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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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## Exercises: Calculus (OpenStax)

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- 1.E: Functions and Graphs (Exercises) These are homework exercises to accompany Chapter 1 of OpenStax's "Calculus" Textmap.
- 2.E: Limits (Exercises) These are homework exercises to accompany Chapter 2 of OpenStax's "Calculus" Textmap.
- 3.E: Derivatives (Exercises) These are homework exercises to accompany Chapter 3 of OpenStax's "Calculus" Textmap.
- 4.E: Applications of Derivatives (Exercises) These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
- 5.E: Integration (Exercises) These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
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- 8.E: Differential Equations (Exercises) These are homework exercises to accompany Chapter 8 of OpenStax's "Calculus" Textmap.
- 9.E: Sequences and Series (Exercises) These are homework exercises to accompany Chapter 9 of OpenStax's "Calculus" Textmap.
- 10.E: Power Series (Exercises) These are homework exercises to accompany Chapter 10 of OpenStax's "Calculus" Textmap.
- 11.E: Parametric Equations and Polar Coordinates (Exercises) These are homework exercises to accompany Chapter 11 of OpenStax's "Calculus" Textmap.
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- 14.E: Differentiation of Functions of Several Variables (Exercise) These are homework exercises to accompany Chapter 14 of OpenStax's "Calculus" Textmap.
- 15.E: Multiple Integration (Exercises) These are homework exercises to accompany Chapter 15 of OpenStax's "Calculus" Textmap.
- 16.E: Vector Calculus (Exercises) These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.
- 17.E: Second-Order Differential Equations (Exercises) These are homework exercises to accompany Chapter 17 of OpenStax's "Calculus" Textmap.

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## Unit 1: Limits and continuity

Unit 2: derivatives: definition and basic rules, unit 3: derivatives: chain rule and other advanced topics, unit 4: applications of derivatives, unit 5: analyzing functions, unit 6: integrals, unit 7: differential equations, unit 8: applications of integrals.

## Math 104: Calculus I – Homework

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- The matrices section contains commands for the arithmetic manipulation of matrices.
- The graphs section contains commands for plotting equations and inequalities.
- The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

## Math Topics

More solvers.

- Add Fractions
- Simplify Fractions

## 10 best homework AI: Free online AI tutoring that helps you study

Boost your assignment and exam grades with these best AI homework helpers.

- March 17, 2024

Did you struggle to grasp the concepts taught in class and find it difficult to finish your homework? This is a common struggle for many students.

While some can seek assistance from a tutor, others may face barriers in accessing such support.

How can we address this issue and ensure equal educational opportunities for everyone? The solution lies in homework AI.

Leveraging the power of AI, homework AI tools provide users with accurate answers and offer detailed explanations, aiding in a deeper comprehension of the subject matter.

If you need help with your studies, explore this guide highlighting the top 10 AI homework helpers that can effectively assist you.

1. Genius Tutor – Best homework AI helper overall

2. ExamUp – Best homework AI helper for improving test grades

3. HIX Tutor – Best homework AI helper for in-depth guides

4. Question AI – Best homework AI helper for affordable tutoring

5. HomeworkAI – Best homework AI helper for all education levels

6. Mathful – Best homework AI helper for accurate math solutions

7. TutorEva – Best homework AI helper for animated AI help

8. Brainly – Best homework AI helper for community access

9. AI Tutors – Best homework AI helper for mobile use

10. OddityAI – Best homework AI helper for better writing

## Genius Tutor – Best homework AI helper overall

Stuck on a homework question? Genius Tutor is designed to give users instantaneous answers to all types of homework problems.

The groundbreaking AI homework help tool provides comprehensive support for many subjects, including math, biology, physics, literature, history, and chemistry.

Powered by the most advanced AI model, it understands the context of your questions.

Regardless of difficulty, it breaks down solutions into manageable steps and offers comprehensive and clear explanations for the answers.

While it is best tailored toward high school and college students, anyone can benefit from using the tool to elevate their education. You can start using Genius Tutor for free with no hidden costs.

## Pros:

- Provides a step-by-step breakdown of homework questions.
- Highlights key formulas, rules, and formulas.
- Fosters learning in a comfortable, judgment-free environment.
- It is a straightforward tool without many extra features.

Explore the capabilities of the Genius Tutor AI homework helper.

## ExamUp – Best homework AI helper for improving test grades

Exams can be a major source of stress for students. ExamUp reduces the pressure placed on students to memorize large amounts of information by simplifying complex concepts.

It breaks down homework problems step by step, explaining the “why” and “how” of each solution.

Additionally, it features multilingual support, enabling users to grasp the concepts in their native languages for enhanced understanding.

The unique AI homework help tool is available 24/7 and acts like an online tutor whenever and wherever students want to study.

ExamUp offers a free plan that students can access at no cost. For advanced features, consider upgrading to a paid subscription.

- Supports over 30 different languages.
- It contains a built-in plagiarism detector to ensure original answers.
- Provides guidance on math, computer science, and more.
- ExamUp is not yet available as an app for smartphones.

Try ExamUp to get a powerful AI homework helper to ace your next exam.

## HIX Tutor – Best homework AI helper for in-depth guides

HIX Tutor is a sophisticated AI homework tool that enables students to gain a deeper understanding of their coursework.

This smart tutoring technology delivers comprehensive, in-depth guides that clearly explain the homework topic in a way that students at all grade levels can understand.

HIX Tutor is a trustworthy study assistant with a 98% accuracy rate, ensuring you get correct answers on your homework.

Start using HIX Tutor for free to see if it works for you. Monthly and annual plans are available for frequent users.

- Covers a wide range of subjects, including foreign languages.
- Supports several inputs, such as image and document uploads.
- Can understand textbook material and assist with practice problem-solving.
- Requires a paid subscription after the initial free trial.

Uncover the academic possibilities of HIX Tutor’s AI homework solver.

## Question AI – Best homework AI helper for affordable tutoring

Many students could benefit from having a personal tutor, but most families cannot afford this luxury.

Question AI serves as a personal AI tutor that makes it easier for students to tackle difficult subjects in school, such as math and science.

The innovative ChatGPT homework tool alternative has a 98% accuracy rate and is more affordable than hiring a human tutor.

Anyone can try Question AI for free. To continue asking questions after the free trial, choose a low-cost subscription.

- Supports more than 100 languages, such as English, German, and French.
- It can be integrated into homework routines to promote learning.
- It helps students learn the material and achieve higher grades in school.
- Students may become too dependent on it.

Use QuestionAI to solve homework question and unlock your learning potential.

## HomeworkAI – Best homework AI helper for all education levels

HomeworkAI is a convenient tool for instant assistance with assignments. The highly accurate AI homework helper is designed to simplify homework and study sessions with instant, one-click solutions.

The user-friendly platform is tailored to help students and lifelong learners enhance their knowledge base at every level.

HomeworkAI is suitable for elementary, middle, high, and college students. Users can try HomeworkAI for free; no credit card is required.

- Get expert help for everything from zoological classifications to algebra problems.
- Reduces student stress by generating instant solutions.
- It can give users an academic advantage.
- Some solutions may lack human-like creativity.

Students at all education levels can benefit from HomeworkAI.

## Mathful – Best homework AI helper for math solutions

Many students agree that Math is one of the most difficult subjects. Mathful is a viable solution for students struggling to do their math homework.

This AI homework help tool is geared toward solving all branches of math, such as calculus, geometry, and algebra.

It doesn’t just give you the answer; it also provides users with in-depth, tailored explanations that help students better understand the math at hand.

It is free to try, and budget-friendly subscriptions are available.

- Supports grades from elementary math to university level.
- It can assist with statistics, word problems, and more.
- The tool uses encrypted transmissions and is safe to use.
- It is limited to math homework help.

Use Mathful to complete your math homework quickly and efficiently.

## TutorEva – Best homework AI helper for animated AI help

TutorEva is an online AI homework solver that offers comprehensive help. It can help you research an academic topic, check your homework for accuracy, and study for an upcoming exam.

It even shows colorful illustrations and animations to make learning more fun. You can try it for free, and premium subscriptions are also available.

- Gives one-stop access to textbook solutions.
- Offers a browser extension to solve problems anywhere on the web.
- Students can join their textbooks and documents together.
- Requires users to sign up to try the tool.

## Brainly – Best homework AI helper for community access

Many students enjoy being part of a community. Brainly offers a vast community of students, parents, and educators who work together to promote learning.

Just ask a question, and the tool will swiftly analyze the problem and prepare a solution. In seconds, you’ll receive the correct answer and explanation. If AI can’t find a solution, ask the community or a live expert.

It provides a 7-day free trial for testing out its features.

- Offers a scan-to-solve feature to save time.
- Students can get expert-verified answers.
- It is free to join.
- Some features are not yet available, such as Test Prep.
- The interface is busy and can be difficult to navigate.

## AI Tutors – Best homework AI helper for mobile use

Many students waste hours completing assignments and studying for exams only to forget the material when it matters most.

AI Tutors transforms how students answer questions and study by creating an interactive platform.

Here, users can learn, ask for feedback, and even request quizzes on the material that they learned. The AI homework help tool is available as an app for convenient smartphone use.

- Helps overcome location barriers for students who need homework AI help.
- Allows students to chat with real tutors 24/7.
- Comes with a built-in plagiarism checker, Google Translator, and more.
- Some solutions may contain plagiarism.

## OddityAI – Best homework AI helper for better writing

OddityAI harnesses the power of AI using advanced algorithms and machine learning capabilities.

This AI homework help tool provides instant assistance with assignments, helps students prepare for major tests, and can be used to verify answers.

It’s also a game-changer for writing help. The AI homework tool offers grammar checkers, literature analysis, and other tools designed to improve writing. Users can try OddityAI for free by signing up.

- Flexible payment plans include subscriptions or credit purchases.
- Can tackle complex subjects.
- Helps generate ideas and verify information in essays.
- May have difficulty providing solutions for niche subjects.

## In conclusion: Choosing the best homework AI helper

Homework is a daunting task for students of all grade levels. The sheer volume of assignments and the difficulty of certain subjects can be overwhelming, especially when students don’t understand the coursework.

Homework AI tools help eliminate the stress and anxiety of homework. If you need help completing an assignment, studying for a test, or learning about a certain subject, opt for one of the best homework AI tools.

1. Can AI homework helpers provide immediate assistance with homework?

Yes! Homework AI tools work quickly to analyze the question and generate accurate and detailed solutions, often within seconds.

2. How reliable are homework AI tools?

While no AI tool is completely flawless, the best AI homework helpers have significantly high accuracy rates of 98% or higher, ensuring correct answers.

3. Which subjects can an AI homework help tool assist me with?

Most of the homework AI tools support a wide range of subjects, such as math, physics, chemistry, history, geography, literature, computer science, foreign languages, and more.

Have any thoughts on this? Drop us a line below in the comments, or carry the discussion to our Twitter or Facebook .

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## AIMath is the most efficient math AI solver for math learning

AIMath is a free online math AI solver designed for people struggling with complex...

## AIMath is the most efficient math AI solver for math learning

O ften perceived as an enigmatic language, mathematics has been revered and feared. For some, it's a thrilling puzzle waiting to be solved; for others, it's a daunting maze of symbols and theorems.

Recognizing the difficulties students encounter in grasping intricate mathematical concepts, AIMath emerges as a supportive solution.

This math AI solver pledges to streamline homework completion and strives to enhance users’ comprehension.

Let’s delve into the realm of AIMath, exploring its potential to reshape the future of math education.

## AIMath overview: What is it?

AIMath represents a revolutionary advance in educational technology, constituting an invaluable aid for students and educators in the field of mathematics.

It is an artificial intelligence-powered system designed to assist with a wide range of mathematical problems.

This AI-driven platform, often called a math AI solver, distinguishes itself by providing users with step-by-step guidance and explanations-not just answers.

Its technology is tailored to enhance and facilitate learning and is capable of addressing topics from basic arithmetic to complex calculus equations.

## How can AIMath enhance your math learning experience?

Now that we understand the essence of AIMath, the next question arises: How can we harness its potential?

## Application in self-study and exam prep

Utilizing AIMath proves highly beneficial for self-study and exam preparation.

This online math AI solver is a dynamic addition to students’ study routines, seamlessly complementing traditional methods and textbooks.

Its adaptability and swift feedback on practice problems make it an indispensable resource, empowering students to deepen their comprehension and achieve improved academic performance.

## Support for curriculum and homework

Moreover, AIMath is a valuable support system for formal curriculums and homework assignments.

The broad spectrum of solutions and detailed explanations provided align seamlessly with educational systems’ diverse requirements.

By offering reliable and immediate assistance, AIMath becomes a crucial ally for students seeking to successfully complete assignments and grasp new concepts introduced in their classes.

## Who is AIMath designed for?

AIMath, designed for a diverse user base, is particularly tailored to:

- Students: The sophisticated technologies embedded within AIMath have a proven track record of improving student test scores. By delivering comprehensive explanations and fostering independent problem-solving skills, this math AI homework solver significantly boosts academic performance among its users. The AI not only equips students with the correct answers but also ensures that they understand the rationale behind each step, resulting in deeper learning and better retention of mathematical concepts.
- Educators: AIMath also serves as a formidable resource for educators. By integrating this AI math homework solver into their teaching methodologies, teachers can offer supplemental support to students outside the classroom. The math AI solver’s in-depth explanations and real-time assistance enable teachers to direct their students towards an additional layer of instruction, reinforcing lesson plans and improving educational outcomes.

## Key features of AIMath

With so many math AI solvers on the market, what sets AIMath apart from its competitors?

## Comprehensive coverage of mathematical topics

AIMath distinguishes itself by its ability to address a wide spectrum of mathematical subjects.

These include arithmetic, algebra, geometry, trigonometry, calculus, and even advanced fields such as discrete math and linear algebra.

By incorporating a comprehensive range of topics, AIMath ensures that users with varying degrees of proficiency can find the help they need, whether they are tackling primary school-level calculations or university-level problems.

## Global accessibility through multi-language support

One of the significant advantages of AIMath is its multi-language support, catering to a global audience.

By providing solutions in numerous languages, this math AI solver bridges the communication gap and ensures that language barriers do not hinder students’ quest for mathematical understanding.

## Real-time problem analysis and solutions

AIMath’s efficiency as an AI math helper derives from its real-time problem analysis capabilities.

This feature enables the system to produce accurate solutions promptly, which is especially beneficial for students requiring immediate assistance with homework or exam preparation.

The state-of-the-art algorithms employed by AIMath allow it to interpret and solve even the most intricate math problems within seconds, drastically reducing students’ time on problem-solving.

## Universal access and 24/7 availability

AIMath’s universal access and 24/7 availability highlight its commitment to constantly supporting learners of all ages and from all locales.

With just an internet connection, users can access the robust features of this AI math homework solver at any time and from anywhere in the world.

## How do you solve your homework problems with AIMath?

Let's walk through the step-by-step guide for using AIMath, the AI-powered math solver:

- Visit the AIMath website.
- Type your math problem into the provided input box.
- Click the button to submit your math problem.
- AIMath will process your query using its AI algorithms. Within seconds, you will receive an accurate answer and a detailed explanation.

## The evolving role of AIMath in bridging education gaps

As we look to the future, what benefits can we anticipate from the continued use of AIMath?

## Addressing geographic and socio-economic barriers

One crucial role of AIMath in education is its remarkable potential to mitigate geographic and socioeconomic disparities in learning.

With AIMath, students in remote or underserved areas gain improved access to quality math education.

It democratizes learning by making high-quality math assistance available to anyone with internet access, regardless of their location or socio-economic status.

## Achievement of educational equity

The ongoing evolution of AIMath actively contributes to the attainment of educational equity.

By ensuring that all students have equal opportunities to understand and excel in mathematics, this math AI solver lays the groundwork for a more equitable educational landscape.

Education is a fundamental right, and the increasing prominence of AI math solvers positions it as a key player in realizing this ideal on a global scale.

## Final thoughts

As an AI math solver, AIMath provides precision and insightful guidance through the complexities of mathematical landscapes.

It effectively addresses comprehension gaps and fosters a deep appreciation for the elegance embedded in numerical intricacies.

Trustworthy and helpful, AIMath stands as a reliable AI Math companion for students, teachers, and anyone eager to learn mathematics.

Have any thoughts on this? Drop us a line below in the comments, or carry the discussion to our Twitter or Facebook .

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## The numbers do not add up for mathematics homework, according to a new study

by University of South Australia

Supporting kids with math homework is a common afterschool activity. But beyond the basics, new curricula and teaching strategies are making it harder for parents to help and it's taking a serious toll on children's confidence and learning.

In a study from the University of South Australia, researchers have found that mathematics homework can sometimes cause more harm than good.

Exploring how homework policies and practices affect families, researchers found that mathematics homework could inadvertently affect a child when it often:

- was too difficult for a child to complete independently, and/or with the support of a parent
- required significant support from parents and seeped into family time
- resulted in a negative experience for the child and their parent, leading to negative associations with mathematics and potentially students' disengagement from the subject
- generated feelings of despair, stress, and negativity among parents who were unable to help
- made students feel inadequate when they struggled with the work.

UniSA researcher Associate Professor Lisa O'Keeffe says such negativity around mathematics has broad implications.

"Homework has long been accepted as a practice that reinforces children's learning and improves academic success," Assoc. Prof. O'Keeffe says.

"But when it is too complex for a student to complete even with parent support, it raises the question as to why it was set as a homework task in the first place. We know that parents play a key role in supporting their children with schooling and homework. When children need help, their parents are often the first people they turn to.

"But many parents are unsure of the current mathematics strategies and approaches that their children are learning as these have changed a lot since they were at school. Like many things, mathematics teaching has evolved over time. But when parents realize that their tried-and-true methods are different to those which their children are learning, it can be hard to adapt, and this can add undue pressure. When children see their parents struggle with mathematics homework, or where mathematics homework becomes a shared site of frustration for families, it can lead to negativity across generations.

"For example, we might hear adults saying things like, 'I wasn't very good at math, so my child won't be either.' Negative interactions with mathematics, and negative discourses like these can lead to reduced confidence, reduced self-efficacy, and can negatively affect children's resilience, persistence, and ultimately their inclination to continue with mathematics."

Any decline in STEM subjects such as math can have long-term impacts for Australia's future. Statistics show that fewer than 10% of students are studying a higher level of math, with math capabilities declining more than 25 points (15-year-olds in 2022 scored at a level that would have been expected of 14-year-olds, 20 years earlier).

Co-researcher, UniSA's Dr. Sarah McDonald, says the research also identified gendered biases.

"Our research showed that it was overwhelmingly mothers who were responsible for managing children's homework. And they often experience frustration or despair when they were unable to understand the math problems," Dr. McDonald says. "When mothers find math hard, there is concern that this may demonstrate to their children , especially their girls, that this is not an area in which they would naturally excel.

"The last thing teachers want to do is disadvantage girls in developing potentially strong mathematical identities. We need a greater understanding of homework policies and expectations.

"The experiences of the families in our study do not support the often-quoted claim by researchers that that homework has potential non-academic benefits such as fostering independence, creating positive character traits, developing good organizational skills, or virtues such as self-discipline and responsibility."

Provided by University of South Australia

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