Advanced Calculus Notes and Guide

lOMoARcPSD|15589855




ADVANCED CALCULUS.




SMTA 021
University of Limpopo (Turfloop Campus)




DR L. Rundora
School of Mathematical and Computer Sciences
Department of Mathematics and Applied Mathematics
University of Limpopo (Turfloop Campus)




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Contents

1 Limits: A Review 1
1.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 1
1.2 Limits: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Limits of Indeterminate Type 6
2.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 6
2.2 Limits of Indeterminate Type . . . . . . . . . . . . . . . . . . . . 6
2.2.1 L’Hospital’s Rules . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Other Indeterminate forms . . . . . . . . . . . . . . . . . . 11

3 Sequences 14
3.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 14
3.2 Properties of convergent sequences . . . . . . . . . . . . . . . . . 16
3.2.1 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . . . 25

4 Infinite series: Convergence and Divergence 29
4.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 29
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Tests for Convergence and divergence . . . . . . . . . . . . . . . . 34
4.3.1 Comparison test . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2 The ratio test . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.3 The Integral Test . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.4 Alternating Series Test . . . . . . . . . . . . . . . . . . . . 41
4.3.5 The nth root test . . . . . . . . . . . . . . . . . . . . . . . 44

5 Power Series 48
5.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 48
5.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . 51

6 Limits and Continuity 55
6.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 55
6.2 Limits of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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6.2.1 Characterization of limits of functions in terms of conver-
gence of sequences . . . . . . . . . . . . . . . . . . . . . . 56
6.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3.1 Some properties of continuous functions . . . . . . . . . . 59

7 Differentiation and Partial Differentiation 63
7.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 63
7.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.2.1 Inverse trigonometric functions . . . . . . . . . . . . . . . 64
7.2.2 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 66
7.2.3 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . 69
7.2.4 Parametric Equations . . . . . . . . . . . . . . . . . . . . . 71
7.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3.1 First partial derivatives . . . . . . . . . . . . . . . . . . . . 73
7.3.2 Second partial derivatives . . . . . . . . . . . . . . . . . . 75
7.3.3 Applications: Differentials . . . . . . . . . . . . . . . . . . 77

8 Integration 81
8.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 81
8.2 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2.1 Riemann Integrable Functions . . . . . . . . . . . . . . . . 84
8.2.2 Properties of the Riemann integral . . . . . . . . . . . . . 85
8.2.3 The fundamental theorem of integral calculus . . . . . . . 87
8.3 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.3.1 The Double integral . . . . . . . . . . . . . . . . . . . . . 88
8.3.2 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . 89
8.3.3 Transformations of multiple integrals . . . . . . . . . . . . 92
8.4 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.5 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.5.1 Gradient, Divergence and Curl . . . . . . . . . . . . . . . . 100

9 Ordinary differential equations 106
9.1 What this Unit is all About . . . . . . . . . . . . . . . . . . . . . 106
9.2 Basic concepts and definitions . . . . . . . . . . . . . . . . . . . . 107
9.3 Types of first order differential equations . . . . . . . . . . . . . . 108
9.3.1 Equations with separable variables . . . . . . . . . . . . . 108

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9.3.2 Equations reducible to variables separable type . . . . . . 112
9.3.3 Homogeneous equations . . . . . . . . . . . . . . . . . . . 114
9.3.4 Linear differential equations . . . . . . . . . . . . . . . . . 117
9.4 Second order linear differential equations with constant coefficients 120
9.4.1 Homogeneous equations of first order . . . . . . . . . . . . 120
9.4.2 Homogeneous equations of second order . . . . . . . . . . . 120




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