Comprehensive Differential calculus 1 with problems and solutions, guaranteed 100% Pass

A Collection of Problems in Differential Calculus
Problems Given At the Math 151 - Calculus I and Math 150 - Calculus I With
Review Final Examinations
Department of Mathematics, Simon Fraser University

2000 - 2010



Veselin Jungic · Petra Menz · Randall Pyke
Department Of Mathematics
Simon Fraser University
c Draft date December 6, 2011

,To my sons, my best teachers. - Veselin Jungic

,Contents

Contents i

Preface 1

Recommendations for Success in Mathematics 3

1 Limits and Continuity 11
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Differentiation Rules 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Tangent Lines and Implicit Differentiation . . . . . . . . . . . . . . . 28

3 Applications of Differentiation 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Differential, Linear Approximation, Newton’s Method . . . . . . . . . 51

i

, 3.6 Antiderivatives and Differential Equations . . . . . . . . . . . . . . . 55
3.7 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . 58
3.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Parametric Equations and Polar Coordinates 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 True Or False and Multiple Choice Problems 81

6 Answers, Hints, Solutions 93
6.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.6 Tangent Lines and Implicit Differentiation . . . . . . . . . . . . . . . 105
6.7 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.8 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.9 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.10 Differential, Linear Approximation, Newton’s Method . . . . . . . . . 126
6.11 Antiderivatives and Differential Equations . . . . . . . . . . . . . . . 131
6.12 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . 133
6.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.14 Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.15 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.16 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.17 True Or False and Multiple Choice Problems . . . . . . . . . . . . . . 146