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 TrigonometryExercisesMATH 221FIRST SEMESTERCALCULUSfall 2007Typeset:August 31, 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 be 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 Fr ee 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 51. What is a number? 5Another reason to believe in√2 6Why are real numbers called real? 7Exercises 72. The real number line and intervals 72.1. Intervals 82.2. Set notation 8Exercises 93. Sets of Points in the Plane 93.1. Cartesian Coordinates 93.2. Sets 93.3. Lines 10Exercises 104. Functions 114.1. Example: Find the domain and range of f(x) = 1/x2114.2. Functions in “real life” 125. The graph of a function 125.1. Vertical Line Property 125.2. Example 126. Inverse functions and Implicit functions 136.1. Example 136.2. Another example: domain of an implicitly defined function 136.3. Example: the equation alone does not determine the function 146.4. Why use implicit functions? 146.5. Inverse functions 156.6. Examples 156.7. Inverse trigonometric functions 16Exercises 16II. Derivatives (1) 187. The tangent to a curve 188. An example – tangent to a parabola 199. Instantaneous velocity 2010. Rates of change 21Exercises 21III. Limits and Continuous Functions 2211. Informal definition of limits 2211.1. Example 2211.2. Example: substituting numbers to guess a limit 2211.3. Example: Substituting numbers can suggest the wrong answer 23Exercise 2312. The formal, authoritative, definition of limit 2312.1. Show that limx→33x + 2 = 11 2512.2. Show that limx→1x2= 1 2612.3. Show that limx→41/x = 1/4 26Exercises 2713. Variations on the limit theme 2713.1. Left and right limits 27413.2. Limits at infinity. 2713.3. Example – Limit of 1/x 2813.4. Example – Limit of 1/x (again) 2814. Properties of the Limit 2815. Examples of limit computations 2915.1. Find limx→2x22915.2. Try the examples 11.2 and 11.3 using the limit properties 3015.3. Example – Find limx→2√x 3015.4. Example – Find limx→2√x 3115.5. Example – The derivative of√x at x = 2. 3115.6. Limit as x → ∞ of rational functions 3115.7. Another example with a rational function 3216. When limits fail to exist 3216.1. The sign function near x = 0 3216.2. The example of the backward sine 3316.3. Trying to divide by zero using a li mi t 3416.4. Using limit properties to show a li mi t does not exist 3416.5. Limits at ∞ which don’t exist 3517. What’s in a name? 3518. Limits and Inequalities 3618.1. A backward cosine sandwich 3719. Continuity 3719.1. Polynomials are continuous 3819.2. Rational functions are continuous 3819.3. Some discontinuous functions 3819.4. How to make functions discontinuous 3819.5. Sandwich in a bow tie 3920. Substitution in Limits 3920.1. Compute limx→3√x3− 3x2+ 2 39Exercises 4021. Two Limits in Trigonometry 40Exercises 425I. Numbers, Points, Lines and Curves1. What is a number?The basic objects that we deal with in calculus are the so-called “real numbers” whichyou have already seen in pre-calculus. To refresh your memory let’s look at the variouskinds of “real” numbers that one runs into.The simplest numbers are the positive integers1, 2, 3, 4, ···the number zero0,and the negative integers··· , −4, −3, −2, −1.Together these form the integers or “whole numbers.”Next, there are the numbers you get by dividing one whole number by another (nonzero)whole number. These