MATH 221 FIRST SEMESTER CALCULUS

1 MATH 221 FIRST SEMESTERCALCULUS fall 2007 Typeset:December 11, 200712 Math 221 1st SEMESTER CalculusLecture notes version (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 LATEX andPythonfiles which were used to produce these notes are available at thefollowing web 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 Free Documentation License, Version 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.

2 3 ContentsI. Numbers, Points, Lines and Curves81. What is a number?8 Another reason to believe in 29 Why are real numbers called real?10 Exercises102. The real number line and Set notation11 Exercises123. Sets of Points in the Cartesian Lines13 Exercises134. Example: Find the domain and range off(x) = 1 Functions in real life 155. The graph of a Vertical Line Example156. Inverse functions and Implicit Another example: domain of an implicitly defined Example: the equation alone does not determine the Why use implicit functions? Inverse Inverse trigonometric functions19 Exercises19II. Derivatives (1)217. The tangent to a curve218. An example tangent to a parabola229. Instantaneous velocity2310. Rates of change24 Exercises24 III. Limits and Continuous Functions2511. Informal definition of Example: substituting numbers to guess a Example: Substituting numbers can suggest the wrong answer26 Exercise2612.

3 The formal, authoritative, definition of Show thatlimx 33x+ 2 = Show thatlimx 1x2= Show thatlimx 41/x= 1/429 Exercises3013. Variations on the limit Left and right Limits at Example Limit of1 Example Limit of1/x(again)3114. Properties of the Limit3115. Examples of limit Findlimx Try the examples and using the limit Example Findlimx 2 Example Findlimx 2 Example The derivative of xatx= Limit asx of rational Another example with a rational function3516. When limits fail to The sign function nearx= The example of the backward Trying to divide by zero using a Using limit properties to show a limit Limits at which don t exist3817. What s in a name?3818. Limits and A backward cosine sandwich4019. Polynomials are rational functions are Some discontinuous How to make functions Sandwich in a bow tie4220.

4 Substitution in Computelimx 3 x3 3x2+ 242 Exercises4321. Two Limits in Trigonometry43 Exercises45IV. Derivatives (2)4722. Derivatives Other notations4723. Direct computation of Example The derivative off(x) =x2isf (x) = The derivative ofg(x) =xisg (x) = The derivative of any constant function is Derivative ofxnforn= 1,2,3, .. Differentiable implies Some non-differentiable functions50 Exercises5224. The Differentiation Sum, product and quotient Proof of the Sum Proof of the Product Proof of the Quotient A shorter, but not quite perfect derivation of the Quotient Differentiating a constant multiple of a Picture of the Product Rule5525. Differentiating powers of Product rule with more than one The Power The Power Rule for Negative Integer The Power Rule for rational Derivative ofxnfor Example differentiate a Example differentiate a rational Derivative of the square root57 Exercises5726.

5 Higher The derivative is a Operator notation59 Exercises5927. Differentiating Trigonometric functions60 Exercises6128. The Chain Composition of A real world Statement of the Chain FIRST Example where you really need the Chain The Power Rule and the Chain The volume of a growing yeast A more complicated The Chain Rule and composing more than two functions67 Exercises6729. Implicit The Dealing with equations of the formF1(x, y) =F2(x, y) Example Derivative of4 1 Another Derivatives of Arc Sine and Arc Tangent71 Exercises on implicit differentiation71 Exercises on rates of change72V. Graph Sketching and Max-Min Problems7530. Tangent and Normal lines to a graph7531. The intermediate value theorem76 Example Square root of 276 Example The equation + sin = 276 Example Solving1/x= 07632. Finding sign changes of a Example7733. Increasing and decreasing functions7734.

6 Example: the parabolay= Example: the hyperbolay= 1 Graph of a cubic A function whose tangent turns up and down infinitely often near the origin 8135. Maxima and Where to find local maxima and How to tell if a stationary point is a maximum, a minimum, or Example local maxima and minima off(x) =x3 A stationary point that is neither a maximum nor a minimum8436. Must there always be a maximum?8537. Examples functions with and without maxima or minima8538. General method for sketching the graph of a Example the graph of a rational function8739. Convexity, Concavity and the Second Example the cubic functionf(x) =x3 The second derivative Example that cubic function When the second derivative test doesn t work9040. Proofs of some of the Proof of the Mean Value Proof of Theorem Proof of Theorem Optimization Example The rectangle with largest area and given Exercises95VI.

7 Exponentials and Logarithms9842. The trouble with powers of negative numbers9943. Logarithms10044. Properties of logarithms10045. Graphs of exponential functions and logarithms10046. The derivative ofaxand the definition ofe10147. Derivatives of Logarithms10348. Limits involving exponentials and logarithms10349. Exponential growth and Half time and doubling DeterminingX0andk105 Exercises105 VII. The Integral10950. Area under a Graph10951. Whenfchanges its sign11152. The Fundamental Theorem of Terminology112 Exercises11253. The indefinite You can always check the About +C Standard Integrals11654. Properties of the Integral11655. The definite integral as a function of its integration bounds11756. Method of Leibniz notation for Substitution for definite Example of substitution in a definite integral120 Exercises1217 VIII. Applications of the integral12557.

8 Areas between graphs125 Exercises12558. Cavalieri s principle and volumes of Example Volume of a General Cavalieri s Solids of revolution12959. Examples of volumes of solids of Problem 1: RevolveRaround Problem 2: RevolveRaround the linex= Problem 3: RevolveRaround the liney= 213160. Volumes by cylindrical Example The solid obtained by rotatingRabout they-axis, again135 Exercises13561. Distance from velocity, velocity from Motion along a Velocity from Free fall in a constant gravitational Motion in the plane parametric The velocity of an object moving in the Example the two motions on the circle from The length of a Length of a parametric The length of the graph of a Examples of length computations14063. Work done by a Work as an Kinetic energy14264. Work done by an electric , Points, Lines and 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 .

9 To refresh your memory let s look at the variouskinds of real numbers that one runs simplest numbers are thepositive integers1,2,3,4, the numberzero0,and thenegative integers , 4, 3, 2, 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 are the so called fractions orrational numberssuch as12,13,23,14,24,34,43, or 12, 13, 23, 14, 24, 34, 43, By definition, any whole number is a rational number (in particular zero is a rationalnumber.)You can add, subtract, multiply and divide any pair of rational numbers and the resultwill again be a rational number (provided you don t try to divide by zero).One day in middle school you were told that there are other numbers besides therational numbers, and the FIRST example of such a number is the square root of two. It hasbeen known ever since the time of the greeks that no rational number exists whose squareis exactly 2, you can t find a fractionmnsuch that`mn 2= 2, , since( )2= is less than 2, and( )2= is more than 2,it seems that there should be some numberxbetween and whose square is exactly2.

10 So, we assume that there is such a number, and we call it the square root of 2, writtenas 2. This raises several questions. How do we know there really is a number and for whichx2= 2? How many other such numbers we are going to assume intoexistence? Do these new numbers obey the same algebra rules as the rational numbers?( when you add three numbersa,bandcthe sum does not depend on the order inwhich you add them.) If we knew precisely what these numbers (like 2) were then wecould perhaps answer such questions. It turns out to be rather difficult to give a precisedescription of what a number is, and in this course we won t try to get anywhere near thebottom of this issue. Instead we will think of numbers as infinite decimal expansions can represent certain fractions as decimal fractions, all fractions can be represented as decimal fractions. For instance, expanding13intoa decimal fraction leads to an unending decimal fraction13= 333 333 333 333 It is impossible to write the complete decimal expansion of13because it contains infinitelymany digits.