Question

A) Domain, B)Intercepts, C) symmetry, D)Asymptotes, E) Intervals, F) Intervals of Increase or Decrease, G) Concavity and Points of Inflection, H) Sketch the Curve y = sinx/(2 + cosx) A) Domain, B)Intercepts, C) symmetry, D)Asymptotes, E) Intervals, F) Intervals of Increase or Decrease, G) Concavity and Points of Inflection, H) Sketch the Curve y = sinx/(2 + cosx) @Mathematics

Answer 1

I believe domain is all real numbers

Answer 2

the numerator domain is all reals, but the denominator cannot equal to zero
so when 2 + cos x = 0 then we have division by zero
cos x = -2 never occurs though, lol

Answer 3

ROFL , so the domain is all reals

Answer 4

intercepts, there are no vertical asymptotes because the denominator is never equal to zero. the numerator is equal to zero when x = pi*k , for k integer

Answer 5

so x intercepts are pi*k or pi*n

Answer 6

Intercepts
\[y = \frac{\sin x}{\cos {x} + 2} \]
\[ \sin x = 0\]
\[ x = \sin^{-1}0\]
\[y = 0\]

Answer 7

3) symmetry, it is an odd function. test it by using f(-x) and see what you get out

Answer 8

ishann, theres more than one x intercept

Answer 9

what is pi*k or pi*n?

Answer 10

yea there is @barry

Answer 11

k stands for an integer, like { ...-3,-2,-1,0,1,2,3,...}

Answer 12

I only solved it for principle case, General case is barry's solution

Answer 13

f(-x) = sin(-x) / ( 2 + cos(-x)) = -sin x / ( 2 + cos x ) = - f(x) , so its odd function
it has symmetry wrt to origin

Answer 14

well can you continue solving using principle case Ishaan94

Answer 15

what about the intervals that are increasing and decreasing, concavity, and points of inflection

Answer 16

ta, why are you talking to ishaan, i have you the COMPLETE answer ?

Answer 17

now i wont help

Answer 18

you can ignore me. i am not helping anymore

Answer 19

sorry barry2, I was just wondering what was going on, can you please help me?

Answer 20

ok we are done with symmetry , since you have no questions i can move on

Answer 21

ok

Answer 22

d) asymptotes, there are no vertical asymptotes because the denominator is never equal to zero. so horizontal asymptote?

Answer 23

0 or is it pi?

Answer 24

plug in infinity and -infinity

Answer 25

how am I supposed to plug in infinity?

Answer 26

you plug it in ( sin oo / ( 2 + cos (oo)

Answer 27

infinity divided by infinity=infinity

Answer 28

sin infinity is not infinity

Answer 29

is it undefine

Answer 30

well it oscillates between -1 and 1

Answer 31

what does that mean?

Answer 32

I thought sinx=0 at pi and 0

Answer 33

we are talking about infinity

Answer 34

so is -1, 1 the horizontal asymptote? If so why I don't get what you mean by oscillates

Answer 35

sin x oscillates between 1 and -1 , the graph y = sin x , as x goes to infinity

Answer 36

what about cos x where does that oscillates?

Answer 37

also between -1 and 1

Answer 38

so what will the critical points be to determine the intervals of increasing and decreasing

Answer 39

wait

Answer 40

ok i checked the graph im pretty sure no horizontal asymptote

Answer 41

I'm about to go to sleep because I got class early in the morning, but I still need help finding the critical points to determine the intervals where in its increasing/decreasing, concavity, and point of inflection

Answer 42

ok so no asymptotes, we're good

Answer 43

dont sleep!

Answer 44

ok, please not to be mean, could you hurry up

Answer 45

i can , but i have no motivation. wait

Answer 46

if you paid i work faster. ok , lets see

Answer 47

y = sinx/(2 + cosx)
y ' = [ 2 cos (x) + 1 ] / ( cos x + 2)^2

Answer 48

y '' = [ 2 sin x ( cos x - 1) ] / ( cos (x) + 2) ^ 3

Answer 49

what is E) intervals?

Answer 50

that just talking about finding the critical points just using to test to determine if the graph is increase or decreasing between the points

Answer 51

ok the critical points occur when y ' = 0 or y ' is undefined

Answer 52

y ' = 0 when 2 cos x + 1 = 0 , cos x = -1/2, this occurs when

Answer 53

x = 2pi/3 +2pi*n , 5pi/3 +2pi*n

Answer 54

those are critical pts

Answer 55

cos at -pi/3 and -4pi/3 =-1/2

Answer 56

thats wrong

Answer 57

cos (-pi/3) = .5

Answer 58

how does 5pi/3 +2pi*n?

Answer 59

cos (5pi/3) = -1/2

Answer 60

and 2pi *n because cos ( x + 2pi*n ) = cos x

Answer 61

so cos (5pi/3 + 2pi*n) = cos (5pi/3) = -1/2 , same with other solution

Answer 62

cos(5pi/3) does not equal -1/2, it equal positive 1/2
cos(4pi/3) equal -1/2

Answer 63

check your calculator, *shrugs*

Answer 64

cosine is negative in quadrant 2 or 3, 5pi/3 is quadrant 4

Answer 65

no 5pi/3 is quad 3

Answer 66

do 5pi/3*180/pi=300

Answer 67

how much more of this problem is there left?

Answer 68

oh sorry

Answer 69

will we still need to determine the intervals of the critcal numbers which I believe is 4pi/3 and 2pi/3

Answer 70

x = 2pi/3 +2pi*n , 4pi/3 +2pi*n

Answer 71

I need to go to sleep what are the intervals its increasing and decreasing, and also concavity?

Answer 72

y = sinx/(2 + cosx)
y ' = [ 2 cos (x) + 1 ] / ( cos x + 2)^2
y '' = [ 2 sin x ( cos x - 1) ] / ( cos (x) + 2) ^ 3

Answer 73

so when is y' > 0 and y' < 0 ,
2 cos x + 1 > 0 , cos x > -1/2 ,

Answer 74

so thats gives a range of -2pi/3< x < 2pi/3 right? for cosx > -1/2

Answer 75

it is easier to find 2 cos x + 1 < 0 , which is 2pi/3 < x < 4pi/3

Answer 76

are those the intervals where its increasing/decreasing?

Answer 77

of course with any multiple of 2pi etc etc.

Answer 78

so for 2 cos x + 1 > 0 , we have x > 4pi/3 or x < 2pi/3

Answer 79

right right.

Answer 80

0 < x < 2pi/3 or when 4pi/3 < x < 2pi it is increasing

Answer 81

2pi*n < x < 2pi/3 + 2pi (n+1) , or 4pi/3 +2pi*n < x < 2pi + 2pi(n+1)

Answer 82

decreasing when
2pi/3 +2pi*n< x < 4pi/3+2pi*(n+1)