Equations and Inequalities


 Griselda Kelley
 3 years ago
 Views:
Transcription
1 Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40
2
3 Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in mthemticl epressions is not enough. Sometimes the vlue of the vrible is needed. An eqution is used to find the vlue of vrible. Answer these questions, before working through the chpter. I used to think: Wht does it men to sy tht n eqution is liner? Wht does it men to mke vrible the subject of n eqution? Wht re the signs of inequlity? Answer these questions, fter working through the chpter. But now I think: Wht does it men to sy tht n eqution is liner? Wht does it men to mke vrible the subject of n eqution? How re equtions solved? Wht do I know now tht I didn t know before? 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1
4 Bsics Wht is Liner Eqution? An eqution is mthemticl epression tht hs two sides seprted by n equls sign (=) nd t lest one vrible (or pronumerl ). If the highest power of the vrible is 1 then the eqution is clled liner eqution. For emple, these re ll equtions becuse = ppers in ech of them. + 4 = 1 b = 15 c + = 8 d = 5 e These re lso ll liner becuse the power of (the vrible) is 1 (there is no or etc). 6 7 = 1 How re Equtions Solved? To solve ANY eqution the gol is lwys to get the vrible by itself. But whtever is done to find the vrible by itself, must lso be done to the other side. Ech liner eqution hs only one solution! Finding by itself If you look t the liner eqution + 4 = 1, it s probbly esy to see = 8. Your brin hs ctully simplified the eqution to hve by itself without you even relising it. This is how: Subtrcting 4 from the left side of the eqution leves by itself. But the sme must be done on the right side. + 4 = = 1 4 = 8 For the eqution in b, to find by itself, both sides must be divided by We must do the sme to both sides = 15 = 15 = 5 For the eqution in c, it tkes two steps to get by itself Subtrct from both sides to leve the term with by itself Divide both sides by to leve the by itself + = 8 + = 8 = 6 = 6 = 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
5 Bsics If the vrible is in the numertor of frction, like in d, then both sides must be multiplied by the denomintor. For the eqution in d, we need to multiply the left hnd side by to find by itself Sometimes the vrible will be in frction. Multiply nd divide both sides to get the vrible by itself. In the eqution in e is the numertor of frction # = 5# = 15 Multiply both sides by the denomintor (lwys remove the denomintor first!) Divide both sides by 6 to leve the by itself 6 # 7 = 1# = 84 6 = = 14 Sometimes the left side will hve to be simplified by collecting like terms: Solve for in the following liner eqution Simplify by collecting like terms Divide both sides by 5 to leve by itself + 5+ = = = = 5 5 = = 5 Wht hppens if the Power of the Vrible is? If the highest power of the vrible is (ie. is in the eqution), then the eqution is not liner, but is clled Qudrtic eqution. To get the vrible by itself on one side, the squre root is used to chnge to. However, the squre root of both sides must be found. Ech qudrtic eqution hs two solutions! Solve for in the following liner eqution Find the squre root of both sides + 5 = = 1 5 = 16 = ^! 4h =! 4 So = 4 or = 4 Subtrct 5 from both sides Becuse 4 = 16 nd ^ 4h = % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC
6 Questions Bsics 1. Choose whether the following equtions re liner or qudrtic by circling the pproprite word (don t solve): + = 6 is liner /qudrtic eqution b = 1 is liner / qudrtic eqution c = 1 is liner / qudrtic eqution d 5 = is liner / qudrtic eqution. Solve these liner equtions: + 4 = 7 b + = 10 c  = 1 d  6 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
7 Questions Bsics. Find the vlue of the vrible in ech of these liner equtions: = 8 b 5 = 5 c 4 = 5 d = 8 e = 9 f 5 = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5
8 Questions Bsics 4. Solve for the vrible in ech liner eqution: 4 + = b  5 = 9 c 5m + 6 = 1 d n = 4 e k = 10 f m + 4 = 6 (Hint: m = 1m) 6 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
9 Questions Bsics 5. Solve for the vrible in ech liner eqution: y 4 = 4 b 5 4 = 15 c p 9 = 6 d 4d 11 = 8 e 7 = 14 f 10m 4 = 5 6. Solve these qudrtic equtions: y  9 = 0 b 4 = % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 7
10 Knowing More Wht hppens if the Vrible is on BOTH sides of the Eqution? If the vribles pper on both sides of the eqution, one side needs to be chnged to hve no vribles in it. REMEMBER! Whtever is done to one side must be done to the other side too. Solve this liner eqution: cn be subtrcted from both sides. Now the right hnd side will hve no vribles. 8 4 = = = = 6+ 4 Add 4 to both sides so tht 5 is by itself Divide both sides by 5 so tht is by itself 5 = 10 5 = = Solve this liner eqution: 4p cn be dded to both sides. Now the right hnd side will hve no vribles. p 1= 16 4p p 1+ 4p = 16 4p+ 4p 7p 1 = 16 7p 1+ 1 = Add 1 to both sides so tht 7p is by itself Divide both sides by 7 so tht p is by itself 7p = 8 7p = p = 4 If the eqution hs brckets, they must be epnded first. Then just solve the eqution the sme wy s before. Solve the following liner eqution: Epnd both brckets ^ h= ^ h 9 = = 4 cn be subtrcted from both sides so tht the right side will hve no vribles Add 9 to both sides so tht is by itself 9+ 9 = 4+ 9 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
11 Knowing More Equtions with Frctions Some shortcuts cn be used when the vrible is prt of frction in the eqution. Type 1: One frction in the eqution Multiply both sides of the eqution by the denomintor. Solve the liner eqution: (multiply both sides by the denomintor) = The denomintor ofthe frction is Multiply both sides by the denomintor # ` j= # + 1 = 18 = 18 1 = 6 Here is nother emple with the vrible in frction: Solve the liner eqution: (multiply both sides by the denomintor) ^5 h = Multiply both sides by the denomintor The denomintor ofthe frction is 5 # c m= # 5 = = 4 4 = 0 = Type : More thn one frction in the eqution Multiply both sides of the eqution by the LCD of the frctions. Solve the liner eqution: (Find the LCD) + 5 = The LCD of the two frctions is 1 Multiply both sides by the LCD 1 # ` + 5 j= 1 # ` j = = = = 1 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 9
12 Knowing More Word Problems Word problems cn be trnsformed into equtions which cn be solved. Trnsforming the words into n eqution is the most difficult prt. The equtions cn be solved using one of the methods lerned in this chpter. When number is hlved the nswer is 5, wht is this number? Let the number be According to the word problem Multiply both sides by the denomintor = 5 # = # 5 = 10 In the bove emple = 5 is the eqution represented by the word problem. To find n eqution from word problem, let vrible equl the missing vlue nd use the informtion in the word problem to crete n eqution. Dniel is 0 cm tller thn Philip nd the sum of their height is 0 cm. How tll is Philip? Let Philip s height be cm So Dniel s height must be ( +0) cm Philip s Height + Dniel s Height = 0 cm + ^+ 0h= = 0 = 0 0 = 00 = 00 = 150 So Phillip is 150 cm tll (nd Dniel is 170 cm tll) Luren is 5 yers older thn her sister Mri. If the sum of their ges is 5, how old is Luren? Let Luren s ge be (Mri) (Luren) So Mri s ge is  5 ^ 5h+ = 5 5 = 5 = 5+ 5 = 0 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
13 Questions Knowing More 1. In these liner equtions the vrible ppers on boths sides. Solve for the missing vlue: u 10 = u b 718 = + 10 c ^ + h= d 4y+ 18 = 1 y e 10^n 6h= ^10 + nh f 6m 4 = 5^m+ h g 8^k4h 5k+ = 4 h 5^y1h6^y h+ = 6 i 8t^t 18h=1 j ^+ h ^+ 4h=10. Find Ivn's mistke when he tried to solve this eqution? ^h+ h= ^h+ 1h+ 5 h+ = h+ + 5 h+ = h+ 7 h+ h = h+ 7 h h = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 11
14 Questions Knowing More. Solve these liner equtions which contin frctions: 1 8 = 4 b n + 4 = 5n c b + 4 = 5 4 d c + c = 1 e 16r + = 10 5 f m + m = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
15 Questions Knowing More 4. Solve these liner equtions which contin frctions: 7q 5 q = 6 5u+ 5 u b = c g g 1 g + = (Hint: Find LCD ofall frctions) d = 6 (Hint: Multiply both sidesbythe denomintor) e 6 8d = 1 f k 45 k = 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1
16 Questions Knowing More 5. Three times number is 45. Wht is the number? 6. Clire, Lenne nd Lindsy re sisters. Clire is two yers older thn Lenne nd Lenne is 4 yers older thn Lindsy. The sum of ll their ges is 54. How old is ech sister? 7. Chrlie hs been collecting stmps which he keeps in two seprte books. The second book hs 7 more thn triple the stmps of the first book. If he hs 5 stmps in totl (from both books) then: How mny stmps re in the first book? b How mny stmps re in the second book? 8. Victor hs bg filled with c nd 5c coins. He hs more coins worth c thn the coins worth 5c. How mny c coins does Victor hve if ll his coins sum to 88c? % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
17 Using Our Knowledge Wht re Liner Inequlities? An inequlity is mthemticl epression with two sides seprted by one of these inequlity signs: greterthn $ greter thn or equlto 1 less thn # less thn or equlto For emple 1 4 nd 8. If there is vrible in the inequlity nd its highest power is 1, then the inequlity is liner inequlity (unless the vrible is in denomintor). These epressions re ll liner inequlities becuse n inequlity ppers in ech of them nd the highest power of the vrible is b $ 18 c $ 18 d e ^ 4 h# 16 How re Inequlities Solved? Just like equtions, the im is to simplify the inequlity to get the vrible by itself on one side. Whtever is done to one side must be done to both sides. Solve these inequlities y + $ 10 b m y + $ 10 m y $ 7 m 1 18 m 1 18 m 1 6 Multiplying or Dividing by Negtive Number Everyone knows tht 1 5 is true. If both sides re multiplied by 1 then This is NOT true. If both sides of n inequlity re multiplied by negtive number then the inequlity sign must be reversed. So 5. Solve these inequlities 5 # # 5 10 b 5 # # 17 5 # $ 4 Inequlity sign is reversed fter multiplying both sides by 1 Inequlity sign is reversed fter multiplying both sides by  100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 15
18 Using Our Knowledge Other thn reversing the inequlity sign when multiplying or dividing by negtive number, inequlities re solved in the sme wy s equtions. Vrible is on both sides 5+ 5 $ $ $ 1 $ $ 1 7 b m+ 1 5m 5 m+ 1 5m 5 m 5m 1 5m 8 5m m 1 8 m 4 Inequlity sign is reversed fter dividing both sides by  Inequlities with brckets ^q + h # 7 b ^y+ 5h ^y+ 5h q + 9 # 7 q # 7 9 q # 18 # 6 y+ 10 y+ 15 y y y y y+ 5 y y 5 y 1 5 Inequlity sign is reversed fter dividing both sides by 1 Inequlities with frctions (multiply by the LCD of ALL the frctions in the inequlity) 6k 8 $ 8 5 6k 8 5 # $ 8# 5 5 6k 8+ 8 $ k $ k $ 8 b t t # # t t ` 4 j# 10 # 1 LCD of the frctions t 8t # 10 5t # 10 t $ 4 Inequlity sign is reversed fter dividing both sides by % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
19 Using Our Knowledge Grphing Inequlities Solutions to inequlities cn be represented on number line. For emple, look t the inequlity >. This mens cn be ny number greter thn, but not equl to. On number line > looks like this: On number line looks like this: Cn you spot the difference in the grphs bove? In the first grph is not included in the inequlity (>), so the circle on the number line is hollow. In the second grph the inequlity includes the number ( ) so the circle is solid. Here re some more emples: <  1 b c >  4 d < e  < < 4 f g  < 4 h 0 < i <  or 0 j 0 or % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 17
20 Questions Using Our Knowledge 1. Identify if the following re true or flse: 6 b c 8 d 1 5 e 4 4 f Solve these inequlities: b 4 $ 5 c m + 7 $ 4 d p 10 # 8 e 5q # 5 f 4h + 51 g h ^ 1h# ^1 h 4h 8 i 1 j $ % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
21 Questions Using Our Knowledge. Grph these inequlities: b $ c # 0 d e 5 # 1 5 f 1 # g 1 0nd $ h 4nd # Write down the inequlity represented by ech of the following grphs: b c d e f 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 19
22 Questions Using Our Knowledge 5. Solve these more complicted liner inequlities, then grph their solution: ^+ h b 7y 4^y+ 4h 5 4 d c # 8 d b b 5 8 # e $ f 6c c % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
23 Thinking More Simultneous Equtions Wht hppens when there re two vribles in equtions? The eqution + y = 7 hs possible solutions: The eqution + y = 18 hs possible solutions: = 1 nd y = 6 = nd y = 9 = nd y= 4 = 10 nd y = ndmnyother possibilities Common Solution The equtions hve the common solution = nd y = 4. This solution solves both equtions simultneously. = nd y = 8 = nd y = 4 = 6 nd y = = 10 nd y = 4 ndmnyother possibilities Since two equtions re solved t the sme time they re clled simultneous equtions. There re three methods to solve simultneous equtions. All of them work with ny question. It s up to you to choose the method you think is esiest. Method 1: Substitution In this method, one of the vribles is mde the subject of the formul nd then substituted into the other eqution. It hs three esy steps. Solve the simultneous equtions using substitution + y = y = 18 Step 1: Mke one of the vribles the subject in the eqution, for emple, using 1 y = 7  Step : Substitute this epression for y into the to mke new eqution, nd solve this new eqution: + (7  ) = = 18  =  = Step : Substitute this vlue into either of the equtions to find the yvlue: Substitute into 1 + y = 7 y = 7 y = 4 Second eqution ^h + y = 18 y = 1 y = 4 So the solutions re = nd y = 4. As you cn see in Step, it doesn t mtter which originl eqution you substitute the vlue of the first vrible into. The sme nswer is found for both. So choose the eqution you think would be esier to use. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1
24 Thinking More Method : Elimintion If the coefficients of one of the pronumerls re the sme in both equtions, then the equtions cn be subtrcted from one nother to eliminte one of the vribles. If the coefficients of both vribles re different in ech eqution we cn mke them the sme by multiplying one of the equtions by n pproprite number. Solve the simultneous equtions using elimintion + y = y = 18 Step 1: Mke sure one of the vribles hs the sme coefficient in both equtions At this stge both nd y hve different coefficients in the two equtions. If 1 is multiplied by then the coefficient of will be the sme s in # + y = 14 1 sme coefficient for s Step : Subtrct equtions with the sme coefficients Subtrct from + y = 18 ^+ y = 14h y = 4 The hs been eliminted nd we only need to solve for y Step : Substitute the vlue of the solved vrible into ny eqution to find the vlue of the vrible which is still unknown Substitute y = 4 into = 7 = So the solutions re = nd y = 4. In Step, y = 4 could hve lso been substituted into choose the eqution you think would be esier to use. nd the sme vlue for would hve been found. So 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
25 Thinking More Method : Grphicl Method If the stright line grphs re drwn for ech liner eqution, then the point where the lines intersect (, y) will be the solution to nd y respectively. Solve the simultneous equtions using the grphicl method + y = y = 18 Step 1: Mke y the subject of the eqution for 1 nd y = y = Step : Drw the grphs of these two equtions on the sme es y Lines intersect Step : Red the point where the lines intersect The lines intersect t the point (, y) = (, 4) So = nd y = 4 Notice tht ll three methods hve the solution = nd y = 4. All three methods work ll the time so just use the one you feel is the esiest. The net pge demonstrtes ll methods with nother emple. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC
26 Thinking More Find the solution to these simultneous equtions: y + = 1 nd y + 4 = Using substitution, b elimintion nd c grphicl method. Substitution Step 1: Mke one of the vribles the subject of 1 =  y Step : Substitute this epression into nd solve: y+ 4^ yh= y+ 1 8y = 5y = 10 y = Step : Substitute this vlue into 1 or to solve for the remining vrible: ^ h+ = = 4 = 1 So = 1 nd y = b Elimintion Step 1: Mke sure one of the vribles hs the sme coefficient in both equtions 4# 1 = 8y+ 4 = 1 hs the sme coefficient for s Step : Subtrct equtions with the sme coefficients to eliminte vrible So 5y = 10 ` y = 8y+ 4 = 1 ^y+ 4 = h 5y = 10 Step : Substitute the vlue of the solved vrible into ny eqution to find the vlue of the vrible which is still unknown Substitute y = into 1 to obtin: So = 1 nd y = ^ h+ = ` = 4 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
27 Thinking More c Grphicl Step 1: Mke y the subject of both equtions y = 1 + y = Step : Drw the grphs of these two equtions on the sme es Step : Red the point where the lines intersect The lines intersect t (1, ) so = 1 nd y = y As you cn see, ll three methods produce the sme solution. Simultneous Equtions Word Problems As with single liner equtions, word problems cn be trnslted into simultneous equtions. Determine which TWO missing vlues re required nd choose vribles to represent these. Write two equtions using these vribles nd then use Substitution, Elimintion or the Grphicl Method to solve the equtions. The sum of two numbers is 1 nd their difference is 6. Find the two numbers: Let nd y represent the numbers. So + y = 1 1 nd  y = 6 These simultneous equtions cn be solved using subsitution, elimintion or the grphicl method. Jun is twelve yers older thn his sister Jmil. In two yers Jmil will be hlf Jun s ge. Find Jun nd Jmil s ge: Let = Jun s ge Let y = Jmil s ge  y = = y These re simultneous equtions which cn be solved using subsitution, elimintion or the grphicl method. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5
28 Questions Thinking More 1. Write down possible solutions for the vribles in these equtions: + y = 4 b + b = 6 c  4y = 10. Solve for the vribles in these simultneous equtions using the substitution method: + y = 1  y = 4 b p + q = 10 q  4p = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
29 Questions Thinking More. Use the grphicl method to solve for these equtions: + y =  y = 6 b y  4 = 4 y + = 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 7
30 Questions Thinking More 4. Solve for the vribles in these simultneous equtions using the elimintion method:  y = 15 y + = 0 b b  4 = 1  b = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
31 Questions Thinking More 5. Solve these simultneous equtions using ny method: 8c  8d = 9c  8d = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 9
32 Questions Thinking More 6. Solve these simultneous equtions using ny method: 7m = 166n n = m % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
33 Questions Thinking More 7. Solve these simultneous equtions using ny method: 6  b  10 = 0 + b  9 = 0 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1
34 Questions Thinking More 8. Find equtions nd solve them for these word problems (using ny method): The sum of two numbers is 1. The sum of the first number, nd double the second number is 16. Wht re the numbers? b Ari is three yers older thn Eric. In three yers from now, Ari will be twice s old s Eric will be. How old re they now? c A resturunt sells two kinds of mels: pizz nd pst. A pizz costs $14 nd pst costs $10. In single dy the resturunt sold 79 mels. If they erned $994 on this dy, how mny of ech mel ws sold? 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
35 Answers Bsics: 1. Liner b Qudrtic c Liner d Liner 4. Knowing More: q = 4 b u = 11 c g = 0 5 d = e d = 1 f k = 45. = b = 7 5. n 15 = c = 15 d = Lenne is 18, Clire is 0 nd Lindsy is 14. = 4 b = 7 c = 0 d = 4 e = 6 f = 7. 7 b 8 4. d 5 = b = 7 c m = 5 n = 4 e k = 5 f m = 8. Victor hs 1 coins worth of 5c nd 14 coins worth of c. 5. d y = 16 b = 1 c p = 7 d = e = 6 f m = Using Our Knowledge: d True Flse b e True Flse c f Flse True 6. b so y = or y = so = 5or = 5. d 1 1 b $ 9 c m $ 11 p # e q # 7 f h 1 Knowing More: g 6 h # 1 i h u = 10 b = 7 c = j $ 40 d g y = 1 e n = 10 f m = 1 k = 11 h y = 1 i t 5 =. j 4 = b. Ivn epnded the brckets incorrectly Finl nswer should be: h 1 = c. = 40 b n = 1 c b = 8 d c = 8 e r = f m = 6 d 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC
36 Answers. Using Our Knowledge: e f g Thinking More: 1. Anything where nd y dd up to 4 Eg =, y = 1 or =, y = or = 4, y = 0 or = 5, y = 1 or =.5, y = 0.5 etc b Anything where twice, plus b, mkes 6. Eg = 1, b = 4, or =, b = or =, b = 0 or = 4, b =  or = 0.5, b = 5 etc 4. h 1 4 b # 4 c Anything where times the first number, minus three times the second number, mkes 10. Eg = 6, y = 4, or = , y = 4 or = 0, y = 0 or =, y = 1 or = 1, y = etc c $1 d  # 1 4. = 1. nd y = 1.4 e 4 1 # 0 f # b p = 7 nd q = b b y 7 c d $ 4 = nd y =  =  nd y = 4 d b $ 4 4. b =  nd y = 6 = 5 nd b = 8 e # 1 5. c = nd d = 11 4 f c n 5 = 7. = 4, b = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
37 Answers Thinking More: 8. b c = 8 nd b = 4 Eric is 0; Ari is. 51 pizzs nd 8 pst mels were sold 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5
38 Notes 6 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning
39
40 Equtions nd Inequlities
ALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationSample pages. 9:04 Equations with grouping symbols
Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationConsolidation Worksheet
Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationExponentials  Grade 10 [CAPS] *
OpenStxCNX module: m859 Exponentils  Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
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 bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
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 informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9 # + 4 ' ) ' ( 9 + 7) ' ' Give this
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X Section 4.4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether  is root of 0. Show
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationAlgebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1
Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS  Algebr Comprehensive PrePost Assessment CRS  Algebr Comprehensive Midterm Assessment Algebr Bsics CRS  Algebr QuikPiks
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 informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
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 informationPreparation for A Level Wadebridge School
Preprtion for A Level Mths @ Wdebridge School Bridging the gp between GCSE nd A Level Nme: CONTENTS Chpter Removing brckets pge Chpter Liner equtions Chpter Simultneous equtions 6 Chpter Fctorising 7 Chpter
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationCH 9 INTRO TO EQUATIONS
CH 9 INTRO TO EQUATIONS INTRODUCTION I m thinking of number. If I dd 10 to the number, the result is 5. Wht number ws I thinking of? R emember this question from Chpter 1? Now we re redy to formlize the
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationthan 1. It means in particular that the function is decreasing and approaching the x
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
More information7h1 Simplifying Rational Expressions. Goals:
h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & , no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd
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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
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 informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationSECTION 94 Translation of Axes
94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationEquations, expressions and formulae
Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
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 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 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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More information2.4 Linear Inequalities and Problem Solving
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
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 informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
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 informationSUMMER ASSIGNMENT FOR PreAP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day!
SUMMER ASSIGNMENT FOR PreAP FUNCTIONS/TRIGONOMETRY Due Tuesdy After Lor Dy! This summer ssignment is designed to prepre you for Functions/Trigonometry. Nothing on the summer ssignment is new. Everything
More information1. Twelve less than five times a number is thirty three. What is the number
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
More informationPHYS Summer Professor Caillault Homework Solutions. Chapter 2
PHYS 1111  Summer 2007  Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement
More information12.1 Introduction to Rational Expressions
. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To
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 information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationFaith Scholarship Service Friendship
Immcult Mthemtics Summer Assignment The purpose of summer ssignment is to help you keep previously lerned fcts fresh in your mind for use in your net course. Ecessive time spent reviewing t the beginning
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationChapter 3 Exponential and Logarithmic Functions Section 3.1
Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re nonlgebric functions. The re clled trnscendentl functions. The eponentil
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 informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationSpring 2017 Exam 1 MARK BOX HAND IN PART PIN: 17
Spring 07 Exm problem MARK BOX points HAND IN PART 0 555=x5 0 NAME: Solutions 3 0 0 PIN: 7 % 00 INSTRUCTIONS This exm comes in two prts. () HAND IN PART. Hnd in only this prt. () STATEMENT OF MULTIPLE
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 informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationBelievethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationAP Calculus AB Summer Packet
AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
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 informationSection 3.2: Negative Exponents
Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.
More informationMatrix Solution to Linear Equations and Markov Chains
Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More information5A5 Using Systems of Equations to Solve Word Problems Alg 1H
5A5 Using Systems of Equtions to Solve Word Problems Alg 1H system of equtions, solve the system using either substitution or liner combintions; then nswer the problem. Remember word problems need word
More informationNAME: MR. WAIN FUNCTIONS
NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
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 informationMathcad Lecture #1 Inclass Worksheet Mathcad Basics
Mthcd Lecture #1 Inclss Worksheet Mthcd Bsics At the end of this lecture, you will be ble to: Evlute mthemticl epression numericlly Assign vrible nd use them in subsequent clcultions Distinguish between
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 informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus Professor Richrd Blecksmith richrd@mth.niu.edu Dept. of Mthemticl Sciences Northern Illinois University http://mth.niu.edu/ richrd/mth229. The Definite Integrl We define
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 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 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 informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
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 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
More information