# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

13 Pages

### L05

Course: MATH MAC1147, Summer 2008
School: University of Florida
Rating:

Word Count: 547

#### Document Preview

5: Lecture Section A.5 Solving Equations Def. An equation in is a statement that two algebraic expressions are equal. To solve an equation is to nd all values of which the equation is true. for Such values of are solutions (or roots, zeros) of the equation. NOTE: If an equation has no solution, the solution set is empty, written as . ex. Solve the equation 1 = 2 Linear Equations Def. A linear equation in one...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Florida >> University of Florida >> MATH MAC1147

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.
5: Lecture Section A.5 Solving Equations Def. An equation in is a statement that two algebraic expressions are equal. To solve an equation is to nd all values of which the equation is true. for Such values of are solutions (or roots, zeros) of the equation. NOTE: If an equation has no solution, the solution set is empty, written as . ex. Solve the equation 1 = 2 Linear Equations Def. A linear equation in one variable an equation of the form + =0 where and are real numbers with = 0. is To solve a linear equation: 1. Remove all parenthesizes and simplify each side of the equation as much as possible. 2. Rewrite the equation by isolating the variable: variable terms on one side, numbers on the other. 3. Solve for the variable and check your solution. ex. Solve 6( 1) + 4 = 3(7 + 1) NOTE: A linear equation has exactly one solution. Solve a linear equation with fractions, multiply both sides by LCD to clear the fraction. 1 + 1 = ( 6) ex. 2 4 To solve rational equations that leads to linear equations 1. Find the domain of the variable. 2. Clear the equation of fractions by multiplying both sides by LCM of the denominator. 3. Solve for the variable. 4. Choose the solutions that are in the domain. ex. Solve: 3 = 2 1 + 1 7 23 +2 ex. Solve: 2 = 24 NOTE: We call 4 24 3 +2 = 2 an extraneous solution. NOTE: An equation with a single fraction on each side can be cleared of denominators by cross multiply. ex. 3 = 2 1 1 Quadratic Equations Def. A quadratic or second-degree equation is an equation that can be written in the general form 2 where , and + + =0 are real numbers and = 0. To solve a quadratic equation: 1. By Factoring: Use the Zero-Factor property: If = 0, then ex. Solve: 3 2 + 18 = 21 NOTE: The right side of the equation has to be 0 before factoring. ex. Solve: ( + 3)( 4) = 8 2. By Square Root Principle: If 2 = , where 0, then ex. Solve: ( + 2)2 = 5 = . 3. By Completing the Square of 2 + + = 0, = 0 1) Make sure = 1. If not, divide each term by . 2) Move the constant the to right side of the equation. 3) Complete the square by adding the square of onehalf of the coecient of to each side of the equation. 4) Factor the left side as a perfect square. 5) Solve for using the Square Root Principle. ex. Solve by completing the square: 3 2 4 2 = 0 4. By Quadratic Formula: If = 0, then 24 = 2 2 + Proof. Complete the square. ex. Solve using the quadratic formula: 2 1) 2 +8 +8=0 2) 2 (3 ) = 3 3) 2 +2 +2=0 + = 0, Def. The quantity 2 4 of the quadratic equation. NOTE: The equation is the discriminant 2 + + = 0, 1) If 2 4 > 0, then the equation has two distinct real number solutions. 2) If 24 = 0, then the equation has one repeated real number solution. 3) If 2 4 < 0, then the equation has no real number solutions. Practice. Use the discriminant to determine the number of real solutions of the equations 2 +9 +1=0 2) 2 2 +8 +8=0 Answer. 1) 2 1) two distinct real solutions 2) one repeated real solution Equations with Radicals or Rational Exponents To nd the real solution of an equation with radicals: 1. Simplify the equation if possible. 2. Isolate the most complicated radical on one side of the equation. 3. Raise both sides of the equation to the index of the radical to eliminate the radical. 4. Check for extraneous solution, since raise both sides to an even power may add extraneous solutions. ex. Solve: 3 + 3 +1= ex. Solve: ( + 3)2/3 = 4 ex. Solve: 3 +1 1=2 Absolute Value Equations Recall the denition: = ex. Solve each equation: 1) 4 + 3 = 2 2) 2 6 = NOTE: If the equation involving rational expressions, radicals, rational exponents or absolute values, you must check your solution(s) in the original equation. Practice. 1) Solve 16( + 1)2 + 8( + 1) + 1 = 0 (Hint: Use substitution = + 1) 2) Solve: 3 =6 2 9 3) Solve: 6 6 3 +9=0 4) Solve: 3 3 2 3 +9=0 5) Solve: (3 + 1)1/2 + 2(3 + 1)1/2 = 0 6) Solve: ( 1)2/3 + ( 1)1/3 12 = 0 5) no solution 6) 63, 28 Answer. 1) 5 4 2) 0, 3 3) 3 3 4) 3, 3, 3
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:

University of Florida - MATH - MAC1147
Lecture 8, part I: Section 1.3Linear Equations in Two VariablesDef. The equation++ = 0 is the generalform of a linear equation in two variables.The slopeofis dened by the formula=( x2 , y 2 )( x1 , y 1 )Slope as a Rate of ChangeFor line ,is
University of Florida - MATH - MAC1147
Lecture 11: Section 1.8Combinations of FunctionsAlgebra of FunctionsLet and be two functions with domains. We dene some new functions:NameDenitionandDomainex. Given ( ) =function3and ( ) =and its domain.. Find theex. If ( ) =and ( ) =func
University of Florida - MATH - MAC1147
Lecture 12: Section 1.9Inverse FunctionsDef. If a function is a set of ordered pairs ( , ),then inverse relation of is the set of orderedpairs ( , ).ex. Find the inverse relation of the following functions. Is the inverse a function?1): cfw_(2, 2),
University of Florida - MATH - MAC1147
Lecture 15: Section 2.4Complex NumbersConsider the equation2= 1.Def. The imaginary unit, , is the number suchthat2= 1 or= 1Power of1=5=2= 13=4=61 ==7=8=Therefore, every integer power of can be written as, 1, , 1.In general, d
University of Florida - MATH - MAC1147
Lecture 16, Part I: Section 2.5Zeros of Polynomial FunctionsLinear Factorization TheoremEvery polynomial function ( ) of degree &gt; 0 canbe factored into linear factors (not necessarilydistinct) of the form( )=where1, 2, . . . ,(1 )(2) ()are co
University of Florida - MATH - MAC1147
Lecture 17: Section 2.6Rational FunctionsHoles and Vertical AsymptotesThe graph of a rational function ( ) =hole at=if()has a()1. both ( ) and ( ) have the common factor( ), and2. the simplied denominator DOES NOT havethe factor ( ).NOTE: To
University of Florida - MATH - MAC1147
Lecture 18: Section 2.7Nonlinear InequalitiesTo Solve a Polynomial Inequality1. Write the inequality with 0 on the right hand sideto obtain one of the following:( )&gt;0( )0( )&lt;0( )02. Find all real zeros of the polynomial ( ). Thesezeros are the c
University of Florida - MATH - MAC1147
Lecture 21: Section 3.2Logarithmic FunctionsRecall the graph of exponential function ( ) =&gt; 1:,(0,1)Since ( ) is one-to-one, it has an inverse function.Def. The logarithmic function with base ,where &gt; 0 and = 1 is written ( ) = log andis dened by
University of Florida - MATH - MAC1147
Lecture 23: Section 3.4Exponential and Logarithmic EquationsTwo basic strategies for solving exponential orlogarithmic equations:1. One-to-One Properties=logif and only if= logif and only if2. Inverse Propertieslog=log=ex. Solve for :1) 2
University of Florida - MATH - MAC1147
Lecture 24: Section 3.5Exponential and Logarithmic ModelsFour common types of mathematical models involving exponential functions and logarithmic functions:1. Exponential growth model:where== amount at time= amount at time 0= relative growth rate,
University of Florida - MATH - MAC1147
11211211212211111111111117212
University of Florida - MATH - MAC1147
Lecture 30: Section 4.7Inverse Trigonometric FunctionsThe Inverse Sine Function11Lets restrict the domain to the interval , .22Then = sin is one-to-one.Def. The inverse sine function is dened by= sin1if and only ifwith domain [1, 1] and range ,
University of Florida - MATH - MAC1147
MAC 2311LECTURE OUTLINE and HOMEWORK EXERCISESStewart: Calculus, Early Transcendentals, 6th editionYou should read the textbook sections covered in each dayslecture before class. After each lecture, review your notes andthe text section to make sure
University of Florida - MATH - MAC1147
Disclaimer:This PDF file is a scanned copy of a sample of pages fromMr. Mahoneys MAC1147 10th Period Lecture.The file cannot be printed but may be viewed. Thepurpose of this file is to allow students to reviewcertain pages that had a lot of detail fi
University of Florida - MATH - MAC1147
Disclaimer:This PDF file is a scanned copy of a sample of pages fromMr. Mahoneys MAC1147 10th Period Lecture.The file cannot be printed but may be viewed. Thepurpose of this file is to allow students to reviewcertain pages that had a lot of detail fi
University of Florida - MATH - MAC1147
Disclaimer:This PDF file is a scanned copy of a sample of pages fromMr. Mahoneys MAC1147 10th Period Lecture.The file cannot be printed but may be viewed. Thepurpose of this file is to allow students to reviewcertain pages that had a lot of detail fi
University of Florida - MATH - MAC1147
Disclaimer:This PDF file is a scanned copy of a sample of pages fromMr. Mahoneys MAC1147 10th Period Lecture.The file cannot be printed but may be viewed. Thepurpose of this file is to allow students to reviewcertain pages that had a lot of detail fi
University of Florida - MATH - MAC1147
MAC1147: Quiz #1 Solutions09/01/20091. LetAA = 0, , 0.67,12, 5, ,3(3)2.State which elements ofare(3)2 = 3)(a) whole numbers; (0,(b) integers; (0, -5,(3)2 = 3)10.67, -5, ,3(,2)(c) rational numbers; (0,(d) irrational numbers.(3)2 = 3)
University of Florida - MATH - MAC1147
MAC1147: Quiz #209/08/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #309/15/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #409/22/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #509/29/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1a
University of Florida - MATH - MAC1147
MAC1147: Quiz #610/06/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #710/20/2009In the top-right corner of a clean sheet of paper, write your name, UFID, and sectionnumber. Please use a pen with blue or black ink. When you are nished, FOLD your paperin half lengthwise and write your name on the back.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #810/27/2009In the top-right corner of a clean sheet of paper, write your name, UFID,and section number. Please use a pen with blue or black ink. When you arenished, FOLD your paper in half lengthwise and write your name on theback.1.
University of Florida - MATH - MAC1147
MAC1147: Quiz #911/3/2009In the top-right corner of a clean sheet of paper, write your name, UFID, and sectionnumber. Please use a pen with blue or black ink. When you are nished, FOLD your paperin half lengthwise and write your name on the back.1. F
University of Florida - MATH - MAC1147
MAC1147: Quiz #1011/17/2009In the top-right corner of a clean sheet of paper, write your name, UFID, and sectionnumber. Please use a pen with blue or black ink. When you are nished, FOLD your paperin half lengthwise and write your name on the back.y
University of Florida - MATH - MAC1147
MAC1147: Quiz #1112/1/2009In the top-right corner of a clean sheet of paper, write your name, UFID, and sectionnumber. Please use a pen with blue or black ink. When you are nished, FOLD your paperin half lengthwise and write your name on the back.1.
University of Florida - MATH - MAC1147
MAC 1147 Review 1, Spring 2011Exam 1 covers Lectures 1-81. Write without absolute value signs and simplify:2 + 2 1(b) 5 3 (a)+1(c) 312 4 + 1 if &gt; 300 3 15 12, , 0.9, 4, , 3 8, , , , , 161/4 , 0.08 .300 5 2List the elements that belong to each
University of Florida - MATH - MAC1147
MAC 1147 Review 2, Spring 2011Exam 2 covers Lectures 9-181. Classify the following functions as even, or odd, or neither even nor odd:12(b) ( ) = (a) ( ) =2+51(c) ( ) = 3 (d) ( ) = 2. Find the average rate of change ( ) =11on the interval [3
University of Florida - MATH - MAC1147
MAC 1147 Review 3, Spring 2011Exam 3 covers Lectures 18-261. Solve the inequalities. Give the answers in interval notation.3+3+4&lt;3(b)(a)2+33(c) log4 + log4 ( 3) &lt; 12. Solve the system:(a)22 3 = 23 2 = 12(b)+3221= 2(c)23 3 + 4 =
University of Florida - MATH - MAC1147
MAC 1147 Review 4, Spring 2011Exam 4 covers Lectures 27-34The problems with an astroid ( ) are from lectures 35 and 36. They willbe in the nal exam, but not exam 4.1. If (4, 8) is on the terminal side of an angle in standard position, nd sin ,cos and
University of Florida - MATH - MAC1147
UCSB - ECON - 171
Professor GarrattEconomics 171Introduction to Game TheoryMW 2:00-3:15, SH 1431www.econ.ucsb.edu/~garratt/Econ171Game theory is the study of the interaction of rational decision makers. This theory has become afundamental tool in the study of social
University of Florida - MATH - MAC1147
APPROVED LIST OF MATHEMATICS TUTORS (VA APPROVED)NameAbernethy, DeannaBhattacharya, SouvikBrennan, JosephDebRoy, SwatiDucey, JoshuaGentimis, ThanosGrizzell, KeithHungerford, JamesHuynh, DucIanuzzi, ArthurJefferson, AzizaKane, YehonatanKeeran
University of Florida - MAC - 1114
Combinatorial aspects of the theory of q-seriesP. R. HammondSubmittedfor the degreeof D. Phil.University of SussexJune, 2006C2oos]DeclarationI herebydeclarethat this thesishas not beensubmitted,either in the sameor differentform, to this or any ot
UCSB - ECON - 171
University of Florida - MAC - 2233
MAC 2233 Calendar, SUMMER C 2008 SUMMER EXAM 1* EXAM 3* BREAKHoliday EXAM 4* EXAM 2*TERM FINAL*NO CLASS
UCSB - ECON - 171
University of Florida - MAC - 2233
MAC 2233 Lecture OutlineReading should be completed before you attend lecture. After lecture, you should reviewyour notes and make sure you understand the main ideas before you work the exercises.This should be done before the next lecture class. If yo
University of Florida - MAC - 2233
Answers Fall 2009 Exam 1 version A1a. 1/101b. 3/41c. 8 years2. A(x) = (3a. p = 55 1 x23b. C (x) = 200 + 26xR(x) = 55x 1 x22P (x) = 200 + 29x 1 x223c. 29 passengers at price \$40.503d. R(p) = 2p(p 55)4a.01DNE0DNE0004b.4c. x = 0 innit
UCSB - ECON - 171
What is a Game? There are many types of games, board games, card games, videogames, field games (e.g. football), etc. We focus on games where: There are 2 or more players. There is some choice of action where strategy matters. The game has one or mo
University of Florida - MAC - 2233
MAC 2233 Review Exam 1 This is intended to be a tool to help you review some of the material that could appear on theexam. It is not inclusive of all topics discussed in lecture. You should review problems frompractice exams, problem sets, projects, an
University of Florida - MAC - 2233
Answers Fall 2009 Exam 2 version A122h0 h 2x + 2h + 12x + 11b. y = 4x + 2dH2.= 250 ft/secdt1a. lim= . =4(2x + 1)23a. g (x) = (x2 1)1/3 [11x2 3]; HTLs: x = 1; x33b. h (x) = ; HTLs: x = 9(2 x 3)24a. ( 0, 3 )4b. t = 3; 18 feet5a. 2 L/mi
UCSB - ECON - 171
Sequential Move GamesUsing Backward Induction(Rollback) to Find EquilibriumSequential Move Class Game:Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses a numberbetween 1 and 10 inclusive. This choice is
University of Florida - MAC - 2233
Review Answers Exam 21. a) -64 units/sec b) -32 units/sec2. -143. -18 cents/unit4. 2.4 in/hr5. a)f (x) =x(10 3x2 ); HT at: x = 0, 5 x2103;y = 7x 232b) f (x) = 2(6x 2)2 (x + 4)(15x + 34); HTL at x = 4, 1 , 34 ; y = 7840x 62403151c) y =1
University of Florida - MAC - 2233
MAC 2233 Review Exam 2 This is intended to be a tool to help you review some of the material that could appear on theexam. It is not inclusive of all topics discussed in lecture. You should review problems frompractice exams, problem sets, projects, an
UCSB - ECON - 171
BargainingBargainingGamesAnApplicationofSequentialMoveGamesTheTheBargainingProblem TheBargainingProblemarisesineconomicsituationswheretherearegainsfromtrade,forexample,whenabuyervaluesanitemmorethanaseller. Theproblemishowtodividethegains,forexam
University of Florida - MAC - 2233
Answers Fall 2009 Exam 3 version A1a. W (x) = 2ex ; W (x) = 2ex ;1b. W (0) = 2 hundred wolves now;increasing and concave downlim W (x) = 4 hundred in the long runx1c. graph is increasing, conc down from ( 0, 2 ) toward horiz asymptote W = 41d. W 1
University of Florida - MAC - 2233
Review Answers Exam 31. Information for the graph: Domain R; Intercepts ( 1, 0 ), ( 2, 0 ), ( 0, 3 4 ) ; No asymptotesor symmetry; lim f (x) = , lim f (x) = ; Increasing on ( , 2 ), ( 0, 1 ), ( 1, ) ;xxDecreasing on ( 2, 0 ) ; Local max at 2 (cusp);
UCSB - ECON - 171
SimultaneousSimultaneousMoveGamesDecision making without knowledgeof the strategy choice of opponentsSimultaneousSimultaneousMoves Arisewhenplayershavetomaketheirstrategychoicessimultaneously,withoutknowingthestrategiesthathavebeenchosenbytheother
University of Florida - MAC - 2233
Exam 3 Review MAC 2233 This is intended to be a tool to help you review some of the material that could appear on theexam. It is not inclusive of all topics discussed in lecture.211. Graph f (x) = (x 1) 3 (x + 2) 3 .Note: f (x) =x21(x 1) 3 (x +
UCSB - ECON - 171
CoordinationGamesandContinuousStrategySpacesMoreComplicatedSimultaneousMoveGamesOtherOtherCoordinationGames Supposeyouandapartnerareaskedtochooseoneelementfromthefollowingsetsofchoices.Ifyoubothmakethesamechoice,youearn\$1,otherwisenothing. cfw_Re
University of Florida - MAC - 2233
Answers Fall 2009 Exam 4 version A1a. v (t) = 300 300(1 + 3t)21b. s(t) = 300t + 100(1 + 3t)1 1001c. 1125/4 microns/sec; 225 microns2a. 502b. k = ln(1/2) = ln(2)2c. 25 ln(1 + 2e2t ) + C ;2d.1[25 ln(1 + 2e10 ) 25 ln(1 + 2e0 )];503a. 2 ln(2) + 2 l
UCSB - ECON - 171
Probability, Expected Payoffs and Expected Utility In thinking about mixed strategies, we will need to make useof probabilities We will therefore review the basic rules ofof probabilities. We will therefore review the basic rules ofprobability and the
University of Florida - MAC - 2233
Review Answers Exam 41. rows from left to right:Row 1: x + 4 x + ln| x | + C 1 (x2 4)2 + C4Row 2:Row 3:2112 11 (212 (2 x)62312x+ e)6 + C12 (e983131)2 (e e132(4 x) 2 +322. (a); x)11 + C163 (435 x) 2 2 (4 x) 2 + C5ln| 1 +
University of Florida - MAC - 2233
Exam 4 Review MAC 2233 This is intended to help you review some of the material that could appear on the exam.It is not inclusive of all topics discussed in lecture.1. Evaluate the following integrals:( x + 1 )2xdxdxx(x2 4)3e2120[ 1 + 2 ln(x
UCSB - ECON - 171
MixedMixedStrategiesKeepem guessingMixedMixedStrategyNashEquilibrium Amixedstrategyisoneinwhichaplayerplayshisavailablepurestrategieswithcertainprobabilities. Mixedstrategiesarebestunderstoodinthecontextofrepeatedgames,whereeachplayersaimistokeept
University of Florida - MAC - 2233
Note: Problems 18 are worth four points each.1. Find all values of y so that the distance between (1, 2) and (1, y ) is 3.A. y = 1 andy = 5B. y = 5C. y = 1D. y = 5 and y = 5E. no such y -values exist2. After t hours of operation, an assembly line
UCSB - ECON - 171
More on Sequential and SimultaneousMove Games So far we have studied two types of games: 1)sequential move (extensive form) games wheresequential move (extensive form) games whereplayers take turns choosing actions and 2)strategic form (normal form)
University of Florida - MAC - 2233
Answers Fall 2008 Exam 11.2.3.4.CDAE5.6.7.8.B9. BA 10. DB 11. CE 12. A1a. 500 bears5+t1d. T (t) =6 + 2t2a. p = 100 1b. 100 bears1c. 5/3 years1e. 500 trout110 x2b. C (x) = 1200 + 5x +R(x) = 100x 3220 x1210 xP (x) = 1200 +