[PDF] Math 150 Lecture Notes Introduction to Functions - TAMU Math
The term function is used to describe a dependence of one quantity on another A function f is a rule that assigns to each element x in set A exactly one element, called f(x), in a set B The set A is called the domain of the function
[PDF] Module 1 Lecture Notes
17 Determining the Domain and Range of a Function Graphically 12 18 Reading the Math 111 Module 1 Lecture Notes Definition 1 A relation is a
[PDF] LECTURE NOTES ON RELATIONS AND FUNCTIONS Contents 1
LECTURE NOTES ON RELATIONS AND FUNCTIONS PETE L CLARK Contents 1 Relations 1 11 The idea of a relation 1 12 The formal definition of a
[PDF] MATH 221 FIRST SEMESTER CALCULUS
This is a self contained set of lecture notes for Math 221 The subject of this course is “functions of one real variable” so we begin by wondering what a real
[PDF] Chapter 10 Functions
one to one and onto (or injective and surjective), how to compose functions, and when they are invertible Let us start with a formal definition Definition 63
[PDF] Part I: Sets, Functions, and Limits
they appeared at the end of the lecture, have been made and repro duced as " Lecture Notes" Consequently, as you proceed through an assignment, there is
[PDF] Lecture Notes 2
Linear Functions (LECTURE NOTES 1) 1 Points, graphs and tables reading ability versus level of illumination Consider the following graph of the set of points
[PDF] Chapter 9: Modeling Our World Lecture notes Math 1030 Section A
These relation ships are described by mathematical tools called functions Section A2 Language and Notation of Functions Definition of function A function
[PDF] Lecture Notes in Calculus - Einstein Institute of Mathematics
Jul 10, 2013 · Lecture Notes in Calculus Raz Kupferman In this course we will cover the calculus of real univariate functions, which was developed during
[PDF] Topic 1: Elementary Linear Functions Lecture Notes: section 11
If a student gets y= 80 for their final grade, how many hours per week did they study? Express x as a function of y The Inverse Function is x = (y 5) 15 solving
[PDF] functions of ingredients worksheet
[PDF] functions of management pdf notes
[PDF] functions of mobile computing
[PDF] functions of propaganda
[PDF] functions of proteins
[PDF] functions of the nervous system
[PDF] functions of the respiratory system
[PDF] functions of the skin
[PDF] functions of theatre in society
[PDF] functions pdf
[PDF] functions pdf notes
[PDF] functions problems and solutions pdf
[PDF] fundamental analytical chemistry calculations
[PDF] fundamental of business intelligence grossmann w rinderle ma pdf
MATH 221
FIRST SEMESTER
CALCULUS
fall 2009Typeset:June 8, 2010
1MATH 221 { 1st SEMESTER CALCULUS
LECTURE NOTES VERSION 2.0 (fall 2009)This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting
from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX andPythonles which were used to produce these notes are available at the following web site http://www.math.wisc.edu/ ~angenent/Free-Lecture-Notes They are meant to be freely available in the sense that \free software" is free. More precisely: Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version1.2 or 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 the section entitled "GNU Free Documentation License".Contents
Chapter 1. Numbers and Functions
51. What is a number?
52. Exercises
73. Functions
84. Inverse functions and Implicit functions
105. Exercises
13Chapter 2. Derivatives (1)
151. The tangent to a curve
152. An example { tangent to a parabola
163. Instantaneous velocity
174. Rates of change
175. Examples of rates of change
186. Exercises
18Chapter 3. Limits and Continuous Functions
211. Informal denition of limits
212. The formal, authoritative, denition of limit
223. Exercises
254. Variations on the limit theme
255. Properties of the Limit
276. Examples of limit computations
277. When limits fail to exist
298. What's in a name?
329. Limits and Inequalities
3310. Continuity
3411. Substitution in Limits
3512. Exercises
3613. Two Limits in Trigonometry
3614. Exercises
38Chapter 4. Derivatives (2)
411. Derivatives Dened
412. Direct computation of derivatives
423. Dierentiable implies Continuous
434. Some non-dierentiable functions
435. Exercises
446. The Dierentiation Rules
457. Dierentiating powers of functions
488. Exercises
499. Higher Derivatives
5010. Exercises
5111. Dierentiating Trigonometric functions
5112. Exercises
5213. The Chain Rule
5214. Exercises
5715. Implicit dierentiation
5816. Exercises
60Chapter 5. Graph Sketching and Max-Min Problems
631. Tangent and Normal lines to a graph
632. The Intermediate Value Theorem
63 3. Exercises64
4. Finding sign changes of a function
655. Increasing and decreasing functions
666. Examples
677. Maxima and Minima
698. Must there always be a maximum?
71 9. Examples { functions with and without maxima or
minima 7110. General method for sketching the graph of a
function 7211. Convexity, Concavity and the Second Derivative
7412. Proofs of some of the theorems
7513. Exercises
7614. Optimization Problems
7 715. Exercises
78Chapter 6. Exponentials and Logarithms (naturally) 81
1. Exponents
812. Logarithms
823. Properties of logarithms
834. Graphs of exponential functions and logarithms
835. The derivative ofaxand the denition ofe84
6. Derivatives of Logarithms
857. Limits involving exponentials and logarithms
868. Exponential growth and decay
869. Exercises
87Chapter 7. The Integral
911. Area under a Graph
912. Whenfchanges its sign92
3. The Fundamental Theorem of Calculus
934. Exercises
945. The indenite integral
956. Properties of the Integral
977. The denite integral as a function of its integration
bounds 988. Method of substitution
999. Exercises
100Chapter 8. Applications of the integral
1051. Areas between graphs
1052. Exercises
1063. Cavalieri's principle and volumes of solids
1064. Examples of volumes of solids of revolution
1095. Volumes by cylindrical shells
1116. Exercises
1137. Distance from velocity, velocity from acceleration
1138. The length of a curve
1169. Examples of length computations
11 710. Exercises
11811. Work done by a force
11812. Work done by an electric current
119Chapter 9. Answers and Hints
121GNU Free Documentation License
1253
1. APPLICABILITY AND DEFINITIONS125
2. VERBATIM COPYING
1253. COPYING IN QUANTITY
1254. MODIFICATIONS
1255. COMBINING DOCUMENTS
1266. COLLECTIONS OF DOCUMENTS
1267. AGGREGATION WITH INDEPENDENT WORKS
1268. TRANSLATION
1269. TERMINATION
12610. FUTURE REVISIONS OF THIS LICENSE
12611. RELICENSING
1264
CHAPTER 1
Numbers and FunctionsThe subject of this course is \functions of one real variable" so we begin by wondering what a real number
\really" is, and then, in the next section, what a function is.1. What is a number?
1.1. Dierent kinds of numbers.The simplest numbers are thepositive integers
1;2;3;4;
the numberzero 0; and thenegative 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 are the so called fractions orrational numberssuch as 12 ;13 ;23 ;14 ;24 ;34 ;43 or 12 ;13 ;23 ;14 ;24 ;34 ;43 By denition, any whole number is a rational number (in particular zero is a rational number.)You can add, subtract, multiply and divide any pair of rational numbers and the result will 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 the rational numbers, and
the rst example of such a number is the square root of two. It has been known ever since the time of the
greeks that no rational number exists whose square is exactly 2, i.e. you can't nd a fractionmn such that mn2= 2;i.e.m2= 2n2:
xx21:21:44
1:31:69
1:41:96<2
1:52:25>2
1:62:56
Nevertheless, if you computex2for some values ofxbetween 1 and 2, and check if you get more or less than 2, then it looks like there should be some numberxbetween 1:4 and1:5 whose square is exactly 2. So, weassumethat there is such a number, and we call it
the square root of 2, written asp2. This raises several questions. How do we know there really is a number between 1:4 and 1:5 for whichx2= 2? How many other such numbers are we going to assume into existence? Do these new numbers obey the same algebra rules(likea+b=b+a) as the rational numbers? If we knew precisely what these numbers (likep2) were then we could perhaps answer such questions. It turns out to be rather dicult to give a precise
description of what a number is, and in this course we won't try to get anywhere near the bottom of this
issue. Instead we will think of numbers as \innite decimal expansions" as follows. One can represent certain fractions as decimal fractions, e.g. 27925=1116100 = 11:16: 5 Not all fractions can be represented as decimal fractions. For instance, expanding 13 into a decimal fraction leads to an unending decimal fraction 13
= 0:333333333333333It is impossible to write the complete decimal expansion of13because it contains innitely many digits.
But we can describe the expansion: each digit is a three. An electronic calculator, which always represents
numbers asnitedecimal numbers, can never hold the number13 exactly. Every fraction can be written as a decimal fraction which may or may not be nite. If the decimal expansion doesn't end, then it must repeat. For instance, 17 = 0:142857142857142857142857::: Conversely, any innite repeating decimal expansion represents a rational number.Areal numberis specied by a possibly unending decimal expansion. For instance,p2 = 1:4142135623730950488016887242096980785696718753769:::
Of course you can never writeallthe digits in the decimal expansion, so you only write the rst few digits
and hide the others behind dots. To give a precise description of a real number (such asp2) you have to
explain how you couldin principlecompute as many digits in the expansion as you would like. During the
next three semesters of calculus we will not go into the details of how this should be done.1.2. A reason to believe in
p2.The Pythagorean theorem says that the hy-
potenuse of a right triangle with sides 1 and 1 must be a line segment of lengthp2. In middle or high school you learned something similar to the following geometric construction of a line segment whose length isp2. Take a square with side of length 1, and construct a new square one of whose sides is the diagonal of the rst square. The gure you get consists of 5 triangles of equal area and by counting triangles you see that the largersquare has exactly twice the area of the smaller square. Therefore the diagonal of the smaller square, being
the side of the larger square, isp2 as long as the side of the smaller square.Why are real numbers called real?
All the numbers we will use in this rst semester of calculus are\real numbers." At some point (in 2nd semester calculus) it becomes useful to assume that there is a number
whose square is1. No real number has this property since the square of any real number is positive, so
it was decided to call this new imagined number \imaginary" and to refer to the numbers we already have
(rationals,p2-like things) as \real."1.3. The real number line and intervals.
It is customary to visualize the real numbers as pointson a straight line. We imagine a line, and choose one point on this line, which we call theorigin. We also
decide which direction we call \left" and hence which we call \right." Some draw the number line vertically
and use the words \up" and \down." To plot any real numberxone marks o a distancexfrom the origin, to the right (up) ifx >0, to the left (down) ifx <0. Thedistance along the number linebetween two numbersxandyisjxyj. In particular, the distance is never a negative number.321 0 1 2 3Figure 1.
To draw the half open interval[1;2)use a lled dot to mark the endpoint which is included and an open dot for an excluded endpoint. 621 0 1 2p2
Figure 2.To ndp2on the real line you draw a square of sides1and drop the diagonal onto the real line.Almost every equation involving variablesx,y, etc. we write down in this course will be true for some
values ofxbut not for others. In modern abstract mathematics a collection of real numbers (or any other
kind of mathematical objects) is called aset. Below are some examples of sets of real numbers. We will use
the notation from these examples throughout this course. The collection of all real numbers between two given real numbers form an interval. The following notation is used (a;b) is the set of all real numbersxwhich satisfya < x < b. [a;b) is the set of all real numbersxwhich satisfyax < b. (a;b] is the set of all real numbersxwhich satisfya < xb. [a;b] is the set of all real numbersxwhich satisfyaxb. If the endpoint is not included then it may be1or1. E.g. (1;2] is the interval of all real numbers (both positive and negative) which are2.1.4. Set notation.
A common way of describing a set is to say it is the collection of all real numbers which satisfy a certain condition. One uses this notationA=xjxsatises this or that condition
Most of the time we will use upper case letters in a calligraphic font to denote sets. (A,B,C,D, ...)
For instance, the interval (a;b) can be described as (a;b) =xja < x < bThe set
B=xjx21>0
consists of all real numbersxfor whichx21>0, i.e. it consists of all real numbersxfor which eitherx >1
orx <1 holds. This set consists of two parts: the interval (1;1) and the interval (1;1).You can try to draw a set of real numbers by drawing the number line and coloring the points belonging
to that set red, or by marking them in some other way. Some sets can be very dicult to draw. For instance,C=xjxis a rational number
can't be accurately drawn. In this course we will try to avoid such sets.Sets can also contain just a few numbers, like
D=f1;2;3g
which is the set containing the numbers one, two and three. Or the setE=xjx34x2+ 1 = 0
which consists of the solutions of the equationx34x2+ 1 = 0. (There are three of them, but it is not easy
to give a formula for the solutions.) IfAandBare two sets thenthe union ofAandBis the set which contains all numbers that belong either toAor toB. The following notation is used