What are the interesting points on the graph of sin(x/2)?

  • Context: Undergrad 
  • Thread starter Thread starter frenkie
  • Start date Start date
  • Tags Tags
    Derivative
Click For Summary

Discussion Overview

The discussion revolves around identifying interesting points on the graph of the function sin(x/2), including its critical points, maxima, minima, and points of inflection. Participants explore the derivatives of the function and their implications within a specified domain.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants discuss the derivative of sin(x/2) as cos(x/2)/2 and attempt to find where this derivative equals zero.
  • There is a suggestion to replace x/2 with t to simplify finding the zeros of cos(t), leading to a discussion about the periodic nature of the cosine function.
  • Participants identify specific values where cos(x/2) is zero, such as Pi/2 and 3Pi/2, but later clarify that these values need to be adjusted based on the relationship x/2 = t.
  • One participant proposes that the domain of interest is [-2Pi, 2Pi], suggesting that -Pi and Pi are critical points, with -Pi as a minimum and Pi as a maximum.
  • Another participant emphasizes the need for proof using first and second derivative tests to justify claims about extrema and points of inflection.
  • There is confusion regarding the second derivative and its zeros, with participants attempting to solve for x when setting the second derivative equal to zero.
  • Some participants express uncertainty about the correct values for critical points and points of inflection, with mentions of Pi, 2Pi, and 0.

Areas of Agreement / Disagreement

Participants generally agree on the need to analyze the function's derivatives to find critical points, but there is disagreement on the specific values and interpretations of those points. The discussion remains unresolved regarding the exact nature of the interesting points on the graph.

Contextual Notes

Participants mention the importance of the first and second derivative tests, but there are unresolved mathematical steps and assumptions about the function's behavior within the specified domain.

frenkie
Messages
37
Reaction score
0
cos(x/2)/2 is my derivative. and when I set it equal to 0, i try to solve for x.

so cos(x/2)/2=0
cos(x/2)=0/2 this is the part where I'm stuck...a friend of mine did the rest and he came up with what's on the bottom, but I don't know how he did it or if its right...help?
cos x=sqrt2/2
 
Physics news on Phys.org
frenkie said:
cos(x/2)/2 is my derivative. and when I set it equal to 0, i try to solve for x.

so cos(x/2)/2=0
cos(x/2)=0/2 this is the part where I'm stuck...a friend of mine did the rest and he came up with what's on the bottom, but I don't know how he did it or if its right...help?
cos x=sqrt2/2
You should have put this in your other thread, and yes that would be a correct derivative, but I don't know what the heck your friend did to find the zeros of that function.. This should really not be that difficult.. What do you KNOW about cos(x), think about that for a second, maybe replace x/2 with t and figure out where cos(t) is zero, if you know anything about trigonometry this should be relatively easy and by know I really hope you know that there are an infinite number of them unless you are being restricted to a specific domain. Once you have the values of t where cos(t) is zero, you know that t = x/2 so from that you can find the values of x that make cos(x/2) zero.
 
Last edited:
cos x is zero at Pi/2 and 3pi/2 on the unit circle so when i plug these two numbers into x in my derivative equation i get o for the answer

so x is Pi/2 and 3pi/2?
 
frenkie said:
cos x is zero at Pi/2 and 3pi/2 on the unit circle so when i plug these two numbers into x in my derivative equation i get o for the answer

so x is Pi/2 and 3pi/2?

NO! Firstly there are more zeros then that, a lot more(read my last post). Second those value are correct for a function of the form y = cos(t), but you have something of the form y = cos(x/2) so then what would the relation between x and t be?

It looks to me like it should be that

x/2 = t.

Do you think you can handle it from there?
 
x=Pi? I'm making this way too hard then it actually is.
 
frenkie said:
x=Pi? I'm making this way too hard then it actually is.

That is ONE of them, but only one. If you would post the original problem it would be much easier to help you, because I would think that you would be given some bounds within which to find the max/min/points of inflection for a function like this if not then well...
 
the teacher said that it would be alright if the domain iz [-2Pi, 2Pi] so i think that -Pi and Pi are the x values...which means that -Pi is a min and Pi is a max and 0 is a PIO.? do u agree?
 
frenkie said:
the teacher said that it would be alright if the domain iz [-2Pi, 2Pi] so i think that -Pi and Pi are the x values...which means that -Pi is a min and Pi is a max and 0 is a PIO.? do u agree?
What are you basing this on? You need to show proof to justify your claims using the first and second derivative tests not just random points where funtions are zero. The first derivative in that domain has only 2 zeros, which means only 2 places that could be extrema, you then need to find the seond derivative to find possible points of inflection.

The zeros of the first derivative you have are correct though. Pi and negative Pi
 
I'm not sure how to solve for x when i set my second derivative equal to 0.

-1sin(x/2)/4=0
if i do PEMDAS in reverse i get
sqrt2/1 multiplied by 4 but i don't think its right
 
  • #10
frenkie said:
I'm not sure how to solve for x when i set my second derivative equal to 0.

-1sin(x/2)/4=0
if i do PEMDAS in reverse i get
sqrt2/1 multiplied by 4 but i don't think its right


Where is sqrt2 coming from?

This is the same as the first problem cos(x/2)=0, but with a sin instead of a cos. Where is sin t = 0?
 
  • #11
pi and 2pi which means that x=pi?
 
  • #12
or pi/2? getting confused all over again.
 
  • #13
You forgot 0 (and negative values). But you are otherwise correct.
 
  • #14
oh nvm my PIO is o and positive and negative 2pi? correct?
 
  • #15
is there any interesting points in the graph of sin(x/2)?
 

Similar threads

Undergrad Questions about these Trigonometry Graphs involving sin() and cos()
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Problem with calculating projections of curl using rotation of contour
  • · Replies 9 ·
Replies
9
Views
1K
High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
5K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad What's my mistake in this integration problem?
  • · Replies 8 ·
Replies
8
Views
3K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Calculating the change of the volume of a sphere using this integral
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Integration of ##e^{-x^2}## with respect to ##x##
  • · Replies 4 ·
Replies
4
Views
3K