677
7.9.
Chapter Exercises (page 437)
Page 678
Solutions.
Page 693
8.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 693
8.2.
Exercise Set 42 (page 452)
Page 693
8.2 .1.
Exercise Set 43 (page 462)
Page 694
8.3.
Exercise Set 44 (page 476)
Page 695
8.4.
Exercise Set 45 (page 487)
Page 696
8.5.
Exercise Set 46 (page 500)
Page 697
8.6.
Chapter Exercises (page 502)
Page 698
Solutions.
Page 701
9.1.
Exercise Set 47 (page 511)
Page 701
9.2.
Exercise Set 48 (page 518)
Page 701
9.3.
Exercise Set 49 (page 523)
Page 702
Solutions.
Page 705
10.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 705
10.2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 705
10.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 705
10.4.
Exercise Set 50 (page 535)
Page 705
10.5.
Exercise Set 51 (page 548)
Page 706
10.6.
Chapter Exercises (page 551)
Page 707
Solutions.
Page 711
11.1.
Exercise Set 52 (page 559)
Page 711
11.2.
Exercise Set 53 (page 563)
Page 711
11.3.
Exercise Set 54 (page 574)
Page 712
11.4.
Exercise Set 55 (page 580)
Page 712
11.5.
Exercise Set 56 (page 584)
Page 712
11.6.
Exercise Set 57 (page 594)
Page 713
Solutions to Problems in the Appendices.
Page 715
1.
APPENDIX A - Exercise Set 58 (page 601)
Page 715
12.2.
APPENDIX B - Exercise Set 59 (page 607)
Page 716
12.3.
APPENDIX C - Exercise Set 60 (page 620)
Page 716
12.4.
APPENDIX D - Exercise Set 61 (page 626)
Page 718
Acknowledgments.
719
Credits.
721
About the Author.
723
Index.
724
The Physical Object
ID Numbers
Work Description
A course on single variable differential and integral calculus with a different twist.
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History
- Created September 13, 2014
- 11 revisions
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| August 22, 2020 | Edited by ISBNbot2 | normalize ISBN |
| September 14, 2014 | Edited by Angelo B. Mingarelli | Edited without comment. |
| September 14, 2014 | Edited by Angelo B. Mingarelli | Edited without comment. |
| September 14, 2014 | Edited by Angelo B. Mingarelli | Edited without comment. |
| September 13, 2014 | Edited by Angelo B. Mingarelli | Update of previous editions, typographical errors fixed, some motivation addition. |