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UW-Madison MATH 221 - MATH 221 Lecture Notes

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I. Numbers, Points, Lines and Curves1. What is a number?Another reason to believe in 2Why are real numbers called real? Exercises2. The real number line and intervals2.1. Intervals2.2. Set notationExercises3. Sets of Points in the Plane3.1. Cartesian Coordinates3.2. Sets3.3. LinesExercises4. Functions4.1. Example: Find the domain and range of f(x) = 1/x24.2. Functions in ``real life''5. The graph of a function5.1. Vertical Line Property 5.2. Example6. Inverse functions and Implicit functions6.1. Example6.2. Another example: domain of an implicitly defined function6.3. Example: the equation alone does not determine the function6.4. Why use implicit functions?6.5. Inverse functions6.6. Examples6.7. Inverse trigonometric functionsExercisesII. Derivatives (1)7. The tangent to a curve8. An example -- tangent to a parabola9. Instantaneous velocity10. Rates of changeExercisesIII. Limits and Continuous Functions11. Informal definition of limits11.1. Example11.2. Example: substituting numbers to guess a limit11.3. Example: Substituting numbers can suggest the wrong answerExercise12. The formal, authoritative, definition of limit12.1. Show that limx33x+2=11 12.2. Show that limx1x2 = 112.3. Show that limx41/x = 1/4Exercises13. Variations on the limit theme13.1. Left and right limits13.2. Limits at infinity. 13.3. Example -- Limit of 1/x 13.4. Example -- Limit of 1/x (again) 14. Properties of the Limit15. Examples of limit computations15.1. Find limx2x215.2. Try the examples 11.2 and 11.3 using the limit properties15.3. Example -- Find limx2x 15.4. Example -- Find limx2x 15.5. Example -- The derivative of x at x=2. 15.6. Limit as x of rational functions15.7. Another example with a rational function 16. When limits fail to exist16.1. The sign function near x=0 16.2. The example of the backward sine16.3. Trying to divide by zero using a limit16.4. Using limit properties to show a limit does not exist16.5. Limits at which don't exist17. What's in a name?18. Limits and Inequalities18.1. A backward cosine sandwich19. Continuity19.1. Polynomials are continuous19.2. Rational functions are continuous19.3. Some discontinuous functions19.4. How to make functions discontinuous19.5. Sandwich in a bow tie20. Substitution in Limits20.1. Compute limx3x3-3x2+2Exercises21. Two Limits in TrigonometryExercisesIV. Derivatives (2)22. Derivatives Defined22.1. Other notations23. Direct computation of derivatives23.1. Example -- The derivative of f(x)= x2 is f'(x) = 2x 23.2. The derivative of g(x) = x is g'(x) =1 23.3. The derivative of any constant function is zero 23.4. Derivative of xn for n=1, 2, 3, … 23.5. Differentiable implies Continuous23.6. Some non-differentiable functionsExercises24. The Differentiation Rules24.1. Sum, product and quotient rules24.2. Proof of the Sum Rule24.3. Proof of the Product Rule24.4. Proof of the Quotient Rule 24.5. A shorter, but not quite perfect derivation of the Quotient Rule 24.6. Differentiating a constant multiple of a function 24.7. Picture of the Product Rule25. Differentiating powers of functions25.1. Product rule with more than one factor25.2. The Power rule 25.3. The Power Rule for Negative Integer Exponents 25.4. The Power Rule for Rational Exponents 25.5. Derivative of xn for integer n 25.6. Example -- differentiate a polynomial 25.7. Example -- differentiate a rational function25.8. Derivative of the square root Exercises26. Higher Derivatives26.1. The derivative is a function26.2. Operator notationExercises27. Differentiating Trigonometric functionsExercises28. The Chain Rule28.1. Composition of functions28.2. A real world example28.3. Statement of the Chain Rule28.4. First example28.5. Example where you really need the Chain Rule28.6. The Power Rule and the Chain Rule28.7. The volume of a growing yeast cell28.8. A more complicated example28.9. The Chain Rule and composing more than two functionsExercisesMATH 221FIRST SEMESTERCALCULUSfall 2007Typeset:September 25, 200712Math 221 – 1st Semester CalculusLecture notes version 1.0 (Fall 2007)This is a self contained set of lecture notes for Math 221. The notes were written bySigurd Angenent, starting from an extensive collection of notes and problems compiled byJoel Robbin.The LATEX and Python files which were used to produce these notes are available at thefollowing web sitewww.math.wisc.edu/∼angenent/Free-Lecture-NotesThey are meant to b e freely available in the sense that “free software” is free. Moreprecisely:Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/ormodify this document under the terms of the GNU Free Documentation License, Version 1.2or any later version published by the Free Software Foundation; with no Invariant Sections,no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in thesection entitled ”GNU Free Documentation License”.3ContentsI. Numbers, Points, Lines and Curves 61. What is a number? 6Another reason to believe in√2 7Why are real numbers called real? 8Exercises 82. The real number line and intervals 82.1. Intervals 92.2. Set notation 9Exercises 103. Sets of Points in the Plane 103.1. Cartesian Coordinates 103.2. Sets 103.3. Lines 11Exercises 114. Functions 124.1. Example: Find the domain and range of f(x) = 1/x2124.2. Functions in “real life” 135. The graph of a function 135.1. Vertical Line Property 135.2. Example 136. Inverse functions and Implicit functions 146.1. Example 146.2. Another example: domain of an implicitly defined function 146.3. Example: the equation alone does not determine the function 156.4. Why use implicit functions? 156.5. Inverse functions 166.6. Examples 166.7. Inverse trigonometric functions 17Exercises 17II. Derivatives (1) 197. The tangent to a curve 198. An example – tangent to a parabola 209. Instantaneous velo ci ty 2110. Rates of change 22Exercises 22III. Limits and Continuous Functions 2311. Informal definition of limits 2311.1. Example 2311.2. Example: substituting numbers to guess a limit 2311.3. Example: Substituting numbers can suggest the wrong answer 24Exercise 2412. The formal, authoritative, definition of limit 2412.1. Show that limx→33x + 2 = 11 2612.2. Show that limx→1x2= 1 2712.3. Show that limx→41/x = 1/4 27Exercises 2813. Variations on the limit theme 2813.1. Left and right limits 28413.2. Limits at infinity. 2813.3. Example – Limit of 1/x 2913.4. Example – Limit of 1/x (again) 2914. Properties of the Limit 2915. Examples of limit computations


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