Differential Calculus

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 Preface Recommendations for Success in Mathematics 1 Limits and Continuity 1.1 1.2 1.3 1.4 i 1 3 11

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 19

2 Differentiation Rules 2.1 2.2 2.3 2.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Tangent Lines and Implicit Differentiation . . . . . . . . . . . . . . . 28 31

3 Applications of Differentiation 3.1 3.2 3.3 3.4 3.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Differential, Linear Approximation, Newton’s Method . . . . . . . . . 51 i

3.6 3.7 3.8

Antiderivatives and Differential Equations . . . . . . . . . . . . . . . 55 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . 58 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 65

4 Parametric Equations and Polar Coordinates 4.1 4.2 4.3 4.4

Introduction . . . . . . . . . ....