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Solutions to OpenStax Calculus Volume 1

Chapter 1: Functions and Graphs

Chapter 2: Limits



				Problem 35	Problem 36	Problem 37	Problem 38	Problem 39	Problem 40
Problem 41	Problem 42	Problem 43	Problem 44	Problem 45	Problem 46	Problem 47	Problem 48	Problem 49	Problem 50
Problem 51	Problem 52	Problem 53	Problem 54	Problem 55	Problem 56	Problem 57	Problem 58	Problem 59	Problem 60
Problem 61	Problem 62	Problem 63	Problem 64	Problem 65	Problem 66	Problem 67	Problem 68	Problem 69	Problem 70
Problem 71	Problem 72	Problem 73	Problem 74	Problem 75	Problem 76	Problem 77	Problem 78	Problem 79	Problem 80
Problem 81	Problem 82	Problem 83	Problem 84	Problem 85	Problem 86	Problem 87	Problem 88	Problem 89	Problem 90
Problem 91	Problem 92	Problem 93	Problem 94	Problem 95	Problem 96	Problem 97	Problem 98	Problem 99	Problem 100
Problem 101	Problem 102	Problem 103	Problem 104	Problem 105	Problem 106	Problem 107	Problem 108	Problem 109	Problem 110
Problem 111	Problem 112	Problem 113	Problem 114	Problem 115	Problem 116	Problem 117	Problem 118	Problem 119	Problem 120
Problem 121	Problem 122	Problem 123	Problem 124	Problem 125	Problem 126	Problem 127	Problem 128	Problem 129	Problem 130
Problem 131	Problem 132	Problem 133	Problem 134	Problem 135	Problem 136	Problem 137	Problem 138	Problem 139	Problem 140
Problem 141	Problem 142	Problem 143	Problem 144	Problem 145	Problem 146	Problem 147	Problem 148	Problem 149	Problem 150
Problem 151	Problem 152	Problem 153	Problem 154	Problem 155	Problem 156	Problem 157	Problem 158	Problem 159	Problem 160
Problem 161	Problem 162	Problem 163	Problem 164	Problem 165	Problem 166	Problem 167	Problem 168	Problem 169	Problem 170
Problem 171	Problem 172	Problem 173	Problem 174	Problem 175	Problem 176	Problem 177	Problem 178	Problem 179	Problem 180
Problem 181	Problem 182	Problem 183	Problem 184	Problem 185	Problem 186	Problem 187	Problem 188	Problem 189	Problem 190
Problem 191	Problem 192	Problem 193	Problem 194	Problem 195	Problem 196	Problem 197	Problem 198	Problem 199	Problem 200
Problem 201	Problem 202	Problem 203	Problem 204	Problem 205	Problem 206	Problem 207	Problem 208	Problem 209	Problem 210
Problem 211	Problem 212	Problem 213	Problem 214	Problem 215	Problem 216	Problem 217	Problem 218	Problem 219	Problem 220
Problem 221	Problem 222	Problem 223	Problem 224	Problem 225	Problem 226	Problem 227	Problem 228	Problem 229	Problem 230
Problem 231	Problem 232

f ( 1 ) = 3 f ( 1 ) = 3 and f ( a + h ) = a 2 + 2 a h + h 2 − 3 a − 3 h + 5 f ( a + h ) = a 2 + 2 a h + h 2 − 3 a − 3 h + 5

Domain = { x | x ≤ 2 } , { x | x ≤ 2 } , range = { y | y ≥ 5 } { y | y ≥ 5 }

x = 0 , 2 , 3 x = 0 , 2 , 3

( f g ) ( x ) = x 2 + 3 2 x − 5 . ( f g ) ( x ) = x 2 + 3 2 x − 5 . The domain is { x | x ≠ 5 2 } . { x | x ≠ 5 2 } .

( f ∘ g ) ( x ) = 2 − 5 x . ( f ∘ g ) ( x ) = 2 − 5 x .

( g ∘ f ) ( x ) = 0.63 x ( g ∘ f ) ( x ) = 0.63 x

f ( x ) f ( x ) is odd.

Domain = ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , range = { y | y ≥ −4 } . { y | y ≥ −4 } .

m = 1 / 2 . m = 1 / 2 . The point-slope form is

y − 4 = 1 2 ( x − 1 ) . y − 4 = 1 2 ( x − 1 ) .

The slope-intercept form is

y = 1 2 x + 7 2 . y = 1 2 x + 7 2 .

The zeros are x = 1 ± 3 / 3 . x = 1 ± 3 / 3 . The parabola opens upward.

The domain is the set of real numbers x x such that x ≠ 1 / 2 . x ≠ 1 / 2 . The range is the set { y | y ≠ 5 / 2 } . { y | y ≠ 5 / 2 } .

The domain of f f is (−∞, ∞). (−∞, ∞). The domain of g g is { x | x ≥ 1 / 5 } . { x | x ≥ 1 / 5 } .

C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3 C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3

Shift the graph y = x 2 y = x 2 to the left 1 unit, reflect about the x x -axis, then shift down 4 units.

7 π / 6 ; 7 π / 6 ; 330°

cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2 cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2

θ = 3 π 2 + 2 n π , π 6 + 2 n π , 5 π 6 + 2 n π θ = 3 π 2 + 2 n π , π 6 + 2 n π , 5 π 6 + 2 n π for n = 0 , ± 1 , ± 2 ,… n = 0 , ± 1 , ± 2 ,…

To graph f ( x ) = 3 sin ( 4 x ) − 5 , f ( x ) = 3 sin ( 4 x ) − 5 , the graph of y = sin ( x ) y = sin ( x ) needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function f f will have a period of π / 2 π / 2 and an amplitude of 3.

f −1 ( x ) = 2 x x − 3 . f −1 ( x ) = 2 x x − 3 . The domain of f −1 f −1 is { x | x ≠ 3 } . { x | x ≠ 3 } . The range of f −1 f −1 is { y | y ≠ 2 } . { y | y ≠ 2 } .

The domain of f −1 f −1 is ( 0 , ∞ ) . ( 0 , ∞ ) . The range of f −1 f −1 is ( − ∞ , 0 ) . ( − ∞ , 0 ) . The inverse function is given by the formula f −1 ( x ) = −1 / x . f −1 ( x ) = −1 / x .

f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 . f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 .

x / ( 2 y 3 ) x / ( 2 y 3 )

A ( t ) = 750 e 0.04 t . A ( t ) = 750 e 0.04 t . After 30 30 years, there will be approximately $ 2 , 490.09 . $ 2 , 490.09 .

x = ln 3 2 x = ln 3 2

x = 1 e x = 1 e

1.29248 1.29248

The magnitude 8.4 8.4 earthquake is roughly 10 10 times as severe as the magnitude 7.4 7.4 earthquake.

( x 2 + x −2 ) / 2 ( x 2 + x −2 ) / 2

1 2 ln ( 3 ) ≈ 0.5493 . 1 2 ln ( 3 ) ≈ 0.5493 .

Section 1.1 Exercises

a. Domain = { −3 , −2 , −1 , 0 , 1 , 2 , 3 } , { −3 , −2 , −1 , 0 , 1 , 2 , 3 } , range = { 0 , 1 , 4 , 9 } { 0 , 1 , 4 , 9 } b. Yes, a function

a. Domain = { 0 , 1 , 2 , 3 } , { 0 , 1 , 2 , 3 } , range = { −3 , −2 , −1 , 0 , 1 , 2 , 3 } { −3 , −2 , −1 , 0 , 1 , 2 , 3 } b. No, not a function

a. Domain = { 3 , 5 , 8 , 10 , 15 , 21 , 33 } , { 3 , 5 , 8 , 10 , 15 , 21 , 33 } , range = { 0 , 1 , 2 , 3 } { 0 , 1 , 2 , 3 } b. Yes, a function

a. −2 −2 b. 3 c. 13 d. −5 x − 2 −5 x − 2 e. 5 a − 2 5 a − 2 f. 5 a + 5 h − 2 5 a + 5 h − 2

a. Undefined b. 2 c. 2 3 2 3 d. − 2 x − 2 x e 2 a 2 a f. 2 a + h 2 a + h

a. 5 5 b. 11 11 c. 23 23 d. −6 x + 5 −6 x + 5 e. 6 a + 5 6 a + 5 f. 6 a + 6 h + 5 6 a + 6 h + 5

a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

x ≥ 1 8 ; y ≥ 0 ; x = 1 8 ; x ≥ 1 8 ; y ≥ 0 ; x = 1 8 ; no y -intercept

x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2 x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2

x ≠ 4 ; y ≠ 0 ; x ≠ 4 ; y ≠ 0 ; no x -intercept; y = − 3 4 y = − 3 4

x > 5 ; y > 0 ; x > 5 ; y > 0 ; no intercepts

Function; a. Domain: all real numbers, range: y ≥ 0 y ≥ 0 b. x = ± 1 x = ± 1 c. y = 1 y = 1 d. −1 < x < 0 −1 < x < 0 and 1 < x < ∞ 1 < x < ∞ e. − ∞ < x < − 1 − ∞ < x < − 1 and 0 < x < 1 0 < x < 1 f. Not constant g. y -axis h. Even

Function; a. Domain: all real numbers, range: −1.5 ≤ y ≤ 1.5 −1.5 ≤ y ≤ 1.5 b. x = 0 x = 0 c. y = 0 y = 0 d. all real numbers all real numbers e. None f. Not constant g. Origin h. Odd

Function; a. Domain: − ∞ < x < ∞ , − ∞ < x < ∞ , range: −2 ≤ y ≤ 2 −2 ≤ y ≤ 2 b. x = 0 x = 0 c. y = 0 y = 0 d. −2 < x < 2 −2 < x < 2 e. Not decreasing f. − ∞ < x < − 2 − ∞ < x < − 2 and 2 < x < ∞ 2 < x < ∞ g. Origin h. Odd

Function; a. Domain: −4 ≤ x ≤ 4 , −4 ≤ x ≤ 4 , range: −4 ≤ y ≤ 4 −4 ≤ y ≤ 4 b. x = 1.2 x = 1.2 c. y = 4 y = 4 d. Not increasing e. 0 < x < 4 0 < x < 4 f. −4 < x < 0 −4 < x < 0 g. No Symmetry h. Neither

a. 5 x 2 + x − 8 ; 5 x 2 + x − 8 ; all real numbers b. −5 x 2 + x − 8 ; −5 x 2 + x − 8 ; all real numbers c. 5 x 3 − 40 x 2 ; 5 x 3 − 40 x 2 ; all real numbers d. x − 8 5 x 2 ; x ≠ 0 x − 8 5 x 2 ; x ≠ 0

a. −2 x + 6 ; −2 x + 6 ; all real numbers b. −2 x 2 + 2 x + 12 ; −2 x 2 + 2 x + 12 ; all real numbers c. − x 4 + 2 x 3 + 12 x 2 − 18 x − 27 ; − x 4 + 2 x 3 + 12 x 2 − 18 x − 27 ; all real numbers d. − x + 3 x + 1 ; x ≠ − 1 , 3 − x + 3 x + 1 ; x ≠ − 1 , 3

a. 6 + 2 x ; x ≠ 0 6 + 2 x ; x ≠ 0 b. 6; x ≠ 0 x ≠ 0 c. 6 x + 1 x 2 ; x ≠ 0 6 x + 1 x 2 ; x ≠ 0 d. 6 x + 1 ; x ≠ 0 6 x + 1 ; x ≠ 0

a. 4 x + 3 ; 4 x + 3 ; all real numbers b. 4 x + 15 ; 4 x + 15 ; all real numbers

a. x 4 − 6 x 2 + 16 ; x 4 − 6 x 2 + 16 ; all real numbers b. x 4 + 14 x 2 + 46 ; x 4 + 14 x 2 + 46 ; all real numbers

a. 3 x 4 + x ; x ≠ 0 , −4 3 x 4 + x ; x ≠ 0 , −4 b. 4 x + 2 3 ; x ≠ − 1 2 4 x + 2 3 ; x ≠ − 1 2

a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.

a. V ( s ) = s 3 V ( s ) = s 3 b. V ( 11.8 ) ≈ 1643 ; V ( 11.8 ) ≈ 1643 ; a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

a. N ( x ) = 15 x N ( x ) = 15 x b. i. N ( 20 ) = 15 ( 20 ) = 300 ; N ( 20 ) = 15 ( 20 ) = 300 ; therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. N ( 15 ) = 225 ; N ( 15 ) = 225 ; therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: 0 ≤ x ≤ 20 ; 0 ≤ x ≤ 20 ; range: [ 0 , 300 ] [ 0 , 300 ] d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

a. A ( t ) = A ( r ( t ) ) = π · ( 6 − 5 t 2 + 1 ) 2 A ( t ) = A ( r ( t ) ) = π · ( 6 − 5 t 2 + 1 ) 2 b. Exact: 121 π 4 ; 121 π 4 ; approximately 95 cm 2 c. C ( t ) = C ( r ( t ) ) = 2 π ( 6 − 5 t 2 + 1 ) C ( t ) = C ( r ( t ) ) = 2 π ( 6 − 5 t 2 + 1 ) d. Exact: 11 π ; 11 π ; approximately 35 cm

a. S ( x ) = 8.5 x + 750 S ( x ) = 8.5 x + 750 b. $962.50, $1090, $1217.50 c. 77 skateboards

Section 1.2 Exercises

a. −1 b. Decreasing

a. 3/4 b. Increasing

a. 4/3 b. Increasing

a. 0 b. Horizontal

y = −6 x + 9 y = −6 x + 9

y = 1 3 x + 4 y = 1 3 x + 4

y = 1 2 x y = 1 2 x

y = 3 5 x − 3 y = 3 5 x − 3

a. ( m = 2 , b = −3 ) ( m = 2 , b = −3 ) b.

a. ( m = −6 , b = 0 ) ( m = −6 , b = 0 ) b.

a. ( m = 0 , b = −6 ) ( m = 0 , b = −6 ) b.

a. ( m = − 2 3 , b = 2 ) ( m = − 2 3 , b = 2 ) b.

a. 2 b. 5 2 , −1 ; 5 2 , −1 ; c. −5 d. Both ends rise e. Neither

a. 2 b. ± 2 ± 2 c. −1 d. Both ends rise e. Even

a. 3 b. 0, ± 3 ± 3 c. 0 d. Left end rises, right end falls e. Odd

a. 13 , −3 , 5 13 , −3 , 5 b.

a. −3 2 , −1 2 , 4 −3 2 , −1 2 , 4 b.

True; n = 3 n = 3

False; f ( x ) = x b , f ( x ) = x b , where b b is a real-valued constant, is a power function

a. V ( t ) = −2733 t + 20500 V ( t ) = −2733 t + 20500 b. ( 0 , 20 , 500 ) ( 0 , 20 , 500 ) means that the initial purchase price of the equipment is $20,500; ( 7.5 , 0 ) ( 7.5 , 0 ) means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years

a. C = 0.75 x + 125 C = 0.75 x + 125 b. $245 c. 167 cupcakes

a. V ( t ) = −1500 t + 26,000 V ( t ) = −1500 t + 26,000 b. In 4 years, the value of the car is $20,000.

96% of the total capacity

Section 1.3 Exercises

4 π 3 rad 4 π 3 rad

− π 3 − π 3

11 π 6 rad 11 π 6 rad

210 ° 210 °

−540 ° −540 °

− 2 2 − 2 2

3 − 1 2 2 3 − 1 2 2

a. b = 5.7 b = 5.7 b. sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4 sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4

a. c = 151.7 c = 151.7 b. sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471 sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471

a. c = 85 c = 85 b. sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84 sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84

a. y = 24 25 y = 24 25 b. sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24 sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24

a. x = − 2 3 x = − 2 3 b. sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7 sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7

sec 2 x sec 2 x

sin 2 x sin 2 x

sec 2 θ sec 2 θ

1 sin t ( = csc t ) 1 sin t ( = csc t )

{ π 6 , 5 π 6 } { π 6 , 5 π 6 }

{ π 4 , 3 π 4 , 5 π 4 , 7 π 4 } { π 4 , 3 π 4 , 5 π 4 , 7 π 4 }

{ 2 π 3 , 5 π 3 } { 2 π 3 , 5 π 3 }

{ 0 , π , π 3 , 5 π 3 } { 0 , π , π 3 , 5 π 3 }

y = 4 sin ( π 4 x ) y = 4 sin ( π 4 x )

y = cos ( 2 π x ) y = cos ( 2 π x )

a. 1 b. 2 π 2 π c. π 4 π 4 units to the right

a. 1 2 1 2 b. 8 π 8 π c. No phase shift

a. 3 b. 2 2 c. 2 π 2 π units to the left

Approximately 42 in.

a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min

≈ 30.9 in 2 ≈ 30.9 in 2

a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

a. Amplitude = 10 ; period = 24 10 ; period = 24 b. 47.4 ° F 47.4 ° F c. 14 hours later, or 2 p.m. d.

Section 1.4 Exercises

Not one-to-one

a. f −1 ( x ) = x + 4 f −1 ( x ) = x + 4 b. Domain : x ≥ −4 , range : y ≥ 0 : x ≥ −4 , range : y ≥ 0

a. f −1 ( x ) = x − 1 3 f −1 ( x ) = x − 1 3 b. Domain: all real numbers, range: all real numbers

a. f −1 ( x ) = x 2 + 1 , f −1 ( x ) = x 2 + 1 , b. Domain: x ≥ 0 , x ≥ 0 , range: y ≥ 1 y ≥ 1

These are inverses.

These are not inverses.

− π 6 − π 6

a. x = f −1 ( V ) = 0.04 − V 500 x = f −1 ( V ) = 0.04 − V 500 b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V . c. 0.1 cm; 0.14 cm; 0.17 cm

a. $31,250, $66,667, $107,143 b. ( p = 85 C C + 75 ) ( p = 85 C C + 75 ) c. 34 ppb

a. ~ 92 ° ~ 92 ° b. ~ 42 ° ~ 42 ° c. ~ 27 ° ~ 27 °

x ≈ 6.69 , 8.51 ; x ≈ 6.69 , 8.51 ; so, the temperature occurs on June 21 and August 15

~ 1.5 sec ~ 1.5 sec

tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; the expression does not equal 2.1 since 2.1 > 1.57 = π 2 2.1 > 1.57 = π 2 —in other words, it is not in the restricted domain of tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , since 2.1 is in the restricted domain of cos x . cos x .

Section 1.5 Exercises

a. 125 b. 2.24 c. 9.74

a. 0.01 b. 10,000 c. 46.42

Domain: all real numbers, range: ( 2 , ∞ ) , y = 2 ( 2 , ∞ ) , y = 2

Domain: all real numbers, range: ( 0 , ∞ ) , y = 0 ( 0 , ∞ ) , y = 0

Domain: all real numbers, range: ( − ∞ , 1 ) , y = 1 ( − ∞ , 1 ) , y = 1

Domain: all real numbers, range: ( −1 , ∞ ) , y = −1 ( −1 , ∞ ) , y = −1

8 1 / 3 = 2 8 1 / 3 = 2

5 2 = 25 5 2 = 25

e −3 = 1 e 3 e −3 = 1 e 3

e 0 = 1 e 0 = 1

log 4 ( 1 16 ) = −2 log 4 ( 1 16 ) = −2

log 9 1 = 0 log 9 1 = 0

log 64 4 = 1 3 log 64 4 = 1 3

log 9 150 = y log 9 150 = y

log 4 0.125 = − 3 2 log 4 0.125 = − 3 2

Domain: ( 1 , ∞ ) , ( 1 , ∞ ) , range: ( − ∞ , ∞ ) , x = 1 ( − ∞ , ∞ ) , x = 1

Domain: ( 0 , ∞ ) , ( 0 , ∞ ) , range: ( − ∞ , ∞ ) , x = 0 ( − ∞ , ∞ ) , x = 0

Domain: ( −1 , ∞ ) , ( −1 , ∞ ) , range: ( − ∞ , ∞ ) , x = −1 ( − ∞ , ∞ ) , x = −1

2 + 3 log 3 a − log 3 b 2 + 3 log 3 a − log 3 b

3 2 + 1 2 log 5 x + 3 2 log 5 y 3 2 + 1 2 log 5 x + 3 2 log 5 y

− 3 2 + ln 6 − 3 2 + ln 6

ln 15 3 ln 15 3

log 7.21 log 7.21

2 3 + log 11 3 log 7 2 3 + log 11 3 log 7

x = 1 25 x = 1 25

x = 4 x = 4

x = 3 x = 3

1 + 5 1 + 5

( log 82 log 7 ≈ 2.2646 ) ( log 82 log 7 ≈ 2.2646 )

( log 211 log 0.5 ≈ − 7.7211 ) ( log 211 log 0.5 ≈ − 7.7211 )

( log 0.452 log 0.2 ≈ 0.4934 ) ( log 0.452 log 0.2 ≈ 0.4934 )

~ 17 , 491 ~ 17 , 491

Approximately $131,653 is accumulated in 5 years.

i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

a. ~ 333 ~ 333 million b. 94 years from 2013, or in 2107

a. k ≈ 0.0578 k ≈ 0.0578 b. ≈ 92 ≈ 92 hours

The San Francisco earthquake was 10 3.4 or ≈ 2512 10 3.4 or ≈ 2512 times more intense than the Japanese earthquake.

Review Exercises

Domain: x > 5 , x > 5 , range: all real numbers

Domain: x > 2 x > 2 and x < − 4 , x < − 4 , range: all real numbers

Degree of 3, y y -intercept: 0, zeros: 0, 3 − 1 , −1 − 3 3 − 1 , −1 − 3

cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1 cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1

0 , ± 2 π 0 , ± 2 π

One-to-one; yes, the function has an inverse; inverse: f −1 ( x ) = 1 y f −1 ( x ) = 1 y

x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7 x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7

a. C ( x ) = 300 + 7 x C ( x ) = 300 + 7 x b. 100 shirts

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

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Authors: Gilbert Strang, Edwin “Jed” Herman
Publisher/website: OpenStax
Book title: Calculus Volume 1
Publication date: Mar 30, 2016
Location: Houston, Texas
Book URL: https://openstax.org/books/calculus-volume-1/pages/1-introduction
Section URL: https://openstax.org/books/calculus-volume-1/pages/chapter-1

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Math Expressions Common Core Grade 1 Answer Key | Houghton Mifflin Math Expressions Grade 1 Answer Key

The Solution Key of Houghton Mifflin Math Expressions Grade 1 will help kids to build a deep understanding of the math concepts. Math Expressions Common Core 1st Grade Homework and Remembering Answer Key cover every unit & lesson-wise integrates mathematical practices and processes.

Students of 1st Standard learn how to view deeper and select their own path to the answers-skills which will take them far beyond the math classroom. Here, you will find the Unit-wise & Lesson-wise Grade 1 Math Expressions Common Core Answers for Volume 1 and Volume 2 PDF links for easy access and download.

Math Expressions Common Core Grade 1 Volume 1 Answer Key | Math Expressions Grade 1 Homework and Remembering Volume 1 Answer Key

By practicing with the Houghton Mifflin Math Expressions Grade 1 Homework and Remembering Volume 1 Answer Key Unit-wise aspirants will hold the ability to progress at their own pace. So, go ahead and start solving the questions from unit-wise concept using the Math Expressions common core 1st Grade volume 1 Solution key.

Math Expressions Grade 1 Answer Key Unit 1 Partners and Number Patterns Through 10

Math Expressions Grade 1 Unit 1 Lesson 1 Answer Key Discuss Numbers 1 – 10
Math Expressions Grade 1 Unit 1 Lesson 2 Answer Key Visualize Numbers as a 5-Group and Ones
Math Expressions Grade 1 Unit 1 Lesson 3 Answer Key Partners of 2 Through 5
Math Expressions Grade 1 Unit 1 Lesson 4 Answer Key Partners of 6
Math Expressions Grade 1 Unit 1 Lesson 5 Answer Key Partners of 7
Math Expressions Grade 1 Unit 1 Lesson 6 Answer Key Partners of 8
Math Expressions Grade 1 Unit 1 Lesson 7 Answer Key Partners of 9
Math Expressions Grade 1 Unit 1 Lesson 8 Answer Key Partners of 10
Math Expressions Grade 1 Unit 1 Lesson 9 Answer Key Focus on Mathematical Practices

Math Expressions Grade 1 Homework and Remembering Answer Key Unit 2 Addition and Subtraction Strategies

Math Expressions Grade 1 Unit 2 Lesson 1 Answer Key Represent Addition
Math Expressions Grade 1 Unit 2 Lesson 2 Answer Key Addition with Circle Drawings
Math Expressions Grade 1 Unit 2 Lesson 3 Answer Key Addition Equations
Math Expressions Grade 1 Unit 2 Lesson 4 Answer Key Addition Equations and Stories
Math Expressions Grade 1 Unit 2 Lesson 5 Answer Key Explore Solution Methods
Math Expressions Grade 1 Unit 2 Lesson 6 Answer Key Addition Strategies: Counting On
Math Expressions Grade 1 Unit 2 Lesson 7 Answer Key Count On from the Greater Number
Math Expressions Grade 1 Unit 2 Lesson 8 Answer Key Addition Game: Unknown Totals
Math Expressions Grade 1 Unit 2 Lesson 9 Answer Key Practice Counting On
Math Expressions Grade 1 Unit 2 Lesson 10 Answer Key Represent Subtraction
Math Expressions Grade 1 Unit 2 Lesson 11 Answer Key Subtraction with Drawings and Equations
Math Expressions Grade 1 Unit 2 Lesson 12 Answer Key Practice with Subtraction
Math Expressions Grade 1 Unit 2 Lesson 13 Answer Key Generate Subtraction Problems
Math Expressions Grade 1 Unit 2 Lesson 14 Answer Key Relate Addition and Subtraction
Math Expressions Grade 1 Unit 2 Lesson 15 Answer Key Mixed Practice with Equations
Math Expressions Grade 1 Unit 2 Lesson 16 Answer Key Focus on Mathematical Practices

Math Expressions Common Core Grade 1 Volume 1 Answer Key Unit 3 Unknown Numbers in Addition and Subtraction

Math Expressions Grade 1 Unit 3 Lesson 1 Answer Key Explore Unknowns
Math Expressions Grade 1 Unit 3 Lesson 2 Answer Key Problems with Unknown Partners
Math Expressions Grade 1 Unit 3 Lesson 3 Answer Key Solve Equations with Unknown Partners
Math Expressions Grade 1 Unit 3 Lesson 4 Answer Key Addition Game: Unknown Partners
Math Expressions Grade 1 Unit 3 Lesson 5 Answer Key Practice with Unknown Partners
Math Expressions Grade 1 Unit 3 Lesson 6 Answer Key Subtraction Strategies
Math Expressions Grade 1 Unit 3 Lesson 7 Answer Key Subtraction Stories and Games
Math Expressions Grade 1 Unit 3 Lesson 8 Answer Key Practice with Subtraction Stories
Math Expressions Grade 1 Unit 3 Lesson 9 Answer Key Relate Addition and Subtraction Situations
Math Expressions Grade 1 Unit 3 Lesson 10 Answer Key Solve Mixed Problems
Math Expressions Grade 1 Unit 3 Lesson 11 Answer Key Practice with Mixed Problems
Math Expressions Grade 1 Unit 3 Lesson 12 Answer Key Focus on Mathematical Practices

Math Expressions Grade 1 Homework and Remembering Volume 1 Answer Key Unit 4 Place Value Concepts

Math Expressions Grade 1 Unit 4 Lesson 1 Answer Key Introduction to Tens Groupings
Math Expressions Grade 1 Unit 4 Lesson 2 Answer Key Explore Teen Numbers
Math Expressions Grade 1 Unit 4 Lesson 3 Answer Key Represent and Compare Teen Numbers
Math Expressions Grade 1 Unit 4 Lesson 4 Answer Key Visualize Teen Addition
Math Expressions Grade 1 Unit 4 Lesson 5 Answer Key Teen Addition Strategies
Math Expressions Grade 1 Unit 4 Lesson 6 Answer Key Investigate Doubles
Math Expressions Grade 1 Unit 4 Lesson 7 Answer Key Understand Tens and Ones
Math Expressions Grade 1 Unit 4 Lesson 8 Answer Key Integrate Tens and Ones
Math Expressions Grade 1 Unit 4 Lesson 9 Answer Key Practice Grouping Ones into Tens
Math Expressions Grade 1 Unit 4 Lesson 10 Answer Key Add with Groups of Ten
Math Expressions Grade 1 Unit 4 Lesson 11 Answer Key Practice with Tens and Ones
Math Expressions Grade 1 Unit 4 Lesson 12 Answer Key Use Place Value to Compare Numbers
Math Expressions Grade 1 Unit 4 Lesson 13 Answer Key Add Tens or Ones
Math Expressions Grade 1 Unit 4 Lesson 14 Answer Key Mixed Addition with Tens and Ones
Math Expressions Grade 1 Unit 4 Lesson 15 Answer Key Counting On Strategy: 2-Digit Numbers
Math Expressions Grade 1 Unit 4 Lesson 16 Answer Key Practice with 2-Digit Numbers
Math Expressions Grade 1 Unit 4 Lesson 17 Answer Key 2-Digit Addition Games
Math Expressions Grade 1 Unit 4 Lesson 18 Answer Key Focus on Mathematical Practices

Math Expressions Grade 1 Homework and Remembering Volume 2 Answer Key | Math Expressions Common Core Grade 1 Volume 2 Answer Key

Experts designed Math Expressions 1st Grade Homework and Remembering Volume 2 Solution Key PDF are easy-to-understand, fun, and engaging for students. Simply tap the links available here and download the respective unit of Houghton Mifflin Math Expressions Grade 1 Homework and Remembering Volume 2 Answers in PDF to score high.

Math Expressions Common Core Grade 1 Answer Key Unit 5 Place Value Situations

Math Expressions Grade 1 Unit 5 Lesson 1 Answer Key Unknown Partners with Teen Totals
Math Expressions Grade 1 Unit 5 Lesson 2 Answer Key Subtraction with Teen Numbers
Math Expressions Grade 1 Unit 5 Lesson 3 Answer Key Mixed Practice with Teen Problems
Math Expressions Grade 1 Unit 5 Lesson 4 Answer Key Small Group Practice with Teen Problems
Math Expressions Grade 1 Unit 5 Lesson 5 Answer Key Teen Problems with Various Unknowns
Math Expressions Grade 1 Unit 5 Lesson 6 Answer Key Problems with Three Addends
Math Expressions Grade 1 Unit 5 Lesson 7 Answer Key Count with Groups of 10
Math Expressions Grade 1 Unit 5 Lesson 8 Answer Key Numbers Through 120
Math Expressions Grade 1 Unit 5 Lesson 9 Answer Key Add and Subtract Tens
Math Expressions Grade 1 Unit 5 Lesson 10 Answer Key Add and Subtract Multiples of 10
Math Expressions Grade 1 Unit 5 Lesson 11 Answer Key Focus on Mathematical Practices

Houghton Mifflin Harcourt Math Expressions Grade 1 Answer Key Unit 6 Comparisons and Data

Math Expressions Grade 1 Unit 6 Lesson 1 Answer Key Explore Representing Data
Math Expressions Grade 1 Unit 6 Lesson 2 Answer Key Organize Categorical Data
Math Expressions Grade 1 Unit 6 Lesson 3 Answer Key Use Stair Steps to Represent Data
Math Expressions Grade 1 Unit 6 Lesson 4 Answer Key Data Sets with Three Categories
Math Expressions Grade 1 Unit 6 Lesson 5 Answer Key Data Collecting
Math Expressions Grade 1 Unit 6 Lesson 6 Answer Key Introduce Comparison Bars
Math Expressions Grade 1 Unit 6 Lesson 7 Answer Key Comparison Bars and Comparing Language
Math Expressions Grade 1 Unit 6 Lesson 8 Answer Key Solve Compare Problems
Math Expressions Grade 1 Unit 6 Lesson 9 Answer Key Focus on Mathematical Practices

Math Expressions Common Core Grade 1 Volume 2 Answer Key Unit 7 Geometry, Measurement, and Equal Shares

Math Expressions Grade 1 Unit 7 Lesson 1 Answer Key Introduction to Time
Math Expressions Grade 1 Unit 7 Lesson 2 Answer Key Tell and Write Time in Hours
Math Expressions Grade 1 Unit 7 Lesson 3 Answer Key Time in Our Day
Math Expressions Grade 1 Unit 7 Lesson 4 Answer Key Tell and Write Time in Half-Hours
Math Expressions Grade 1 Unit 7 Lesson 5 Answer Key Practice Telling and Writing Time
Math Expressions Grade 1 Unit 7 Lesson 6 Answer Key Squares and Other Rectangles
Math Expressions Grade 1 Unit 7 Lesson 7 Answer Key Triangles and Circles
Math Expressions Grade 1 Unit 7 Lesson 8 Answer Key Equal Shares
Math Expressions Grade 1 Unit 7 Lesson 9 Answer Key Compose 2-Dimensional Shapes
Math Expressions Grade 1 Unit 7 Lesson 10 Answer Key 3-Dimensional Shapes
Math Expressions Grade 1 Unit 7 Lesson 11 Answer Key Compose 3-Dimensional Shapes
Math Expressions Grade 1 Unit 7 Lesson 12 Answer Key Order by Length
Math Expressions Grade 1 Unit 7 Lesson 13 Answer Key Measure with Length Units
Math Expressions Grade 1 Unit 7 Lesson 14 Answer Key Focus on Mathematical Practices

Math Expressions Grade 1 Homework and Remembering Volume 2 Answer Key Unit 8 Two-Digit Addition

Math Expressions Grade 1 Unit 8 Lesson 1 Answer Key Explore 2-Digit Addition
Math Expressions Grade 1 Unit 8 Lesson 2 Answer Key Methods of 2-Digit Addition
Math Expressions Grade 1 Unit 8 Lesson 3 Answer Key Addition of Tens and Ones
Math Expressions Grade 1 Unit 8 Lesson 4 Answer Key Discuss Solution Methods
Math Expressions Grade 1 Unit 8 Lesson 5 Answer Key Practice 2-Digit Addition
Math Expressions Grade 1 Unit 8 Lesson 6 Answer Key Focus on Mathematical Practices

Benefits of Math Expression Homework and Remembering Grade 1 Volume 1 & 2 Answer Keys

Math Expressions is a certified Pre-K–6 common core curriculum that benefits kids be logical of math through exploring, discussing, and demonstrating their understanding of fundamental concepts.
By using the California math expressions grade 1 answer key, children learn how to understand, solve, and use the concepts of preschool maths.
Also, Students use actual examples to make sense of mathematics after practicing from Math Expressions Common Core Grade 1 Homework and Remembering Answer Key.
Math Expressions is based on NSF-funded research so you will be offering Contextual Learning, Multiple Strategies, Manageable Instruction for grade 1 math expressions common core volume 1 & 2 Answer key.

Final Thoughts

We believe that the information of Math Expressions Common Core Grade 1 Answer Key Pdf Homework and Remembering aid students in their remote learning needs. If you find any queries about the subject concepts or else download links, let us know via comments and our subject experts respond to you asap. Stay connected with mathexpressionsanswerkey.com to get the latest news on Houghton Mifflin Math Expressions Grade 1 Volume 1 and Volume 2 Answers.

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Dynamo 1 - Module 1 (La rentrée) - Homework booklet

Subject: French

Age range: 11-14

Resource type: Unit of work

Last updated

4 September 2024

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This is a homework booklet for the French textbook Dynamo 1, Module 1 (La rentrée).

The booklet includes:

All the vocab from the module (2 pages).
10 homeworks (one for each unit - each homework has a couple of activities for students to complete).
All answers provided.

The document is completely editable. It includes a Power Point version and a PDF version.

Please note that I am not a French native speaker and my knowledge of French is intermediate. If you spot any mistakes, you can let me know and I am always happy to amend and re-upload the resource. The resource is also editable so that you can make any changes you want.

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Dynamo 1
This is a homework booklet for the French textbook Dynamo 1, Module 1 (La rentrée). The booklet includes: All the vocab from the module (2 pages). 10 homeworks (one for each unit - each homework has a couple of activities for students to complete). All answers provided. The document is completely editable.
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