Copyright © 2011 Pearson Education, Inc. … › sample › Solution-Manual-for-Calculus...y = 1 33.8(12) = 0 .355 y = mx y = m · x y =1.609x x = 261 y =1.609(261) = 419 .95 1 =1.609

3 2 −4

|x− (−1)| = 3 x+ 1 = ±3 x = 2 x = −4

|− 5− 2| = |− 7| = 7

|2− 7| = | − 5| = 5 | − 5− 7| = | − 12| = 12

2x − 4 = ±6 2x = 10 x = 5 2x = −2

x = −1

x− 3 = ±2 x = 5 x = 1

2x+ 3 = ±5 2x = 2 x = 1 2x = −8 x = −4

−2

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± (2x+ 4) = ± (5x− 2) 2x +

4 = 5x− 2 2x+ 4 = − (5x− 2) x = 2 x = −27

5−3u = ± (3 + 2u)

5−3u = 3+2u u = 25 5−3u = −3−2u

u = 8

4 + t2 = ± (

32 t− 2

)t = 6

t = −1

2s−3 = ± (7− s) s = 103 s = −4

−4 ≤ 5x− 2 ≤ 4

−4 + 2 ≤ 5x ≤ 4 + 2

−2

5≤ x ≤ 6

5

1 − 3x < −8 1 − 3x > 8

9 < 3x x > 3 −3x > 7 −3

x < −73

7x+ 4 ≥ 3 x ≥ −17

7x+ 4 ≤ −3 7x ≤ −7 x ≤ −1

|6− 5x| < 7

−7 < 6− 5x < 7

−7− 6 < −5x < 7− 6

−13 < −5x < 1

−15 < x < 13

5




−6 < 2x+ 3 < 6 3 −9 < 2x < 3 −92 < x < 3

2

3−4x ≥ 2 3−4x ≤ −2 x ≤ 14

x ≥ 54

−1 ≤ x+ 5 ≤ 1 −6 ≤ x ≤ −4

x

y − y0 = m(x − x0)

y − 4 = −13(x − 2) 3 3(y − 4) = −(x − 2)

3y − 12 = −x+ 2 x+ 3y − 14 = 0

(x0, y0) = (1,−2) m = 2

y − y0 = m(x− x0)

y − (−2) = 2(x− 1)

y + 2 = 2x− 2

y − 2x+ 2 + 2 = 0

−2x+ y + 4 = 0

A = −2 B = 1 C = 4 2x− y − 4 = 0 A = 2 B = −1 C = −4

(x0, y0) = (0,−2) m = −3

y − y0 = m(x− x0)

y − (−2) = −3(x− 0)

y + 2 = −3x

3x+ y + 2 = 0 A = 3 B = 1 C = 2

(x0, y0) = (−3, 5) m = 12

y − y0 = m(x− x0)

y − 5 = 12(x− (−3))

y − 5 = 12(x+ 3)

2(y − 5) = x+ 3

2y − 10 = x+ 3




−x− 3 + 2y − 10 = 0

−x+ 2y − 13 = 0 A = −1 B = 2 C = −13

x− 2y + 13 = 0 A = 1 B = −2 C = 13

(x, y1) = (−2,−3)

(x2, y2) = (1, 4)

m = y2−y1x2−x1

= (4)−(−3)(1)−(−2) =

4+31+2

m = 73

y − y0 = m(x− x0)

(x0, y0) (x1, y1) (x2, y2)

y − 4 = 73(x− 1)

3(y − 4) = 7(x− 1)

3y − 12 = 7x− 7

3y − 7x− 12 + 7 = 0

−7x+ 3y − 5 = 0

7x− 3y + 5 = 0

(x1, y1) = (−1, 4) (x2, y2) = (2,−12)

m = y2−y1x2−x1

=(− 1

2)−(4)

(2)−(−1) =− 1

2−4

3

m =− 1

2− 8

23 =

− 923 = −9

2 · 13 = −3

2

y − y0 = m(x− x0)

y − (4) = −32(x− (−1))

y − 4 = −32(x+ 1)

2(y − 4) = −3(x+ 1)

2y − 8 = −3x− 3

3x+ 2y − 5 = 0 A = 3 B = 2 C = −5

(x1, y1) = (0, 4)

(x2, y2) = (3, 0)

m = y2−y1x2−x1

= (0)−(4)(3)−(0) = −4

3




y − y0 = m(x− x0)

y − 4 = −43(x− 0)

3(y − 4) = −4x

3y − 12 = −4x

4x+ 3y − 12 = 0

(x1, y1) = (1,−1) (x2, y2) = (4, 5)

m = y2−y1x2−x1

= (5)−(−1)(4)−(1) = 6

3 = 2

y − y0 = m(x− x0)

y − 5 = 2(x− 4)

y − 5 = 2x− 8

−2x+ y + 3 = 0 2x− y − 3 = 0

y = k

(3, 32) y = 32 2y − 3 = 0

y = k (0,−1) y = −1

y + 1 = 0

x = h (−1, 72) x = −1

x+ 1 = 0

x = h (2,−3) x = 2

x− 2 = 0

m = 3 y (0, 2) y = mx+ b

y = 3x+ 2 3x− y + 2 = 0

m = −1 y (0,−3)

y = mx+ b

y = −1x− 3 x+ y + 3 = 0

m = 12 y (0, 2)

y = mx+ b

y = 12x+ 2

x− 2y + 4

m = −13 y (0,−1)

y = mx+ b




y = −13x− 1

x+ 3y + 3 = 0

m = −2 x (1, 0)

y − y0 = m(x− x0)

y − 0 = −2(x− 1)

y = −2x+ 2

2x+ y − 2 = 0

m = 1 x (−2, 0)

y − y0 = m(x− x0)

y − 0 = 1(x− (−2))

y = x+ 2

−x+ y − 2 = 0 x− y + 2 = 0

m = −14 x (3, 0)

y − y0 = m(x− x0)

y − 0 = −14(x− 3)

y = −14x+ 3

4 x+ 4y − 3 = 0

m = 15 x (−1

2 , 0)

y − y0 = m(x− x0)

y − 0 = 15(x− (−1

2))

y = 15(x+ 1

2)

y = 15x+ 1

10

2x− 10y + 1 = 0

(2,−3) x+ 2y − 4 = 0

x+ 2y − 4 = 0

x+ 2y − 4 = 0

2y − 4 = −x

2y = −x+ 4

2y2 = −x+4

2

y = −12x+ 2 m = −1

2




m = −12 (2,−3)

y − y0 = m(x− x0)

y − (−3) = −12(x− 2)

y + 3 = −12(x− 2)

y + 3 = −12x+ 1

2(y + 3) = (−12x+ 1)2

2y + 6 = −x+ 2

x + 2y + 4 = 0 x + 2y − 4 = 0

(2,−3)

(1, 2) x− 3y − 6 = 0

x− 3y − 6 = 0

−3y − 6 = −x

−3y = −x+ 6

y = −x+6−3

y = 13x− 2 m = 1

3

y − y0 = m(x− x0) (1, 2)

y − 2 = 13(x− 1)

3y − 6 = x− 1

−x+ 3y − 6 + 1 = 0

−x+ 3y − 5 = 0

x− 3y + 5 = 0

(−1,−1) (0, 1) (3, 0)

m = y2−y1x2−x1

m = (0)−(1)(3)−(0) = −1

3

y − y0 = m(x− x0)

y − (−1) = −13(x− (−1))

y + 1 = −13(x+ 1)

−3(y + 1) = x+ 1

−3y − 3 = x+ 1




x+ 3y + 4 = 0

(2,−3) (0,−1) (2, 1)

m = y2−y1x2−x1

= (1)−(−1)(2)−(0)

m = 22 = 1

y − y0 = m(x− x0) (2,−3)

y − (−3) = 1(x− 2)

y + 3 = x− 2

−x+ y + 3 + 2 = 0

−x+ y + 5 = 0

x− y − 5 = 0

(1, 4) 2y − 5x+ 7 = 0

2y − 5x+ 7 = 0

2y = 5x− 7

2y2 = 5x−7

2

y = 52x− 7

2 m = 52

m⊥ = − 1m m1 ·m2 = −1)

m⊥ = −1( 52)= −2

5

(1, 4)

y − y0 = m⊥(x− x0)

y − 4 = −25(x− 1)

5(y − 4) = −2(x− 1)

5y − 20 = −2x+ 2

2x− 2 + 5y − 20 = 0

2x+ 5y − 22 = 0

(−1,−1) x− y + 3 = 0

x− y + 3 = 0

−y = −x− 3

y = x+ 3 m = 1 m1 ·m2 = −1




m⊥ = − 1m = −1

1 = −1

(−1,−1)

y − y0 = m⊥(x− x0)

y − (−1) = −1(x− (−1))

y + 1 = −(x+ 1)

y + 1 = −x− 1

y = −x− 1− 1

y = −x− 2

x+ y + 2 = 0

(5,−1) (−2, 1) (1,−2)

m = y2−y1x2−x1

= (−2)−(1)(1)−(−2) = −3

3

m = −1

m⊥ = − 1m = −1

−1 = 1

(5,−1)

y − y0 = m⊥(x− x0)

y − (−1) = 1(x− 5)

y + 1 = x− 5

y = x− 6

x− y − 6 = 0

(4,−1) (−2, 0)

(1, 1)

m = y2−y1x2−x1

= (1)−(0)(1)−(−2) =

13

m⊥ = −1( 13)= −3

(4,−1)

y − y0 = m⊥(x− x0)

y − (−1) = −3(x− (4))

y + 1 = −3(x− 4)

y + 1 = −3x+ 12

y = −3x+ 11

3x+ y − 11 = 0




(4, 2) y = 2

y − 2 = 0

(−1, 5)

y = 5 y − 5 = 0

x = −1

x+ 1 = 0

x = 3

x− 3 = 0

x = 1

x− 1 = 0

x = 4

x− 4 = 0

y = 3

y − 3 = 0

y = 5

y − 5 = 0

y = 30.5x y = mx x y

y x m

x

y

y = 30.5x x = 6

y = 30.5(6) = 183

y = 30.5x x = 3 2 212 ≈ 0.167

x = 3.167

y = 30.5(3.167) = 96.58

y = 30.5x x = 1 7

7 = 712 ≈ 0.583




x = 1.583

y = 30.5(1.583) = 48.29

y = 30.5x y = 173

173 = 30.5x

x = 5.67

y = 30.5x y = 75

75 = 30.5x

x = 2.459

y = 48

y = 30.5x

48 = 30.5x

x = 1.574

1 2.20

y = 2.20x y = mx x y

63

y = 2.20x

63 = 2.20x

x = 28.64

150 = 2.20x

x = 68.18

2.5

y = 2.20(2.5)

y = 5.5




140

y = 2.2(140)

y = 308

· y = m · x14

··

10mi0.25hrs =

mileshour

y = mx y x

x = y =

17 = m(217)

17217 = m

y = 17217 · x

2

2

2

2

1hectare1acre = 10000.00m2

4046.86m2 = 2.471

y = mx m = 1liter33.81ounces

y = m · x

y = 133.81 · x

x = 12




y = 133.8(12) = 0.355

y = mx

y = m · x

y = 1.609x

x = 261 y = 1.609(261) = 419.95

1 = 1.609

(55miles

hour

) · (1.609kilometers1mile

)= 88.5kilometers

hour(130kilometers

hour

) · ( 1mile1.609kilometers

)= 80.8miles

hour

x = ◦ y = ◦

(0, 32) (100, 212)

m = 212−32100−0 = 180

100 = 1810 = 9

5

y

y = 95x+ 32 F = 9

5C + 32

97.6◦F 99.6◦F

C 59(F − 32) = C 97.6◦F

59(97.6− 32) = 36.4◦C 97.6◦F 5

9(99.6− 32) = 37.6◦C

K C

K = mC + b

m

1◦C

K = C + b




(−273.15, 0) b = 273.15

K = C + 273.15

77.4K 90.2K

C = K − 273.15

77.4 − 273.15 = −195.75◦ 90.2 − 273.15 = −182.95◦

F = 95C + 32 ◦

F = 95(−195.75) + 32 = −320.35◦F

F = 95(−182.95) + 32 = −297.31◦F

θ1 θ2 [0, π)

θ2

x l1 l2

θ1 = θ2 +π2

l2

m2 =Δy

Δx= tan θ2

l1

m1 =Δy

Δx= − tan(π − θ1) = tan θ1

−θ2 =π2 − θ1

m1 = tan θ1 =1

cot θ1=

1

tan (π2 − θ1)=

1

tan (−θ2)=

1

− tan θ2= − cot θ2.

m1m2 = − cot θ2 · tan θ2 = −1




1m1

= 1tanθ1

= cotθ1 = tan(π2 − θ1) = tan(−θ2) = −tanθ2 = −m2

m1 ·m2 = −1

r2 = (x− x0)2 + (y − y0)

2

r = 3, (x0, y0) = (−1, 4)

9 = (x− (−1))2 + (y − 4)2

9 = (x+ 1)2 + (y − 4)2

r2 = (x− x0)2 + (y − y0)

2

r = 4, (x0, y0) = (2, 3)

42 = (x− 2)2 + (y − 3)2

16 = (x− 2)2 + (y − 3)2

r2 = (x− x0)2 + (y − y0)

2

r = 3 (x0, y0) = (2, 5)

32 = (x− 2)2 + (y − 5)2

9 = (x− 2)2 + (y − 5)2

y

x = 0 y

9 = (x− 2)2 + (y − 5)2

9 = (0− 2)2 + (y − 5)2

9 = 4 + (y − 5)2

5 = (y − 5)2

√5 = y − 5

5±√5 = y

x y = 0 x

9 = (x− 2)2 + (y − 5)2

9 = (x− 2)2 + (0− 5)2

9 = (x− 2)2 + 25




−16 = (x− 2)2

x

3 ≤ r < 6

r ≥ 6

(x− 2)2 + y2 = 16

(x− 2)2 + (y − 0)2 = 42

(x0, y0) = (2, 0) r = 4

(x+ 1)2 + (y − 3)2 = 9

(x− (−1))2 + (y − 3)2 = 32

(x0, y0) = (−1, 3) r = 3

0 = x2 + y2 − 4x+ 2y − 11

0 = (x2 − 4x+ 4) + (y2 + 2y + 1)− 11− 5

0 = (x− 2)2 + (y + 1)2 − 16

16 = (x− 2)2 + (y + 1)2

16 = (x− 2)2 + (y − (−1))2

r = 4 (x0, y0) = (2,−1)

x2 + y2 + 2x− 4y + 1 = 0

(x2 + 2x+ 1) + (y2 − 4y + 4) + 1− 1− 4 = 0

(x+ 1)2 + (y − 2)2 = 4

(x− (−1))2 + (y − 2)2 = 22

(x0, y0) = (−1, 2) r = 2




(75◦)(π

180◦) =

5

12πradians.

(17

12π)(

180◦

π) =

3060◦

12= 255◦

(−15◦)(π

180◦) =

−π

12

23

12π

34π

(3

4π)(

180◦

π) =

540◦

4= 135◦

sin

(−5

4π

)=

√2

2

cos

(5

6π

)=

−√3

2

tan(π3

)=

√3

sin

(3

4π

)=

1√2=

1√2·√2√2=

√2

2

cos

(−13

6π

)=

√3

2




tan

(4π

3

)=

−√3

−1=

√3

α ∈ [0, 2π)

sinα = −1

2

√3

π +π

3=

4

3π

2π − π

3− 5

3π

α =4

3π

5

3π

α ∈ [0, 2π)

tanα =√3

π3 π + π

3

α =π

3

4

3π

α ∈ [0, 2π)

cosα = −1

2

√2

π − π

4=

3

4π

π +π

4=

5

4π

α =3

4π

5

4π




secα = 2

π

32π − π

3=

5

3π

α =π

3

5

3π

sin2 θ + cos2 θ = 1

sin2 θ + cos2 θ

cos2 θ= 1

cos2 θcos2 θ




sin2 θ

cos2 θ+

cos2 θ

cos2 θ= 1

cos2 θ

(sin θ

cos θ= tan θ,

1

cos θ= sec θ

)

tan2 θ + 1 = sec2 θ

sin2 θ + cos2 θ = 1

sin2 θ + cos2 θ

sin2 θ= 1

sin2 θsin2 θ

sin2 θ

sin2 θ+

cos2 θ

sin2 θ= 1

sin2 θ

(cos θ

sin θ= cot θ,

1

sin θ= csc θ

)

1 + cot2 θ = csc2 θ

2 cos θ sin θ = sin θ, [0, 2π)

2 cos θ sin θ

sin θ= sin θ

sin θ sin θ �= 0 θ �= 0 θ �= π

2 cos θ = 1

cos θ = 12

θ =π

32π − π

3

θ = 0, π,π

3

5

3π




sec2 x =√3 tanx+ 1 sec2 x = 1 + tan2 x

1 + tan2 x =√3 · tanx+ 1

tan2 x =√3 tanx

tanx · tanxtanx

=√3·tanxtanx tanx �= 0 x �= 0

tanx =√3, x = π

3 π + π3 .

x = 0, π343π

434−2/3

aras = ar+s

43+(−2/3) = 47/3 = 16 3√4

3231/2

3−1/2= 35/2

3−1/2 = 35/2 · 31/2

= 35/2+1/2 = 36/2 = 33 = 27

5k52k−1

51−k= 5k · 52k−1 · 5k−1

5k+2k−1+k−1 = 54k−2




(24 · 2−3/2)2

(28/2 · 2−3/2)2

(28/2−3/2)2

(25/2)2 = 25 = 32

(65/2 · 62/3

61/3

)3

=(65/2+2/3

61/3

)3

=(615/6+4/6

62/6

)3=(619/6

62/6

)3=(6(19/6)−(2/6)

)3= (617/6)3 = 6(17/6)·3 = 617/2

(3−2k+3

34+k

)3

=(3−2k+3−4−k

)3(3−3k−1

)3= 3(−3k−1)(3) = 3−9k−3,

33(−3k−1) =(33)−3k−1

= 27−3k−1

log4 x = −2

x = 4−2 = 142

= 116

log1/3 x = −3

x =(13

)−3= 1

( 13)

3 = 1

( 127)

= 27




log10 x = −2

x = 10−2 = 1102

= 1100

log1/2 x = −4

x =

(1

2

)−4

=1−4

2−4=

1(124

) =1(116

) = 16

log1/4 x = 2

x =

(1

4

)2

=12

44=

1

16=

1

16

log5 x = 3 x = 53 = 125

log1/2 32 = x

32 =(12

)x= 1x

2x = 12x

2x = 132

2x = 2−5

x = −5

log1/3 81 = x

81 =(13

)x= 1x

3x

3x = 181 = 1

34= 3−4

x = −4




log10 0.001 = x

10x = 0.001 = 11000 = 1

103

10x = 10−3

x = −3

log4 64 = x

64 = 4x

43 = 4x

x = 3

log1/5 625 = x

625 =(15

)x54 = (5−1)x

54 = 5−x

−x = 4

x = −4

log10 10, 000 = x

10, 000 = 10x

104 = 10x

x = 4




− ln(13

)= ln

[(13

)−1]= ln 3

log4(x2 − 4) = log4[(x− 2)(x+ 2)] = log4(x− 2) + log4(x+ 2)

log2 4(3x−1) = (3x− 1) log2(2

2) = (3x− 1)(2 · log2 2) = (3x− 1)(2 · 1) = 6x− 2

− ln(15

)= ln

[(15

)−1]= ln 5

ln(x2−y2√

x

)= ln(x− y) + ln(x+ y)− ln(x1/2) = ln(x− y) + ln(x+ y)− 1

2 lnx

log3 32x+1 = (2x+ 1)(log3 3) = 2x+ 1

e3x−1 = 2

ln e(3x−1) = ln 2

(3x− 1)(ln e) = ln 2 ln e = 1

3x− 1 = ln 2

3x = 1 + ln 2

x = 1+ln 23

e−2x = 10

ln e(−2x) = ln 10

(−2x)(ln e) = ln 10

−2x = ln 10

x = ln 10−2 = −1

2 · ln 10x = ln 10−1/2 = ln

(1√10

)




ex2−1 = 10

ln e(x2−1) = ln 10

(x2 − 1)(ln e) = ln 10 (ln e = 1)

x2 − 1 = ln 10

x2 = 1 + ln 10

x = ±√1 + ln 10

3x = 81

3x = 34

x = 4

92x+1 = 27

(32)(2x+1) = 33

3(2)(2x+1) = 33

2(2x+ 1) = 3

4x+ 2 = 3

4x = 1

x = 14

105x = 1000

105x = 103

5x = 3

x = 35




ln(x− 3) = 5

eln(x−3) = e5, elnx = x

x− 3 = e5

x = e5 + 3

ln(x+ 2) + ln(x− 2) = 1

ln(x+ 2)(x− 2) = 1

eln[(x+2)(x−2)] = e

(x+ 2)(x− 2) = e

x2 − 4 = e

x2 = 4 + e

x =√4 + e

log3 x2 − log3 2x = 2

log3x2

2x= 2

x2

2x= 32

x

2= 9

x = 18

ln(2x− 3) = 0




eln(2x−3) = e0, elnx = x

2x− 3 = 1

2x = 4

x = 2

log2(1− x) = 3

2log2(1−x) = 23

1− x = 8

x = −7

lnx3 − lnx2 = 1

lnx3

x2= 1

lnx = 1

x = e

3− 2i− (−2 + 5i) = 3− 2i+ 2− 5i = 5− 7i

(7 + i)− 4 = 7− 4 + i = 3 + i

(4− 2i) + (9 + 4i) = 4 + 9− 2i+ 4i = 13 + 2i

(6− 4i) + (2 + 5i) = 6 + 2− 4i+ 5i = 8 + i

3(5 + 3i) = 15 + 9i

(2− 3i)(5 + 2i) = 2 · 5 + 2 · 2i− 3 · 5i− 6i2

= 10 + 4i− 15i− 6(−1) i2 = −1

= 10 + 4i− 15i+ 6

= 16− 11i




(6− i)(6 + i) = 36− i2 = 36− (−1) = 37

(−4− 3i)(4 + 2i) = −4 · 4− 4 · 2i− 3i · 4− 6i2 i2 = −1

= −16− 8i− 12i− 6(−1)

= −16− 20i+ 6

= −10− 20i

z = 3− 2i

z = 3 + 2i

z + u = (3− 2i) + (−4 + 3i)

= 3− 2i− 4 + 3i

= −1 + i

z + v = 3− 2i+ 3 + 5i = 3 + 2i+ 3− 5i = 6− 3i

v − w = 3 + 5i− (1− i) = 3− 5i− (1 + i)

= 3− 5i− 1− i = 2− 6i

vw = (3 + 5i)(1− i) = 8 + 2i = 8− 2i

uz = (−4 + 3i)(3− 2i) = −6 + 17i = −6− 17i

z = a+ bi

z + z = a+ bi+ a+ bi

= a+ bi+ a− bi = 2a




z − z = (a+ bi)− (a+ bi)

= a+ bi− (a− bi)

= a+ bi− a+ bi = 2bi

z = a+ bi z = a− bi

(z) = (a− bi) = a+ bi = z

z = (z)

2x2 − 3x+ 2 = 0 a = 2 b = −3 c = 2

x1,2 = −b±√b2−4ac2a

x1,2 =−(−3)±

√(−3)2−4(2)(2)

2(2)

x1,2 = 3±√9−164

x1,2 = 3±√−74 = 3±

√7i2

4

i2 = −1√ab =

√a · √b

x1,2 = 3±√7·√i2

4 = 3±√7i

4

x1 = 3+√7i

4 x2 =3−√

7i4

a = 3 b = −2 c = 1

x1,2 = −b±√b2−4ac2a

x1,2 =−(−2)±

√(−2)2−4(3)(1)

2(3)

x1,2 = 2±√4−126

x1,2 = 2±√−86

x1,2 = 2±√8i

6

x1,2 = 2±2√2·i

2·3




x1,2 = 1±√2·i

3

x1 = 1+√2i

3

x2 = 1−√2i

3

a = −1, b = 1 c = 2

x1,2 = −b±√b2−4ac2a

x1,2 =−(1)±

√(1)2−4(−1)(2)

2(−1)

x1,2 = −1±√1+8

−2

x1,2 = −1±√9

−2 = −1±3−2

x1 = −1+3−2 = 2

−2 = −1

x2 = −1−3−2 = −4

−2 = 2

−2x2 + x+ 3 = 0

a = −2, b = 1, c = 3

x1,2 = −b±√b2−4ac2a

x1,2 =−(1)±

√12−4(−2)(3)

2(−2)

x1,2 = −1±√1+24

−4

x1,2 = −1±√25

−4 = −1±5−4

x1 = −1+5−4 = 4

−4 = −1

x2 = −1−5−4 = −6

−4 = 32

4x2 − 3x+ 1 = 0

a = 4, b = −3, c = 1

x1,2 = −b±√b2−4ac2a




x1,2 =−(−3)±

√(−3)2−4(4)(1)

2(4)

x1,2 = 3±√9−168

x1,2 = 3±√−78 = 3±

√7i2

8 i2 = −1

x1,2 = 3±√7i

8

x1 = 3+√7i

8

x2 = 3−√7i

8

−2x2 + 4x− 3 = 0

a = −2, b = 4, c = −3

x1,2 = −b±√b2−4ac2a

x1,2 =−(4)±

√42−4(−2)(−3)

2(−2)

x1,2 = −4±√16−24

−2·2

x1,2 = −4±√−8−4

x1,2 = −4±2√2i

−4

x1,2 = 2∓√2i

2

x1,2 = 1±√22 i

x1 = 1 +√22 i,

x2 = 1−√22 i

3x2 − 4x− 7 = 0 b2 − 4ac

a = 3, b = −4, c = −7

(−4)2 − 4(3)(−7) = 16 + 84 = 100 > 0

−1 73

3x2 − 4x+ 7 = 0 b2 − 4ac

a = 3, b = −4, c = 7




(−4)2 − 4(3)(7) = 16− 12(7) = 16− 84 = −68 < 0

2+√17i

3

2−√17i

3

−x2 + 2x− 1 = 0 b2 − 4ac

a = −1, b = 2, c = −1

(2)2 − 4(−1)(−1)

4− 4 = 0

1

4x2 − x+ 1 = 0

a = 4, b = −1, c = 1

b2 − 4ac

(−1)2 − 4(4)(1) = 1− 16 = −15 < 0

1+√15i

8

1−√15i

8

3x2 − 5x+ 6 = 0

a = 3, b = −5, c = 6

b2 − 4ac

(−5)2 − 4(3)(6)

25− 72 = −47 < 0

5+√47i

6

5−√47i

6

−x2 + 7x− 2 = 0

a = −1, b = 7, c = −2




b2 − 4ac

(7)2 − 4(−1)(−2)

49− 8 = 41 > 0

7+√41

27−√

412

z = a+ bi z = a− bi

(z) = (a− bi) = a+ bi = z.

z = (z)

z + w = z + w

z = a+ bi w = c+ di

z + w = (a+ bi) + (c+ di) = (a+ c) + (b+ d)i

= (a+ c)− (b+ d)i.

z = a+ bi, w = c+ di

z = a− bi, w = c− di

z + w = a− bi+ c− di = (a+ c)− (b+ d)i.

z + w = (a+ c)− (b+ d)i = z + w

z = a+ bi

w = c+ di

zw = z · w

z · w = (a+ bi)(c+ di) = (ac− bd) + (ad+ bc)i

= (ac− bd)− (ad+ bc)i




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Copyright © 2011 Pearson Education, Inc. … › sample › Solution-Manual-for-Calculus...y = 1 33.8(12) = 0 .355 y = mx y = m · x y =1.609x x = 261 y =1.609(261) = 419 .95 1 =1.609