2 Pages

#### math115-110notes

Course: MATH 115, Fall 2009

School: UConn

Word Count: 547

Rating:

###### Document Preview

Indeterminates We Calculating have implicitly made use of the following theorem, which is an almost immediate consequence of the preliminary denition of a limit. Theorem 1. If f (x) = g(x) for x = c and either f or g has a limit at c, then both must have a limit at c and their limits must be the same. In eect, this is a reiteration of the fact that the value of a function at a limit point has no eect on a limit....

##### Unformatted Document Excerpt
Coursehero >> Connecticut >> UConn >> MATH 115

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Indeterminates We Calculating have implicitly made use of the following theorem, which is an almost immediate consequence of the preliminary denition of a limit. Theorem 1. If f (x) = g(x) for x = c and either f or g has a limit at c, then both must have a limit at c and their limits must be the same. In eect, this is a reiteration of the fact that the value of a function at a limit point has no eect on a limit. Typical Calculation We use this theorem in the calculation of the limit almost every time we are faced with an indeterminate. The calculation generally looks like the following: limxc f (x) = = limxc g(x) = L. The idea is that we start with an indeterminate f (x) and keep simplifying, each time getting either a new form of f (x) or another function that is equal to f (x) except at c, until we have found some function g(x) whose limit is easy to nd. Example For example, the calculation in the previous example might look like the following: x2 + x 12 (x 3)(x + 4) limx3 = limx3 = limx3 (x + 4) = 7. x3 x3 To repeat, in a nutshell, we usually wind up calculating limits of indeterminates by simplifying until weve found a function, equal to the original except at the limit point, which is not indeterminate. Types of Manipulations Most of the time, the algebraic manipulation we need to do to simplify is in one of the following three categories. (1) Factor and cancel. (2) Rationalize and cancel. (3) Simplifying a complex fractional expression The example we saw fell in the rst category, factor and cancel. Consider the following examples in the other two categories. limx9 = limx9 x3 x3 x+3 = limx9 x9 x9 x+3 x9 1 1 = limx9 = . 6 (x 9)( x + 3) x+3 1 Example Rationalize and Cancel 2 Note: The algebraic manipulation used here was essentially the same as the technique of Rationalizing the Denominator often taught in elementary algebra, but we actually wind up rationalizing the numerator! Simplifying a Complex Fractional Expression Here, complex is used in the sense of complicated, where we have a rational expression (an algebraic expression that may be written or rewritten as a quotient of polynomials) where either the numerator or denominator is itself an expression involving fractions. Consider limz4 1 4 . The simplication here may be approached in z4 at least two dierent ways. Both are illustrated. 1 z Technique 1 z 4z 4z 4z = limz4 = limz4 4z = limz4 limz4 z4 z4 z4 4z 1 1 1 = limz4 = . z4 4z 16 Here, we looked at the numerator as a dierence of fractions and calculated that dierence by rst getting a common denominator and then subtracting the numerators. We then looked at what was left as a quotient and calculated the quotient by inverting the denominator and multiplying. 1 z 1 4 4 4z Technique 2 limz4 1 4 4z 4z 1 = limz4 = limz4 = limz4 = z4 z 4 4z 4z(z 4) 4z 1 z 1 4 1 z 1 . 16 Here, we observe the complications arose from the denominators of 4 4z . and z in the numerator and simply multiplied by 1 in the form 4z There will be other manipulations that will come up, and sometimes the simplication involved using these techniques may not be quite as straightforward, but most of the examples we run across will fall into one of these three cases.

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UConn - MATH - 1132
Parametric EquationsWe sometimes have several equations sharing an independent variable. In those cases, we call the independent variable a parameter and call the equations parametric equations. In many cases, the domain of the parameter is restrict
UConn - MATH - 1132
Integration By PartsIntegration by Parts is a technique that enables us to calculate integrals of functions which are derivatives of products. Its genesis can be seen by dierentiating a product and then ddling around. Write out the formula for the
UConn - MATH - 106
The Natural Logarithm Function and The Exponential FunctionOne specic logarithm function is singled out and one particular exponential function is singled out. Denition 1. e = limx0 (1 + x)1/x Denition 2 (The Natural Logarithm Function). ln x = loge
UConn - MATH - 105
Solving Word ProblemsThe strategy for solving word problems, presented in written form, may be summarized in three words:Read the Question!Well expand on how to read the question.Strategy Look for variables and unknowns represent them by symb
UConn - MATH - 115
Dierent Types of LimitsBesides ordinary, two-sided limits, there are one-sided limits (lefthand limits and right-hand limits), innite limits and limits at innity. Consider limx5 One-Sided Limitsx2 4x 5.One might think that since x2 4x 5 0
UConn - MATH - 115
The Product RuleThe formula for the derivative of a product turns out to be more complicated than the formulas for derivatives of sums and dierences. In plain language, The derivative of a product is equal to the rst factor times the derivative of t
UConn - MATH - 115
LimitsDenition 1 (Limit). If the values f (x) of a function f : A B get very close to a specic, unique number L when x is very close to, but not necessarily equal to, a limit point c, we say the limit of f (x), as x approaches c, is L and write lim
UConn - MATH - 231
Mathematics 231. Chapter 3: Example of Tree Diagram Probability Calculations If a sample space is subdivided into a finite or countably infinite collection of pairwise mutually exclusive events, then the probability of any event can be found by summi
UConn - MATH - 3160
Math 3160 (Math 231) Section 001 SyllabusInstructor: Krzysztof Kubacki, MSB 326, kubacki@math.uconn.edu, Phone: (860)486-3844 Lectures: MWF 11:00 11:50 section 001 Office hours: MWF 1:00 2:00, and by arrangement http:/www.math.uconn.edu/~kubacki/
UConn - MATH - 1030
Math 1030Q - Example Quiz - 10 ptsGrade:Name:You must show all your work to receive full credit, guessing is frowned upon. Question 1. Suppose three candidates - Whitney, Pronzini, and OKane - are running for mayor and the preference ranking of
UConn - MATH - 3160
Section 4.3 Expected Value of a Random Variable (1) Given a random variable X with a finite range of values, the expected value of X is easy to define: If x1 , . . . , xn are the distinct values of X and if the probability mass functionnof X is p(
UConn - MATH - 231
Section 4.3 Expected Value of a Random Variable (1) Given a random variable X with a finite range of values, the expected value of X is easy to define: If x1 , . . . , xn are the distinct values of X and if the probability mass functionnof X is p(
UConn - MATH - 3634
Random numbers EA ValdezRandom numbersMath 3634 Actuarial Models Spring 2009 semesterIntroductionPseudorandom number generationEvaluating integralsIllustrative example 1 Using substitution Illustrative example 2 Substitution with innite limit
UConn - MATH - 3231
EXAMPLE OF EUCLIDS ALGORITHM FOR POLYNOMIALS We will use Euclids algorithm and back-substitution to determine the (monic!) gcd of T 8 1 and T 5 + 2T 3 + 2 in F3 [T ] and to express the gcd as an F3 [T ]-linear combination of the two polynomials. Rem
UConn - MATH - 1070
Section F.41Section F.4: Present Value of Annuities and Amortization Example 1: Suppose your rich uncle has agreed to give you a lump sum of money that you will use to pay for your child's daycare until they are school age in five years. You will
UConn - CSE - 2500
University of Connecticut CSE 2500 Introduction to Discrete Systems Tue/Thu 9:30-10:45 at Castleman 201 Instructor. Aggelos Kiayias. Ofce: ITE Building 243. Ofce Hours: Tuesday 2pm - 4pm and by appointment. Contact Information: http:/www.cse.uconn.ed
UConn - CSE - 244
Chapter 3: Lexical AnalysisAggelos Kiayias Computer Science &amp; Engineering Department The University of Connecticut 371 Fairfield Road, Box U-1155 Storrs, CT 06269aggelos@cse.uconn.edu http:/www.cse.uconn.edu/~akiayiasCSE244CH3.1Lexical Analys
UConn - CSE - 281
Computer Security Spring 2008Botnets, Scripts &amp; Web BrowsersAggelos Kiayias University of ConnecticutBotnetBOTNETCSE281- Computer Security (Spring 2008) University of Connecticut2006-8 Aggelos KiayiasCSE281- Computer Security (Spring 20
UConn - ICDE - 04
Advanced Technology Seminar Similarity Search in Multimedia DatabasesOverview1. 2. 3.Daniel A. Keim and Benjamin BustosDatabases, Data Mining, and Visualization University of Konstanz, Germany4. 5.Introduction Efficiency Effectiveness Appli
UConn - CSE - 207
Spring 2001 CS 207 Project 1The goal is to design a simple BCD calculator that accepts input data in decimal form, encodes the data in BCD, performs computations on the BCD data and outputs the result in decimal form. For example, if the 2-digit de
UConn - OPIM - 203
School of Business Board Room (321) 5:30-7:00 free food, prize drawings! MIS career information!School of Business Board Room Room 321 - 5:30-7:00!&quot;Using Technology to Enable Global Enterprise
UConn - OPIM - 203
Excel Exam ReviewSeptember 18, 2006Formatting Font Size Bold,Italics, etc. Backgroundcolors for cells Merge and Center Freezing rows and columnsSummarizing Data Acrossmultiple worksheetsWhat happens when a worksheet is inserted
UConn - OPIM - 203
OPIM 203C.04 Business Information SystemsMatt DeanSyllabus Office Hours: MW 10:45AM 11:45AM Email: matthew.dean@business.uconn.edu 4 exams total MS Excel 25% Database Design 20% MS Access 30% Final Exam 25% Bonus points are available
UConn - OPIM - 203
Chapter 8 Information Systems Development and Acquisitionwww.prenhall.com/jessup8-1Learning Objectives1. Understand the process used by organizations to manage the development of information systems 2. Describe each major phase of the system dev
UConn - OPIM - 203
Objectives for Monday, Oct 2 Bonus Questions, anyone? Chapter 1 from Grauer bookFilters and Sorting Employees.mdb Relationships revisited Look Ahead.mdb Chapter 2 from Grauer bookCreating a table Creating forms Chapter 3 from Grauer b
UConn - OPIM - 203
NamePhone TypePhone NumberDate EnteredGeorgeMobile860.486.675838972.90381FredDorm860.487.457838970.03819MaggieHome203.717.847736927.81MelissaMobile804.512.969637212.381MelindaHome203.215.543732651.1SusieMobile757.487.646534
UConn - STAT - 100
12. Numerical Descriptive Measures of Central Tendency and Variability.Measures of Central Tendency Usually, we focus our attention on two aspects or measures of central location: We need to have some representative value for the data set which i
UConn - STAT - 5361
Local search Combinatorial Optimization Maximize f (), , where is a nite set of parameters. Example Suppose p predictors x1 , . . . , xp are available to build a linear model for Y . Each candidate model is s X Y = ij xij + j=1First, dene a ne
UConn - STAT - 100
111. Confidence Interval for a Population Proportion^ Sampling Distribution of p^ p=Xn(where n is number of trials and X is the total number of successes) is an unbiased consistent estimatorof p X has the binomial distribution b(n, p)^
UConn - STAT - 100
110. Introduction to Estimation, Confidence Interval for a Population MeanSuppose we have a known distribution, but we don't know what the parameter values (e.g. , 2 , etc.) are. Now how can we estimate these population parameters using a sample
UConn - STAT - 5361
EM OptimizationsExample Let a population be a nite mixture of subpopulations, each one following a distribution p(x | k ) and having a population fraction qk . Both q = (q1 , . . . , qK ), = (1 , . . . , K )are unknown. Let x1 , . . . , xn be a s
UConn - STAT - 220
1Normal DistributionThis is the most important continuous distribution considered here. Many random variables can be properly modeled as normally distributed. Many distributions can be approximated by a normal distribution. The normal distribu
UConn - STAT - 1000
Example. Exercise 2.97How to read data? 1. 2. 3. 4. To read data from the data CD-ROM (that goes with the textbook) click first File Open Worksheet. It opens the Open Worksheet window. Choose Data(*.dat) as Files of type. The data file is on the da
UConn - STAT - 220
1ProbabilityRandom experiment a random experiment is a process or course of action, whose outcome is uncertain performing the same random experiment repeatedly, may result in different outcomes,therefore, the best we can do is talk about the pr
UConn - STAT - 1000
113. Small-Sample Test of Hypothesis about Test of Hypothesis about a Population ProportionSmall-Sample Test of Hypothesis about a Population Mean1. Setup Null Hypothesis H 0 : = 0 Alternative Hypothesis H a : &gt; 0 or H a : &lt; 0 or H a :
CSU Long Beach - SKIM - 43
STAT 475 : Data Analysis with SAS, Note 6 Sung E. Kim, California State University-Long Beach, Dept of Math and Statistics_1NOTE #6: Descriptive and Univariate Statistics IPROC MEANS;PROC MEANS &lt;DATA=mydata&gt; &lt;list of statistics&gt; &lt;options&gt;; VAR
CSU Long Beach - SOC - 320
Soc.320: The Family / Summer-3 2008 / Tues-Thur 01:15-05:00pm (PSY-155) Section: 60 Class# 11380 Last Update: 08/27/2008NOTE:After grades are posted, the University doesn't allow grade changes except for clerical error. It does not allow students
CSU Long Beach - ACARTER - 3
CSU Long Beach - ACARTER - 3
CSU Long Beach - ACARTER - 3
doi:10.1111/j.1420-9101.2005.01054.xResponse of uctuating and directional asymmetry to selection on wing shape in Drosophila melanogaster C. PE LABON,* T. F. HANSEN, A. J. R. CARTER &amp; D. HOULE*Department of Biology, Norwegian University of Scienc
CSU Long Beach - ACARTER - 3
CSU Long Beach - ACARTER - 3
Received 5 December 2001 Accepted 28 January 2002 Published online 22 April 2002Evolution of functionally conserved enhancers can be accelerated in large populations: a populationgenetic model Ashley J. R. Carter and Gunter P. Wagner*Department o
CSU Long Beach - ACARTER - 3
Bio 260 Assignment #5, Due MAR 18One-sample t testsName:_(1) Consider an experimental drug that is being studied to reduce red blood cell (RBC) counts in individuals. The mean number of RBC in normal individuals is 5 million cells per microliter
CSU Long Beach - ACARTER - 3
Bio 260 Assignment #2, Due FEB 11 Print this document and provide answers in space provided. Write clearly to ensure full credit for your answers.Name:_(1) ROULETTE is a casino game where a numbered wheel spins and a steel ball falls into a locat
CSU Long Beach - SOC - 335
Soc.335i: Social Psychology / Spring 2008 / Tuesdays 06:30pm -09:15 pm (LA5-150) S ection:06 C lass # 6341 Las t U pdat e: 05/ 28/ 2008NOTE: After grades are posted, the University doesn't allow grade changes except for clerical error. It does not
CSU Long Beach - CHAPTER - 1
Samuael de Champlain's 1605 map of the Malle Barre (Nauset Harbor, Massachusetts) shows domed and barrel-vaultedwigwams with smoke holes sitting adjacent to cornfields (labeled &quot;L&quot;). After the summer harvest, the Indians of coastal New England wou
CSU Long Beach - CS - 528
CECS 528 Quiz 1, Spring 2005, Professor Ebert Name: Note: all answers involving explanations and/or proofs should use complete sentences and proper use of mathematical notation. Points will be lost otherwise.1. A computer CPU has a 2.5 GHz clock, a
CSU Long Beach - CS - 528
CECS 528 Exam 2, Spring 2005, Professor Ebert Name: Note: all answers involving explanations and/or proofs should use complete sentences and proper use of mathematical notation. Points will be lost otherwise.1. Let X = {A, B, C, D, E} and with resp
CSU Long Beach - CS - 228
Logical ReasoningLogical reasoning is at the core of mathematical reasoning and proof. Moreover, for a scientist or engineer to be able to reason about problems in his or her eld, he or she must rst have a mastery of the rules of logical reasoning,
CSU Long Beach - CS - 528
Introduction to N P and N P-CompletenessIn this lecture we study problems that in the complexity class N P. These are problems that are decidable by some nondeterministic Turing machine for which each branch of its computation tree is bounded by som
CSU Long Beach - CS - 528
CECS 528 Exam 3, Spring 2004, Professor Ebert Name: Show all work to receive full credit. 1a. Suppose you have at your disposal a Turing machine M which can decide in polynomial time whether or not a graph G = (V, E) has a clique of size k, for some
CSU Long Beach - ORD - 3
Appendix1. Description of f the Lower Dawan Formation of Huanghuachang sectionDawan Formation Middle Unit (The lowest part) 24.Yellow green shale with limestone lenses, yielding Azygograptus suecicus, Phyllograptus(Shod-29) 1.5m 23. Purplish red mi
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3
CSU Long Beach - ORD - 3