Precalculus - Lecture notes chapter 1 to 8 PDF

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Math 2

First day in class problems - Rev i e w/Pr e vi ew of T opi cs in t his class

Solve. 1) x 4 - 3x 2 + 2 = 0 A ) 1, 2

1) B) ±1, ± 2

C) ±1, ±2 2

D) ±1, ±2

2) 6x 2/5 + 8x 1/5 + 2 = 0

2)

Sol v e t he e q u at ion . 1 3) 2 (5 - 3x) = 16

3)

4) log (x - 8) + log (x - 8) = 1 9 9

4)

5) ln (x - 2) + ln (x + 5) = ln 30

5)

6) 5 log x = (log x)2

6)

7) 5 3x - 5 (3x + 1) = -100

7)

Solve.

Solve. 8) H ow long will it ta ke for the popul at ion of a certa in c oun t r y to doubl e if its a nnu al grow th rate is 7.9%? Round t o t he nearest year . Use t he exponential growth model P(t) = P e kt . 0 A ) 4 yr B) 1 yr C) 9 yr D) 25 yr N ame t h e q u a dr ant in w hich t h e angl e Ό li es. 9) sec Ό < 0, ta n Ό < 0 A) I B) II 10) tan Ό > 0, A) I

8)

9) C) III

D) IV

C) III

D) IV

sin Ό < 0

10) B) II

1

Solv e t he p robl em. 11) Deter mine the sign of t he t r igonom et ri c va lu es listed below. (i) sin 250° (ii) tan 330° (iii) cos(- 40°)

11)

Use th e p rop erties of the t rigonom et ri c f un ct ion s to f in d t h e exact va lu e of t he e xp r ess ion . D o no t use a calc u l at or . 12) sin 2 65° + cos2 65° 12) 13) sec2 35° - ta n 2 35° 14) ta n 55° cot 55° A) 1

13) 14) B) 55

C) 0

Find the exact va lu e of the i n dicated t r igonom et ri c f un ct ion of Ό. 7 3Δ , 15) cos Ό = < Ό < 2Δ Find cot Ό. 25 2 Find t he exact va l u e of the e xpr ess ion . śΔ 16) cos 12

D) - 1

15)

16)

17) sin 10° cos 50° + c os 10° sin 50°

17)

2

A n sw er K e y Testname: M 2 FIRST H W SU19

1) B 2) - 1, -

1 243

{3} 11 5 1, 100,000 2 7) 3 3) 4) 5) 6)

8) C 9) B 10) C 11) (i) negat iv e (ii) negat iv e (iii) po sit iv e 12) 1 13) 1 14) A 7 15) - 24 2( 3 - 1) 4

16) 17)

3 2

3

Math 2

First day in class problems - Revi e w/Pr e vi ew of Topi cs in t hi s class

Solve. 1) x 4 - 12x 2 + 27 = 0 A) 3, 3

1) B) ±3, ± 3

C) ±3, ±2 3

D) ±3, ±3

2) (t + 2)2/3 + 3(t + 2)1/3 - 10 = 0 A) 127, 6

2)

B) - 5, 2

C)

3

- 5,

3

2

D) - 127, 6

Solve. 3) 4 x = 32(2x + 3)

3)

4) log4 (x - 4) + log 4 (x - 4) = 1

4)

5) ln (x - 7) + ln (x + 3) = ln 24

5)

6) 6 log x = (log x)2

6)

7) 3 3x - 3 (3x + 1) = -18

7)

Solve. 8) H ow long will it take for the popul at ion of a certa in coun t r y to doubl e if its a nnu al grow th rate is 3.9%? Round t o t he nearest year . Use t he exponential growth model P(t) = P e kt . 0 A) 8 yr B) 1 yr C) 18 yr D) 51 yr Sol v e the e q u at ion . 9) b - 3 - 8 = 1

9)

10) 8m + 4 + 9 = 14

10)

Sol v e t he abs ol u te-va lu e in e q u a li ty. 11) 3x + 2 < 8 10 ,2 A) 3 C) - Q, -

8)

11) 10 B) - Q, 3

10 1 2, Q 3

D) - Q, 3

12) b + 4 - 9 > 6 A) (- Q, - 19) 1 (11, Q) C) (- Q, 7 ) 1 (19, Q)

12) B) (- Q, - 19) 1 (- 7, Q) D) (- 19, 11)

1

N ame th e q u a dr ant in w h ich t h e angl e Ό li es. 13) sin Ό > 0, cos Ό < 0 14) cos Ό > 0,

13)

csc Ό < 0

14)

Solv e the probl em. 15) Whi ch of the follow ing t rigonom et ri c va lu es are negat iv e? I. sin(- 292°) II . ta n( - 193°) III. cos(- 207°) IV. cot 222°

15)

Use th e p rop erties of the t rigonom et ri c f un ct ion s to f in d t h e exact va lu e of t he e xp r ess ion . D o no t use a calc u l at or . 16) sin 2 50° + cos2 50° 16) 17) sec2 40° - ta n 2 40°

17)

Find the exact va lu e of the i n dicated t r igonom et ri c f un ct ion of Ό. 15 3Δ 18) cos Ό = , < Ό < 2Δ Find cot Ό. 17 2

18)

Find t he exact va l u e of the e xpr ess ion . Δ 19) sin 12

19)

20) sin 25° cos 35° + c os 25° sin 35°

20)

Use the in f orm at ion gi v en a bou t the a ngl e Ό, 0 K Ό K 2Δ, t o f ind t h e e x act valu e of t h e i n d icated t rigonometri c f u nct ion . 3 Find cos(2Ό). 21) csc Ό = - , ta n Ό > 0 21) 2 Find t he exact va lu e un d er the gi v en condi t ion s. 5 15 Δ Δ < ΅ < Δ; cos Ά = , 0 < Ά < , 22) sin ΅ = 2 17 2 13

Find sin (΅ - Ά).

Find t he exact va l u e of the e xpr ess ion . 1 1 23) cos sin - 1 - tan - 1 2 3

22)

23)

2

A n sw er K e y T estname: M 2 FIRST D A Y I N C L A SS F 19

1) B 2) D 15 8

3) -

4) 6 5) 9 6) 1, 1,000,000 2 7) 3 8) C 9) {12, - 6} 1 9 10) ,8 8 11) A 12) A 13) II 14) IV 15) II a nd III 16) 1 17) 1 15 18) 8 2( 3 - 1) 4

19) 20)

3 2

21)

1 9

22)

171 221

23)

4 10 + 15

5

3

Math 2

Sect 1.1 - 1.2

F in d th e di stance d(P1 , P2 ) bet w een t h e poin ts P1 a nd P2 . 1) P1 = (6, - 4 ); P2 = (2, - 2) A) 12

1)

B) 12 3

C) 6

D) 2 5

D ecide w het her or no t t h e poin ts are the v ert ices of a righ t t ri a ngl e. 2) (- 8, 7 ), (- 6, 11 ), (- 4, 10 )

2)

Sol ve t he p roblem. 3) Find a ll the poin ts ha v ing an x - coordin ate of 9 w ho se di stance from the poin t (3, - 2) is 10.

3)

4) A mo t or cycle a n d a car l eave an in t ersect ion at the same tim e. T h e mo t or cyc l e heads nor th at a n avera ge speed of 20 miles per ho ur , wh ile t he ca r heads east at a n a ver a ge speed of 48 miles per hour . F in d an ex pr ess ion for the ir di stance ap art in mil es at the e n d of t ho ur s. A) 52 t miles B) t 68 miles C) 2t 13 miles D) 52t miles F i nd th e m i dpoi nt of t he l in e se gm e nt join ing t he poin ts P1 a nd P2 . 5) P1 = (4, 9); P2 = (8, 1) 6) P1 = (a, 2 ); P2 = (0, 1) a A) - , 1 2

4)

5) 6)

B) a,

3 2

C)

a 3 , 2 2

D) a, 3

Sol ve t he p roblem. 7) If (4, - 7 ) is the e nd poin t of a lin e se gm ent, a nd ( - 1, - 5) is its midpoin t, fin d t he other endpoint. A) (8, - 17) B) (- 6, - 9) C) (14, - 11) D) (- 6, - 3)

7)

D eter mi e w heth er the gr a p h of the eq uat ion is sy m metric w it h respect to the x -axi s, th e y - xi s, a n d/or th e orig in . 8) x 2 + y - 36 = 0 8) 9) 4x 2 + 16y 2 = 64

10) y =

9)

7x 2 x + 49

10)

Sol ve t he p roblem. 11) If a gr a ph is s y mm et ri c wi th respect to the y - a xi s a n d it con ta in s the poin t (5, - 6), w hi ch of the follo w ing poin ts is also on the gr a p h ? A) (5, - 6) B) (- 5, - 6) C) (- 6, 5) D) (- 5, 6)

F i nd an d si mp l if y t he di f ference q u o t i ent of f ,

f(x+ h) - f(x) , hJ 0, for t h e f u n ct ion . h

12) f(x) = 8x 2

12)

1

11)

A n s w er K e y T estname: M2 SE C T 1.1,1.2 F18

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

D Y es (9, 6), (9, - 10 ) D (6, 5) C D y - axis x - axi s, y - a xi s, origin origin D 8(2x+h)

2

Math 2

Sect 1.1 - 1.2 h w

Find the di sta nce d(P 1 , P2 ) bet w ee n t h e poi n ts P1 a nd P2 . 1) P1 = (2, -7); P2 = (5, -1)

1)

Decide w het her or no t t he poi n ts are t he vert ices of a righ t t ri a ngl e. 2) (-5, 1), (1 , 3), (0 , -2) A) Yes B) No

2)

Solv e the probl em. 3) Find a ll the poin ts ha ving an x - coordin ate of 9 w ho se di stance from the poin t (3, - 2) is 10. 4) Find all va lu es of k so that the giv en poin ts are (- 5, 5), (k , 0)

29 uni ts apart.

3) 4)

5) A mid dle school 's baseball pl a ying fi eld is a squ are, 60 feet on a sid e. H ow far is it dir ect ly from hom e pl ate to second base (the di a gon al of the squ are)? If necessa ry , round to th e n earest foo t. A) 84 fee t B) 85 fee t C) 92 fee t D) 86 fee t

5)

6) A mo t or cycle a nd a car leave an in tersect ion at the same tim e. T h e mo t or cycle heads nor th at a n average speed of 20 miles per hour , w hile t he ca r heads east at a n a verage speed of 48 miles per hour . Find an e xpr ession for the ir di stance apart in mil es at the e nd of t hour s. A) 52 t miles B) t 68 miles C) 2t 13 miles D) 52t miles

6)

Find the mi dpoin t of the lin e se gm e nt joi ning the poin ts P1 a n d P2 .

7) P1 = (7, 1); P2 = (- 16, - 16)

7)

8) P1 = (a, 2 ); P2 = (0, 1) A) a, 3

8) B)

a 3 , 2 2

C) -

a ,1 2

D) a,

Solv e the probl em. 9) If (- 2, - 3) is t he endpoin t of a line segmen t, a nd (2, - 4) is its midpoin t, find t he other endpoint.

3 2

9)

Li st the i n tercepts for the gr a p h of the e q u at ion . 10) 4x 2 + 9 y 2 = 36

10)

A) (- 4, 0), (- 9, 0), (9, 0), (4, 0) C) (- 2, 0), (- 3, 0), (3, 0), (2, 0)

B) (- 9, 0), (0, - 4), (0, 4 ), (9, 0) D) (- 3, 0), (0, - 2), (0, 2 ), (3, 0)

D etermin e w hether the gr a p h of the e q u at ion is sy mm et ri c w ith resp ect to t h e x-axi s, the y -a xi s, a nd/or the origi n . 11) x 2 + y - 36 = 0 11)

12) y =

7x 2 x + 49

12)

1

Solv e the probl em. 13) If a gr a ph is s ymm et ri c w i th respect to the y - a xi s a nd it c on ta in s the poin t (5, - 6), w hi ch of the follow ing poin ts is also on the gr a ph ? A) (5, - 6) B) (- 5, - 6) C) (- 6, 5) D) (- 5, 6)

Find a nd s imp lif y the dif f erence quo tient of f,

f(x+ h) - f(x) , hJ 0, for t h e f un ct ion . h

14) f(x) = 6x 2

14)

G rap h the e q u at ion by p lo tt i ng poi n ts. 15) y = x 3

15)

16) x = y 2

16)

2

13)

A n sw er K e y Testname: M2 SE C T 1.1,1.2 H W

3 5 B (9, 6), (9, - 10) - 3, - 7 B D 9 15 7) - , 2 2

1) 2) 3) 4) 5) 6)

8) B 9) (6, - 5) 10) D 11) y - axis 12) origin 13) D 14) 6(2x+h) 15)

16)

3

Name:

Semester:

Class: Math 2 3A 3B 4A

8 11 253

Yr:

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Time:

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