See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/238705476
Calculus I, Notes
Book · January 2007
CITATIONS READS
0 80,596
1 author:
Eugen Ionascu
Columbus State University
75 PUBLICATIONS 369 CITATIONS
SEE PROFILE
All content following this page was uploaded by Eugen Ionascu on 29 January 2022.
The user has requested enhancement of the downloaded file.
, Calculus, Class Notes
Eugen J. Ionascu © Draft dated January 27, 2022
,Contents
Contents i
Preface vii
1 Limits and The Main Elementary Functions 3
1.1 Basic Elementary Functions and Elementary Functions . . . . . . . . . . . 3
1.2 Sequences and their limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Limits in the geometry of curves . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 Identifying f (x) with ex . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Fundamental Limits of real valued function . . . . . . . . . . . . . . . . . . 18
1.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Continuity and piecewise functions . . . . . . . . . . . . . . . . . . . . . . 33
1.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.2 Solutions to 1.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . 36
1.4.3 Sample Test 1 and Solutions . . . . . . . . . . . . . . . . . . . 36
2 Derivatives and the rules of differentiation 41
2.1 Derivatives of the basic elementary functions . . . . . . . . . . . . . . . . . 41
2.2 Derivatives under algebraic operations . . . . . . . . . . . . . . . . . . . . 44
2.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
i
, ii CONTENTS
2.4 Derivatives of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Related rates problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Newton’s Approximation Scheme . . . . . . . . . . . . . . . . . . . . . . . 56
3 Applications 61
3.1 Fermat’s Theorem, Rolle’s Theorem, Mean Value Theorem, Cauchy Theorem 61
3.1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Sketching Graphs of Elementary Functions . . . . . . . . . . . . . . . . . . 70
4 Definite Integral 73
4.1 Antiderivative and some previous formulae . . . . . . . . . . . . . . . . . . 73
4.1.1 More Homework Problems . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Integration using a substitution . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 Sample Final Exam and solutions . . . . . . . . . . . . . . . . . . . 84
4.5 The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Parametric Equations 89
5.1 Some classical parametrizations . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Curves in space, Curvature and TNB-frame 95
7 Inequalities 99
Bibliography 103
Calculus I, Notes
Book · January 2007
CITATIONS READS
0 80,596
1 author:
Eugen Ionascu
Columbus State University
75 PUBLICATIONS 369 CITATIONS
SEE PROFILE
All content following this page was uploaded by Eugen Ionascu on 29 January 2022.
The user has requested enhancement of the downloaded file.
, Calculus, Class Notes
Eugen J. Ionascu © Draft dated January 27, 2022
,Contents
Contents i
Preface vii
1 Limits and The Main Elementary Functions 3
1.1 Basic Elementary Functions and Elementary Functions . . . . . . . . . . . 3
1.2 Sequences and their limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Limits in the geometry of curves . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 Identifying f (x) with ex . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Fundamental Limits of real valued function . . . . . . . . . . . . . . . . . . 18
1.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Continuity and piecewise functions . . . . . . . . . . . . . . . . . . . . . . 33
1.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.2 Solutions to 1.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . 36
1.4.3 Sample Test 1 and Solutions . . . . . . . . . . . . . . . . . . . 36
2 Derivatives and the rules of differentiation 41
2.1 Derivatives of the basic elementary functions . . . . . . . . . . . . . . . . . 41
2.2 Derivatives under algebraic operations . . . . . . . . . . . . . . . . . . . . 44
2.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
i
, ii CONTENTS
2.4 Derivatives of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Related rates problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Newton’s Approximation Scheme . . . . . . . . . . . . . . . . . . . . . . . 56
3 Applications 61
3.1 Fermat’s Theorem, Rolle’s Theorem, Mean Value Theorem, Cauchy Theorem 61
3.1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Sketching Graphs of Elementary Functions . . . . . . . . . . . . . . . . . . 70
4 Definite Integral 73
4.1 Antiderivative and some previous formulae . . . . . . . . . . . . . . . . . . 73
4.1.1 More Homework Problems . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Integration using a substitution . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 Sample Final Exam and solutions . . . . . . . . . . . . . . . . . . . 84
4.5 The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Parametric Equations 89
5.1 Some classical parametrizations . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Curves in space, Curvature and TNB-frame 95
7 Inequalities 99
Bibliography 103
