2.4 Linear Inequalities and Problem Solving


 Maude Todd
 1 years ago
 Views:
Transcription
1 Section.4 Liner Inequlities nd Problem Solving 77.4 Liner Inequlities nd Problem Solving S 1 Use Intervl Nottion. Solve Liner Inequlities Using the Addition Property of Inequlity. 3 Solve Liner Inequlities Using the Multipliction nd the Addition Properties of Inequlity. 4 Solve Problems Tht Cn Be Modeled by Liner Inequlities. Reltionships mong mesurble quntities re not lwys described by equtions. For exmple, suppose tht slesperson erns bse of $600 per month plus commission of 0% of sles. Suppose we wnt to find the minimum mount of sles needed to receive totl income of t lest $1500 per month. Here, the phrse t lest implies tht n income of $1500 or more is cceptble. In symbols, we cn write income Ú 1500 This is n exmple of n inequlity, nd we will solve this problem in Exmple 8. A liner inequlity is similr to liner eqution except tht the equlity symbol is replced with n inequlity symbol, such s 6, 7,, or Ú. Liner Inequlities in One Vrible x 3x + 5 Ú 4 y 6 0 3(x  4) 7 5x 3 5 c c c c is greter thn is less is greter is less thn or equl to thn thn or equl to Liner Inequlity in One Vrible A liner inequlity in one vrible is n inequlity tht cn be written in the form x + b 6 c where, b, nd c re rel numbers nd 0. In this section, when we mke definitions, stte properties, or list steps bout n inequlity contining the symbol 6, we men tht the definition, property, or steps lso pply to inequlities contining the symbols 7,, nd Ú. 1 Using Intervl Nottion A solution of n inequlity is vlue of the vrible tht mkes the inequlity true sttement. The solution set of n inequlity is the set of ll solutions. Notice tht the solution set of the inequlity x 7, for exmple, contins ll numbers greter thn. Its grph is n intervl on the number line since n infinite number of vlues stisfy the vrible. If we use open/closedcircle nottion, the grph of 5x x 7 6 looks like the following In this text intervl nottion will be used to write solution sets of inequlities. To help us understnd this nottion, different grphing nottion will be used. Insted of n open circle, we use prenthesis. With this new nottion, the grph of 5x x 7 6 now looks like nd cn be represented in intervl nottion s 1,. The symbol is red infinity nd indictes tht the intervl includes ll numbers greter thn. The left prenthesis indictes tht is not included in the intervl. When is included in the intervl, we use brcket. The grph of 5x x Ú 6 is below nd cn be represented s [,. The following tble shows three equivlent wys to describe n intervl: in set nottion, s grph, nd in intervl nottion.
2 78 CHAPTER Equtions, Inequlities, nd Problem Solving Set Nottion Grph Intervl Nottion 5x x , 5x x 7 6 1, 5x x 6 1 , ] 5x x Ú 6 [, 5x 6 x 6 b6 b (, b) 5x x b6 b [, b] 5x 6 x b6 b (, b] 5x x 6 b6 b [, b) Helpful Hint Notice tht prenthesis is lwys used to enclose nd . CONCEPT CHECK Explin wht is wrong with writing the intervl 15, ]. EXAMPLE 1 Grph ech set on number line nd then write in intervl nottion.. 5x x Ú 6 b. 5x x 616 c. 5x x 36 Solution. [, b. 1 , c. 0.5 (0.5, 3] Grph ech set on number line nd then write in intervl nottion.. 5x x b. 5x x Ú 36 c. 5x 1 x 6 46 Answer to Concept Check: should be 15, since prenthesis is lwys used to enclose Solving Liner Inequlities Using the Addition Property We will use intervl nottion to write solutions of liner inequlities. To solve liner inequlity, we use process similr to the one used to solve liner eqution. We use properties of inequlities to write equivlent inequlities until the vrible is isolted.
3 Section.4 Liner Inequlities nd Problem Solving 79 Addition Property of Inequlity If, b, nd c re rel numbers, then 6 b nd + c 6 b + c re equivlent inequlities. In other words, we my dd the sme rel number to both sides of n inequlity, nd the resulting inequlity will hve the sme solution set. This property lso llows us to subtrct the sme rel number from both sides. EXAMPLE nottion. Solve: x Grph the solution set nd write it in intervl Solution x x Add to both sides. x 6 7 Simplify. The solution set is 5x x 6 76, which in intervl nottion is 1 , 7. The grph of the solution set is Solve: x Grph the solution set nd write it in intervl nottion. Helpful Hint In Exmple, the solution set is 5x x This mens tht ll numbers less thn 7 re solutions. For exmple, 6.9, 0, p, 1, nd re solutions, just to nme few. To see this, replce x in x with ech of these numbers nd see tht the result is true inequlity. EXAMPLE 3 Solve: 3x + 4 Ú x  6. Grph the solution set nd write it in intervl nottion. Solution 3x + 4 Ú x  6 3x x Ú x x Subtrct x from both sides. x + 4 Ú 6 x Ú 64 x Ú 10 Combine like terms. Subtrct 4 from both sides. Simplify. The solution set is 5x x Ú 106, which in intervl nottion is [10,. The grph of the solution set is Solve: 3x + 1 x  3. Grph the solution set nd write it in intervl nottion. 3 Solving Liner Inequlities Using the Multipliction nd Addition Properties Next, we introduce nd use the multipliction property of inequlity to solve liner inequlities. To understnd this property, let s strt with the true sttement nd multiply both sides by.
4 80 CHAPTER Equtions, Inequlities, nd Problem Solving Multiply by. True The sttement remins true. Notice wht hppens if both sides of re multiplied by Multiply by . Flse The inequlity is flse sttement. However, if the direction of the inequlity sign is reversed, the result is true True These exmples suggest the following property. Multipliction Property of Inequlity If, b, nd c re rel numbers nd c is positive, then * b nd c * bc re equivlent inequlities. If, b, nd c re rel numbers nd c is negtive, then * b nd c + bc re equivlent inequlities. In other words, we my multiply both sides of n inequlity by the sme positive rel number nd the result is n equivlent inequlity. We my lso multiply both sides of n inequlity by the sme negtive number nd reverse the direction of the inequlity symbol, nd the result is n equivlent inequlity. The multipliction property holds for division lso, since division is defined in terms of multipliction. Helpful Hint Whenever both sides of n inequlity re multiplied or divided by negtive number, the direction of the inequlity symbol must be reversed to form n equivlent inequlity. EXAMPLE 4 intervl nottion x 3 8 Solve nd grph the solution set. Write the solution set in b. .3x Helpful Hint The inequlity symbol is the sme since we re multiplying by positive number. Solution. 1 4 x # 1 4 x 4 # 3 8 x 3 Multiply both sides by 4. Simplify. The solution set is e x ` x 3 f, which in intervl nottion is , 3 d. The grph of the solution set is w
5 Section.4 Liner Inequlities nd Problem Solving 81 Helpful Hint The inequlity symbol is reversed since we divided by negtive number. b. .3x x x 73 Divide both sides by .3 nd reverse the inequlity symbol. Simplify. The solution set is 5x x 736, which is 13, in intervl nottion. The grph of the solution set is Solve nd grph the solution set. Write the solution set in intervl nottion.. 5 x Ú 4 15 b. .4x CONCEPT CHECK In which of the following inequlities must the inequlity symbol be reversed during the solution process?. x 7 7 b. x c. x x 6 7 d. x To solve liner inequlities in generl, we follow steps similr to those for solving liner equtions. Solving Liner Inequlity in One Vrible Step 1. Step. Step 3. Step 4. Step 5. Cler the inequlity of frctions by multiplying both sides of the inequlity by the lest common denomintor (LCD) of ll frctions in the inequlity. Use the distributive property to remove grouping symbols such s prentheses. Combine like terms on ech side of the inequlity. Use the ddition property of inequlity to write the inequlity s n equivlent inequlity with vrible terms on one side nd numbers on the other side. Use the multipliction property of inequlity to isolte the vrible. Helpful Hint Don t forget tht 5 x mens the sme s x Ú 5. Answer to Concept Check:, d EXAMPLE 5 Solve: 1x x x. Solution 1x x x x x x 5  x 7x x + x 7x x 5 8x x x 0 8 8x 8 5 x, or x Ú 5 Apply the distributive property. Combine like terms. Add x to both sides. Combine like terms. Add 15 to both sides. Combine like terms. Divide both sides by 8. Simplify.
6 8 CHAPTER Equtions, Inequlities, nd Problem Solving The solution set written in intervl nottion is c 5, b nd its grph is e 5 Solve: 14x x x. Grph nd write the solution set in intervl nottion. EXAMPLE 6 Solve: 1x  6 Ú x Solution 1x  6 Ú x c 1x  6d Ú 51x x  6 Ú 51x  1 x  1 Ú 5x  53x  1 Ú 53x Ú 73x x Multiply both sides by 5 to eliminte frctions. Apply the distributive property. Subtrct 5x from both sides. Add 1 to both sides. Divide both sides by 3 nd reverse the inequlity symbol. Simplify. The solution set written in intervl nottion is ,  7 d nd its grph is Solve: 3 1x  3 Ú x  7. Grph nd write the solution set in intervl 5 nottion. EXAMPLE 7 Solve: 1x x + 1. Solution 1x x + 1 x x + 1 x x 7 x x Distribute on the left side. Subtrct x from both sides. Simplify is true sttement for ll vlues of x, so this inequlity nd the originl inequlity re true for ll numbers. The solution set is 5x x is rel number6, or 1 , in intervl nottion, nd its grph is Solve: 41x  6 4x + 5. Grph nd write the solution set in intervl nottion.
7 Section.4 Liner Inequlities nd Problem Solving 83 4 Solving Problems Modeled by Liner Inequlities Appliction problems contining words such s t lest, t most, between, no more thn, nd no less thn usully indicte tht n inequlity is to be solved insted of n eqution. In solving pplictions involving liner inequlities, we use the sme procedure s when we solved pplictions involving liner equtions. EXAMPLE 8 Clculting Income with Commission A slesperson erns $600 per month plus commission of 0% of sles. Find the minimum mount of sles needed to receive totl income of t lest $1500 per month. Solution 1. UNDERSTAND. Red nd rered the problem. Let x = mount of sles.. TRANSLATE. As stted in the beginning of this section, we wnt the income to be greter thn or equl to $1500. To write n inequlity, notice tht the slesperson s income consists of $600 plus commission (0% of sles). commission In words: Ú , of sles T T T Trnslte: x Ú SOLVE the inequlity for x x Ú x Ú x Ú 900 x Ú INTERPRET. Check: The income for sles of $4500 is , or Thus, if sles re greter thn or equl to $4500, income is greter thn or equl to $1500. Stte: The minimum mount of sles needed for the slesperson to ern t lest $1500 per month is $4500 per month. 8 A slesperson erns $900 month plus commission of 15% of sles. Find the minimum mount of sles needed to receive totl income of t lest $400 per month. EXAMPLE 9 Finding the Annul Consumption In the United Sttes, the nnul consumption of cigrettes is declining. The consumption c in billions of cigrettes per yer since the yer 000 cn be pproximted by the formul c = 9.4t where t is the number of yers fter 000. Use this formul to predict the yers tht the consumption of cigrettes will be less thn 00 billion per yer. Solution 1. UNDERSTAND. Red nd rered the problem. To become fmilir with the given formul, let s find the cigrette consumption fter 0 yers, which would be the yer , or 00. To do so, we substitute 0 for t in the given formul. c = = 43
8 84 CHAPTER Equtions, Inequlities, nd Problem Solving Thus, in 00, we predict cigrette consumption to be bout 43 billion. Vribles hve lredy been ssigned in the given formul. For review, they re c = the nnul consumption of cigrettes in the United Sttes in billions of cigrettes nd t = the number of yers fter 000. TRANSLATE. We re looking for the yers tht the consumption of cigrettes c is less thn 00. Since we re finding yers t, we substitute the expression in the formul given for c, or 9.4t SOLVE the inequlity. 9.4t t t t 7 pproximtely INTERPRET. Check: Substitute number greter thn 4.6 nd see tht c is less thn 00. Stte: The nnul consumption of cigrettes will be less thn 00 billion more thn 4.6 yers fter 000, or pproximtely for = 05 nd ll yers fter. 9 Use the formul given in Exmple 9 to predict when the consumption of cigrettes will be less thn 75 billion per yer. Vocbulry, Rediness & Video Check Mtch ech grph with the intervl nottion tht describes it , b. 15,  c. 1, 5 d. 1 , # c 7 4, .5b b. .5, 7 4 d c. c .5, 7 4 b d. 7 4, .5b (, 114 b. 111, c. 311, ) d. 1 , 11 j c , 0.b b. 0., d c , 0. d d. c 0., b Use the choices below to fill in ech blnk. 1 , 0.4 (, , ) 10.4, (, The set 5x x Ú written in intervl nottion is. 6. The set 5x x written in intervl nottion is. 7. The set 5x x written in intervl nottion is. 8. The set 5x x written in intervl nottion is.
9 Section.4 Liner Inequlities nd Problem Solving 85 MrtinGy Interctive Videos See Video.4 Wtch the section lecture video nd nswer the following questions Using Exmple 1 s reference, explin how the grph of the solution set of n inequlity cn help you write the solution set in intervl nottion. 10. From the lecture before Exmple 3, explin the ddition property of inequlity in your own words. Wht equlity property does it closely resemble? 11. Bsed on the lecture before Exmple 4, complete the following sttement. If you multiply or divide both sides of n inequlity by the nonzero negtive number, you must the direction of the inequlity symbol. 1. Wht words or phrses in the Exmple 7 sttement tell you this is n inequlity ppliction (besides the lst line telling you to use n inequlity)?.4 Exercise Set Grph the solution set of ech inequlity nd write it in intervl nottion. See Exmple x x x x x x Ú x x x 7 x6 6. 5x 7 Ú x6 7. 5x  6 x x 5 Ú x 716 Solve. Grph the solution set nd write it in intervl nottion. See Exmples through x  7 Ú x x 6 6x x 6 10x x  7 7x x Ú x Ú x x x Ú 9. 4x Ú 8 Solve. Write the solution set using intervl nottion. See Exmples 5 through x + 7 Ú x x Ú 4x x 6 6x x x Ú x x x x x x 3  x x Ú x 5 x x 3 7 x Ú x  1 Ú 6x x x x x  14 MIXED Solve. Write the solution set using intervl nottion. See Exmples 1 through x x x x x x Ú x Ú x x x x x x  7 Ú x x + 1 x x + 0.6x Ú x  x x x  4 Ú 31x x x x x x x  6 Ú 61x y  19y + 51y x  1 Ú 4x x x x 55. 3x x x 6 71x x x +
10 86 CHAPTER Equtions, Inequlities, nd Problem Solving x x x x x x x x x x x x x x x x 3 Ú x x x x x x Solve. See Exmples 8 nd 9. For Exercises 69 through 76,. nswer with n inequlity nd b., in your own words, explin the mening of your nswer to prt (). Exercises 69 nd 70 re written to help you get strted. Exercises 71 nd 7 re written to help you write nd solve the inequlity. 69. Shurek Wshburn hs scores of 7, 67, 8, nd 79 on her lgebr tests.. Use n inequlity to find the scores she must mke on the finl exm to pss the course with n verge of 77 or higher, given tht the finl exm counts s two tests. b. In your own words, explin the mening of your nswer to prt (). 70. In Winter Olympics 5000meter speedskting event, Hns Holden scored times of 6.85, 7.04, nd 6.9 minutes on his first three trils.. Use n inequlity to find the times he cn score on his lst tril so tht his verge time is under 7.0 minutes. b. In your own words, explin the mening of your nswer to prt (). 71. A clerk must use the elevtor to move boxes of pper. The elevtor s mximum weight limit is 1500 pounds. If ech box of pper weighs 66 pounds nd the clerk weighs 147 pounds, use n inequlity to find the number of whole boxes she cn move on the elevtor t one time. Strt the solution: 1. UNDERSTAND the problem. Rered it s mny times s needed.. TRANSLATE into n inequlity. (Fill in the blnks below.) Let x = number of boxes elevtor Clerk s number weight of + times mximum weight of boxes ech box weight T T T T T T T + x # Finish with: 3. SOLVE nd 4. INTERPRET 7. To mil lrge envelope first clss, the U.S. Post Office chrges 88 cents for the first ounce nd 17 cents per ounce for ech dditionl ounce. Use n inequlity to find the number of whole ounces tht cn be miled for no more thn $.00. Price of first ounce Strt the solution: 1. UNDERSTAND the problem. Rered it s mny times s needed.. TRANSLATE into n inequlity. (Fill in the blnks below.) Let x = number of dditionl ounces (fter first ounce) + number of price per 00 dditionl times dditionl mximum ounces ounce cents T T T T T T T + x # Finish with: 3. SOLVE nd 4. INTERPRET 73. A smll plne s mximum tkeoff weight is 000 pounds or less. Six pssengers weigh n verge of 160 pounds ech. Use n inequlity to find the luggge nd crgo weights the plne cn crry. 74. A shopping mll prking grge chrges $1 for the first hlfhour nd 60 cents for ech dditionl hlfhour. Use n inequlity to find how long you cn prk if you hve only $4.00 in csh. 75. A cr rentl compny offers two subcompct rentl plns. Pln A: $36 per dy nd unlimited milege Pln B: $4 per dy plus $0.15 per mile Use n inequlity to find the number of dily miles for which pln A is more economicl thn pln B.
11 Section.4 Liner Inequlities nd Problem Solving Northest Telephone Compny offers two billing plns for locl clls. Pln 1: $5 per month for unlimited clls Pln : $13 per month plus $0.06 per cll Use n inequlity to find the number of monthly clls for which pln 1 is more economicl thn pln. 77. At room temperture, glss used in windows ctully hs some properties of liquid. It hs very slow, viscous flow. (Viscosity is the property of fluid tht resists internl flow. For exmple, lemonde flows more esily thn fudge syrup. Fudge syrup hs higher viscosity thn lemonde.) Glss does not become true liquid until tempertures re greter thn or equl to 500 C. Find the Fhrenheit tempertures for which glss is liquid. (Use the formul F = 9 5 C + 3.) 78. Stibnite is silvery white minerl with metllic luster. It is one of the few minerls tht melts esily in mtch flme or t tempertures of pproximtely 977 F or greter. Find the Celsius tempertures t which stibnite melts. [Use the formul C = 5 1F  3.] Although beginning slries vry gretly ccording to your field of study, the eqution s = 184.7t + 48,133 cn be used to pproximte nd to predict verge beginning slries for cndidtes with bchelor s degrees. The vrible s is the strting slry nd t is the number of yers fter 000. (Source: Sttisticl Abstrct of the U.S.). Approximte the yer in which beginning slries for cndidtes will be greter thn $70,000. (Round your nswer up nd use it to clculte the yer.) b. Determine the yer you pln to grdute from college. Use this yer to find the corresponding vlue of t nd pproximte beginning slry for bchelor s degree cndidte Use the formul in Exmple 9 to estimte the yers tht the consumption of cigrettes will be less thn 50 billion per yer. b. Use your nswer to prt () to describe the limittions of your nswer. The verge consumption per person per yer of whole milk w cn be pproximted by the eqution w = .11t where t is the number of yers fter 000 nd w is mesured in pounds. The verge consumption of skim milk s per person per yer cn be pproximted by the eqution s = 0.4t where t is the number of yers fter 000 nd s is mesured in pounds. The consumption of whole milk is shown on the grph in blue nd the consumption of skim milk is shown on the grph in red. Use this informtion to nswer Exercises 81 through 90. Averge consumption per person per yer (in pounds) Whole Milk versus Skim Milk Whole milk Skim milk Yers fter Source: Bsed on dt from U.S. Deprtment of Agriculture, Economic Reserch Service 81. Is the consumption of whole milk incresing or decresing over time? Explin how you rrived t your nswer. 8. Is the consumption of skim milk incresing or decresing over time? Explin how you rrived t your nswer. 83. Predict the consumption of whole milk in 015. (Hint: Find the vlue of t tht corresponds to 015.) 84. Predict the consumption of skim milk in 015. (Hint: Find the vlue of t tht corresponds to 015.) 85. Determine when the consumption of whole milk will be less thn 45 pounds per person per yer. 86. For 000 through 010, the consumption of whole milk ws greter thn the consumption of skim milk. Explin how this cn be determined from the grph. 87. Both lines hve negtive slope, tht is, the mount of ech type of milk consumed per person per yer is decresing s time goes on. However, the mount of whole milk being consumed is decresing fster thn the mount of skim milk being consumed. Explin how this could be. 88. Do you think it is possible tht the consumption of whole milk will eventully be the sme s the consumption of skim milk? Explin your nswer. 89. The consumption of skim milk will be greter thn the consumption of whole milk when s 7 w.. Find when this will occur by substituting the given equivlent expression for w nd the given equivlent expression for s nd solving for t. b. Estimte to the nerest whole the first yer when this will occur. 90. How will the two lines in the grph pper if the consumption of whole milk is the sme s the consumption of skim milk? REVIEW AND PREVIEW List or describe the integers tht mke both inequlities true. See Section x 6 5 nd x x Ú 0 nd x x Ú  nd x Ú 94. x 6 6 nd x 65
12 88 CHAPTER Equtions, Inequlities, nd Problem Solving Solve ech eqution for x. See Section x  6 = x  1 = x + 7 = 5x x  4 = x  4 CONCEPT EXTENSIONS Ech row of the tble shows three equivlent wys of describing n intervl. Complete this tble by filling in the equivlent descriptions. The first row hs been completed for you. Ech inequlity below (Exercises ) is solved by dividing both sides by the coefficient of x. Determine whether the inequlity symbol will be reversed during this solution process x x x Ú x Solve: x  3 = Solve: x Solve: x Set Nottion 5x x 636 Grph 3 Intervl Nottion 1 , Red the equtions nd inequlities for Exercises 109, 110, nd 111 nd their solutions. In your own words, write down your thoughts When grphing the solution set of n inequlity, explin how you know whether to use prenthesis or brcket Explin wht is wrong with the intervl nottion 16, x x Explin how solving liner inequlity is similr to solving liner eqution x x Explin how solving liner inequlity is different from solving liner eqution (, , 4) Integrted Review LINEAR EQUATIONS AND INEQUALITIES Sections.1.4 Solve ech eqution or inequlity. For inequlities, write the solution set in intervl nottion x = 0. 4x x 4 Ú 4. 5x + 3 Ú + 4x 5. 61y  4 = 31y x x Ú y + 4 = 41y x 6 71x  5x x = x + 4 = x  x = x x x x 5  x 4 = x x  1 = 81x x x = b b  1 = 5b x x x 1. 3t = 5 + t x x = 81x  35x 3. x 6 + 3x y y 5 = y x x 7 31x x  17x 13x x +  x Ú 3 8 1x x x 7 5 1x
Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationReview Factoring Polynomials:
Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationArithmetic & Algebra. NCTM National Conference, 2017
NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationMATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs
MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationDesigning Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For , () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationAdd and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.
TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s
More informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More information11.1 Exponential Functions
. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationCS 310 (sec 20)  Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20)  Winter 2003  Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationPhysics 9 Fall 2011 Homework 2  Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework  s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationThermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report
Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block
More informationHomework Solution  Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution  et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte nonfinl.
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationSeries: Elementary, then Taylor & MacLaurin
Lecture 3 Series: Elementry, then Tylor & McLurin This lecture on series strts in strnge plce, by revising rithmetic nd geometric series, nd then proceeding to consider other elementry series. The relevnce
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationMATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions.
MATH FIELD DAY 2012 Contestnts Insructions Tem Essy 1. Your tem hs forty minutes to nswer this set of questions. 2. All nswers must be justified with complete explntions. Your nswers should be cler, grmmticlly
More informationCalculus  Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus  Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More informationCS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)
CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More information5.3 Nonlinear stability of RayleighBénard convection
118 5.3 Nonliner stbility of RyleighBénrd convection In Chpter 1, we sw tht liner stbility only tells us whether system is stble or unstble to infinitesimllysmll perturbtions, nd tht there re cses in
More information6.3 Nonlinear stability of RayleighBénard convection
6.3. NONLINEAR STABILITY OF RAYLEIGHBÉNARD CONVECTION115 6.3 Nonliner stbility of RyleighBénrd convection In Chpter 1, we sw tht liner stbility only tells us whether system is stble or unstble to infinitesimllysmll
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More information