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Unformatted text preview: Math 221 – 1st Semester Calculus Lecture Notes for Fall 2006. Prof J. Robbin December 21, 2006 All references to “Thomas” or “the textbook” in these notes refer to Thomas’ CALCULUS 11th edition published by Pearson Addison Wesley in 2005. These notes may be downloaded from http://www.math.wisc.edu/~robbin/221dir/lecs221.pdf . Some portions of these notes are adapted from http://www.math.wisc.edu/~angenent/FreeLectureNotes/ . Some problems come from a list compiled by Arun Ram, others come from the WES program, and others come from the aforementioned Thomas text or from Stewart, Calculus (Early Transcendentals) , 3rd Edition. 1 Contents I What you need to know to take Calculus 221 5 1 Algebra 5 2 Coordinate Geometry 7 3 Functions 9 4 Trigonometry 12 5 Additional Exercises 15 II Limits 17 6 Tangent and Velocity 17 7 Limits 19 8 Two Limits in Trigonometry 26 9 Continuity 27 III Differentiation 31 10 Derivatives Defined 31 11 Higher Derivatives and Differential Notation 37 12 Implicit Functions 41 13 The Chain Rule 43 14 Inverse Functions 47 15 Differentiating Trig Functions 50 16 Exponentials and Logarithms 53 17 Parametric Equations 59 18 Approximation* 63 19 Additional Exercises 67 2 IV Applications of Derivatives 68 20 The Derivative as A Rate of Change 68 21 Related Rates 71 22 Some Theorems about Derivatives 74 23 Curve Plotting 81 24 Max Min Word Problems 84 25 Exponential Growth 87 26 Indeterminate Forms (l’Hˆopital’s Rule) 91 27 Antiderivatives 93 28 Additional Problems 95 V Integration 96 29 The Definite Integral 96 30 The Fundamental Theorem of Calculus 104 31 Averages 107 32 Change of Variables 108 VI Applications of Definite Integrals 112 33 Plane Area 112 34 Volumes 114 35 Arc Length 118 36 Surface Area 122 37 Center of Mass 125 VII Loose Ends 132 38 The Natural Log Again* 132 3 39 Taylor Approximation* 133 40 Newton’s Method* 135 VIII More Problems 136 IX Notes for TA’s 153 4 Chapter I What you need to know to take Calculus 221 In this chapter we will review material from high school mathematics. We also teach some of this material at UW in Math 112, Math 113, and Math 114, and some of it will be reviewed again as we need it. This material should be familiar to you. If it is not, you may not be ready for calculus. Pay special attention to the definitions. Important terms are shown in boldface when they are first defined. 1 Algebra This section contains some things which should be easy for you. (If they are not, you may not be ready for calculus.) § 1.1. Answer these questions. 1. Factor x 2 6 x + 8. 2. Find the values of x which satisfy x 2 7 x + 9 = 0. (Quadratic formula.) 3. x 2 y 2 =? Does x 2 + y 2 factor? 4. True or False: √ x 2 + 4 = x + 2?...
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This note was uploaded on 03/27/2008 for the course MATH 221 taught by Professor Denissou during the Fall '07 term at University of Wisconsin Colleges Online.
 Fall '07
 Denissou
 Calculus, The Land

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