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Exercises: Calculus (OpenStax)

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These are homework exercises to accompany OpenStax's "Calculus" Textmap.

• 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.
• 6.E: Applications of Integration (Exercises) These are homework exercises to accompany Chapter 6 of OpenStax's "Calculus" Textmap.
• 7.E: Techniques of Integration (Exercises) These are homework exercises to accompany Chapter 7 of OpenStax's "Calculus" Textmap.
• 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.
• 12.E: Vectors in Space (Exercises) These are homework exercises to accompany Chapter 12 of OpenStax's "Calculus" Textmap.
• 13.E: Vector-Valued Functions (Exercises) These are homework exercises to accompany Chapter 13 of OpenStax's "Calculus" Textmap.
• 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.

Thumbnail: The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. (GNU Free Documentation License, Version 1.3 and CC- SA-BY 3.0; Wikipedia).

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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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

Section 004 - Spring 2014

• Lecture Videos

Course ID: rimmer21998 For instructions on how to create a login, follow the directions here . All homework will be due at midnight (actually 11:59) on the day listed.

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