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150notes - Calculus I with Review Differential Calculus...

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Calculus I with Review Differential Calculus Lecture Notes Veselin Jungic & Jamie Mulholland Department of Mathematics Simon Fraser University c Jungic/Mulholland, August 18, 2011 License is granted to print this document for personal/educational use.
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Contents Contents i Preface iii Greek Alphabet v 1 Review: Functions and Models 1 1.1 Four Ways to Define a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Mathematical Models: A Catalog of Essential Functions . . . . . . . . . . . . . . . . . . . . . 9 1.3 New Functions From Old Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Exponential Functions & Inverse Functions and Logarithms . . . . . . . . . . . . . . . . . . . 23 Review: Preparation for Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Limits and Derivatives 37 2.1 The Tangent and Velocity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Calculating Limits Using the Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 The Precise Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6 Limits at Infinity: Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Review: Problem Solving and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.7 Derivatives and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.8 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Differentiation Rules 89 3.1 Derivatives of Polynomials and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 90 3.2 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.5 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 i
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ii CONTENTS 3.6 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.7 Rates of Change in the Natural and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . 120 3.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Review: Preparation for Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.9 Related rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.10 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3.11 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4 Applications of the Derivative 153 4.1 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3 How Derivatives Affect the Shape of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.4 Indeterminate Forms and L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.5 Summary of Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.6 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.8 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5 Parametric Equations and Polar Coordinates 195 5.1 Curves Defined by Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.3 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 5.4 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6 Review Material 225 6.1 Midterm 1 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.2 Midterm 2 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.3 End of Term Review Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.4 Final Exam Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.5 Final Exam Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Bibliography 263 Index 264
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Preface This booklet contains our notes for courses Math 150/151 - Calculus I at Simon Fraser University. Stu- dents are expected to bring this booklet to each lecture and to follow along, filling in the details in the blanks provided, during the lecture. Definitions of terms are stated in orange boxes and theorems appear in blue boxes . Next to some examples you’ll see [ link to applet ]. The link will take you to an online interactive applet to accompany the example - just like the ones used by your instructor in the lecture. Clicking the link above will take you to the following website containing all the applets: http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html Try it now.
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