APPENDIX D Precalculus Review

Transcription

1 APPENDIX D Preclculus Review SECTION D. Rel Numers nd the Rel Line Rel Numers nd the Rel Line Order nd Inequlities Asolute Vlue nd Distnce Rel Numers nd the Rel Line Rel numers cn e represented y coordinte system clled the rel line or -is (see Figure A.). The rel numer corresponding to point on the rel line is the coordinte of the point. As Figure D. shows, it is customry to identify those points whose coordintes re integers The rel line Figure D Rtionl numers Figure D Irrtionl numers Figure D.3 e π The point on the rel line corresponding to zero is the origin nd is denoted y. The positive direction (to the right) is denoted y n rrowhed nd is the direction of incresing vlues of. Numers to the right of the origin re positive. Numers to the left of the origin re negtive. The term nonnegtive descries numer tht is positive or zero. The term nonpositive descries numer tht is negtive or zero. Ech point on the rel line corresponds to one nd only one rel numer, nd ech rel numer corresponds to one nd only one point on the rel line. This type of reltionship is clled one-to-one-correspondence. Ech of the four points in Figure D. corresponds to rtionl numer one tht cn e epressed s the rtio of two integers. (Note tht 4. 9 nd.6 3. Rtionl numers cn e represented either y terminting decimls such s.4, or y repeting decimls such s Rel numers tht re not rtionl re irrtionl. Irrtionl numers cnnot e represented s terminting or repeting decimls. In computtions, irrtionl numers re represented y deciml pproimtions. Here re three fmilir emples e (See Figure D.3.) D

2 D APPENDIX D Preclculus Review < if nd only if lies to the left of. Figure D.4 Order nd Inequlities One importnt property of rel numers is tht they cn e ordered. If nd re rel numers, is less thn if is positive. This order is denoted y the inequlity <. The sttement is greter thn is equivlent to sying tht is less thn. When three rel numers,, nd c re ordered such tht < nd < c, we sy tht is etween nd c nd < < c. Geometriclly, < if nd only if lies to the left of on the rel line (see Figure D.4). For emple, < ecuse lies to the left of on the rel line. The following properties re used in working with inequlities. Similr properties re otined if < is replced y nd > is replced y. (The symols nd men less thn or equl to nd greter thn or equl to, respectively.) Properties of Inequlities Let,, c, d, nd k e rel numers.. If < nd < c, then < c. Trnsitive Property. If < nd c < d, then c < d. Add inequlities. 3. If <, then k < k. Add constnt. 4. If < nd k >, then k < k. Multiply y positive constnt.. If < nd k <, then k > k. Multiply y negtive constnt. NOTE Note tht you reverse the inequlity when you multiply y negtive numer. For emple, if < 3, then 4 >. This lso pplies to division y negtive numer. Thus, if > 4, then <. A set is collection of elements. Two common sets re the set of rel numers nd the set of points on the rel line. Mny prolems in clculus involve susets of one of these two sets. In such cses it is convenient to use set nottion of the form {: condition on }, which is red s follows. The set of ll such tht certin condition is true. { : condition on } For emple, you cn descrie the set of positive rel numers s : >. Set of positive rel numers Similrly, you cn descrie the set of nonnegtive rel numers s :. Set of nonnegtive rel numers The union of two sets A nd B, denoted y A B, is the set of elements tht re memers of A or B or oth. The intersection of two sets A nd B, denoted y A B, is the set of elements tht re memers of A nd B. Two sets re disjoint if they hve no elements in common.

3 APPENDIX D Preclculus Review D3 The most commonly used susets re intervls on the rel line. For emple, the open intervl, : < < Open intervl is the set of ll rel numers greter thn nd less thn, where nd re the endpoints of the intervl. Note tht the endpoints re not included in n open intervl. Intervls tht include their endpoints re closed nd re denoted y, :. Closed intervl The nine sic types of intervls on the rel line re shown in the tle elow. The first four re ounded intervls nd the remining five re unounded intervls. Unounded intervls re lso clssified s open or closed. The intervls, nd, re open, the intervls, nd, re closed, nd the intervl, is considered to e oth open nd closed. Intervls on the Rel Line Intervl Nottion Set Nottion Grph Bounded open intervl, : < < Bounded closed intervl, : Bounded intervls (neither open nor closed),, : < : < Unounded open intervls,, : < : > Unounded closed intervls,, : : Entire rel line, : is rel numer NOTE The symols nd refer to positive nd negtive infinity. These symols do not denote rel numers. They simply enle you to descrie unounded conditions more concisely. For instnce, the intervl, is unounded to the right ecuse it includes ll rel numers tht re greter thn or equl to.

4 D4 APPENDIX D Preclculus Review EXAMPLE Liquid nd Gseous Sttes of Wter Descrie the intervls on the rel line tht correspond to the temperture (in degrees Celsius) for wter in. liquid stte. gseous stte. Solution. Wter is in liquid stte t tempertures greter thn nd less thn, s shown in Figure D.()., : < <. Wter is in gseous stte (stem) t tempertures greter thn or equl to, s shown in Figure D.()., : () Temperture rnge of wter (in degrees Celsius) () Temperture rnge of stem (in degrees Celsius) Figure D. A rel numer is solution of n inequlity if the inequlity is stisfied (is true) when is sustituted for. The set of ll solutions is the solution set of the inequlity. EXAMPLE Solving n Inequlity Solve < 7. Solution < 7 Originl inequlity < 7 Add to oth sides. < < < 6 Multiply oth sides y. The solution set is, 6. If, then ( ) If, then () NOTE In Emple, ll five inequlities listed s steps in the solution re clled equivlent ecuse they hve the sme solution set. If , then ( 7) Checking solutions of < 7 Figure D Once you hve solved n inequlity, check some -vlues in your solution set to verify tht they stisfy the originl inequlity. You should lso check some vlues outside your solution set to verify tht they do not stisfy the inequlity. For emple, Figure D.6 shows tht when or the inequlity < 7 is stisfied, ut when 7 the inequlity < 7 is not stisfied.

5 APPENDIX D Preclculus Review D EXAMPLE 3 Solving Doule Inequlity Solve 3. Solution [, ] Solution set of 3 Figure D The solution set is,, s shown in Figure D.7. Originl inequlity Sutrct. Divide y nd reverse oth inequlities. The inequlities in Emples nd 3 re liner inequlities tht is, they involve first-degree polynomils. To solve inequlities involving polynomils of higher degree, use the fct tht polynomil cn chnge signs only t its rel zeros (the numers tht mke the polynomil zero). Between two consecutive rel zeros, polynomil must e either entirely positive or entirely negtive. This mens tht when the rel zeros of polynomil re put in order, they divide the rel line into test intervls in which the polynomil hs no sign chnges. Thus, if polynomil hs the fctored form r r < r < r 3 <... r... r n, < r n the test intervls re, r, r, r,..., r n, r n, nd r n,. To determine the sign of the polynomil in ech test intervl, you need to test only one vlue from the intervl. EXAMPLE 4 Solving Qudrtic Inequlity Solve < 6. Choose ( 3)( 3. ) Choose 4. ( 3 )( ) Solution < 6 6 < 3 < Originl inequlity Write in stndrd form. Fctor. 3 Choose ( 3) ( Testing n intervl Figure D.8. ) 3 4 The polynomil 6 hs nd 3 s its zeros. Thus, you cn solve the inequlity y testing the sign of 6 in ech of the test intervls,,, 3, nd 3,. To test n intervl, choose ny numer in the intervl nd compute the sign of 6. After doing this, you will find tht the polynomil is positive for ll rel numers in the first nd third intervls nd negtive for ll rel numers in the second intervl. The solution of the originl inequlity is therefore, 3, s shown in Figure D.8.

6 D6 APPENDIX D Preclculus Review Asolute Vlue nd Distnce If is rel numer, the solute vlue of is,, The solute vlue of numer cnnot e negtive. For emple, let 4. Then, ecuse 4 <, you hve 4 if if < Rememer tht the symol does not necessrily men tht is negtive. Opertions with Asolute Vlue Let nd e rel numers nd let n e positive integer..., n n NOTE You re sked to prove these properties in Eercises 73, 7, 76, nd 77. Properties of Inequlities nd Asolute Vlue Let nd e rel numers nd let k e positive rel numer... k if nd only if k k. 3. k if nd only if k or k. 4. Tringle Inequlity: Properties nd 3 re lso true if is replced y <. EXAMPLE Solving n Asolute Vlue Inequlity Solve 3. units units 4 4 Solution set of 3 Figure D Solution Using the second property of inequlities nd solute vlue, you cn rewrite the originl inequlity s doule inequlity. 3 Write s doule inequlity Add 3. The solution set is,, s shown in Figure D.9.

7 APPENDIX D Preclculus Review D7 EXAMPLE 6 A Two-Intervl Solution Set (, ) (, ) Solution set of 3 < Figure D. d d + d Solution set of d d d + d Solution set of d Figure D. d d Solve 3 <. Solution Using the third property of inequlities nd solute vlue, you cn rewrite the originl inequlity s two liner inequlities. 3 < < or or < 3 < The solution set is the union of the disjoint intervls, nd,, s shown in Figure D.. Emples nd 6 illustrte the generl results shown in Figure D.. Note tht if d >, the solution set for the inequlity is single intervl, wheres the solution set for the inequlity d d is the union of two disjoint intervls. The distnce etween two points nd on the rel line is given y d. The directed distnce from to is nd the directed distnce from to is, s shown in Figure D.. Distnce etween nd Directed distnce from to Directed distnce from to Figure D. EXAMPLE 7 Distnce on the Rel Line Distnce = Figure D.3 4. The distnce etween 3 nd 4 is (See Figure D.3.). The directed distnce from 3 to 4 is c. The directed distnce from 4 to 3 is or The midpoint of n intervl with endpoints nd is the verge vlue of nd. Tht is, Midpoint of intervl,. To show tht this is the midpoint, you need only show tht is equidistnt from nd.

8 D8 APPENDIX D Preclculus Review E X E R C I S E S F O R APPENDIX D. In Eercises, determine whether the rel numer is rtionl or irrtionl In Eercises 4, epress the repeting deciml s rtio of integers using the following procedure. If , then Sutrcting the first eqution from the second produces or Given <, determine which of the following re true. () (c) (e) < > > 6. Complete the tle with the pproprite intervl nottion, set nottion, nd grph on the rel line. Intervl Nottion, 4, 7 In Eercises 7, verlly descrie the suset of rel numers represented y the inequlity. Sketch the suset on the rel numer line, nd stte whether the intervl is ounded or unounded. In Eercises 4, use inequlity nd intervl nottion to descrie the set.. y is t lest 4. Set Nottion () (d) 7. 3 < < < 8 (f) : 3 3 < < < Grph. q is nonnegtive. 3. The interest rte on lons is epected to e greter thn 3% nd no more thn 7%. 4. The temperture T is forecst to e ove 9 tody. < In Eercises 44, solve the inequlity nd grph the solution on the rel line < 3 < < 9. 3 > 3. > 3. < 3. 3 > > 3 3. <, > < < 9 3 In Eercises 4 48, find the directed distnce from to, the directed distnce from to, nd the distnce etween nd () 6, 7 () 6, () 9.34,.6 () 6, 7 In Eercises 49, find the midpoint of the intervl = = = = = = = = () 7,. () 6.8, 9.3 () 8.6,.4 () 4.6,.3

9 APPENDIX D Preclculus Review D9 In Eercises 3 8, use solute vlues to define the intervl or pir of intervls on the rel line = = 3 3 = 3 = () All numers tht re t most ten units from. () All numers tht re t lest ten units from. 8. () y is t most two units from. () y is less thn units from c. 9. Profit The revenue from selling units of product is R.9 nd the cost of producing units is C 9 7. = = = = To mke (positive) profit, R must e greter thn C. For wht vlues of will the product return profit? 6. Fleet Costs A utility compny hs fleet of vns. The nnul operting cost of ech vn is estimted to e where C is mesured in dollrs nd m is mesured in miles. The compny wnts the nnul operting cost of ech vn to e less thn $,. To do this, m must e less thn wht vlue? 6. Fir Coin To determine whether coin is fir (hs n equl proility of lnding tils up or heds up), n eperimenter tosses it times nd records the numer of heds. The coin is declred unfir if For wht vlues of will the coin e declred unfir? 6. Dily Production The estimted dily production p t refinery is p,, <, C.3m where p is mesured in rrels of oil. Determine the high nd low production levels. In Eercises 63 nd 64, determine which of the two rel numers is greter. 63. () or () or 73 () or () or 6. Approimtion Powers of The speed of light is meters per second. Which est estimtes the distnce in meters tht light trvels in yer? () 9. () 9. (c) 9. (d) Writing The ccurcy of n pproimtion to numer is relted to how mny significnt digits there re in the pproimtion. Write definition for significnt digits nd illustrte the concept with emples. True or Flse? In Eercises 67 7, determine whether the sttement is true or flse. If it is flse, eplin why or give n emple tht shows it is flse. 67. The reciprocl of nonzero integer is n integer. 68. The reciprocl of nonzero rtionl numer is rtionl numer. 69. Ech rel numer is either rtionl or irrtionl. 7. The solute vlue of ech rel numer is positive. 7. If <, then. 7. If nd re ny two distinct rel numers, then < or >. In Eercises 73 8, prove the property Hint: 7., n,, 3, n n, 79. k, if nd only if k k, k >. 8. if nd only if k or k, k >. k Find n emple for which nd n emple for which Then prove tht. for ll,. 8. Show tht the mimum of two numers nd is given y the formul m,. Derive similr formul for min, >,