Solve f(x)=sin(x/2): Find Roots, POI, Min, Max

[frenkie]

original equation: f(x)=sin(x/2)
need: show work on how to find roots, POI, min, max.
intervals of increase/descrease.
intervals of concavity, end behavior.
please help..any help is greatly appreciated.

 

[arildno]

Need: Show your own work first.

 

[frenkie]

to find a root i plugged the equation into the calculator in the y= and i graphed it and pushed 2nd trace and pushed #2, and then it asked me to pick a number less then o and greater then o so I picked -2 and 2 and it gave me a root at x=0.

for the POI, i graphed it and looked at where the concavity changes. (not sure if its correct)

for limits i just looked at the graph and saw that as x goes to infinity y goes to 1 and as x goes to negative infinity y goes to -1.

and for the mins and max's i also looked at the graph and everytime thre was a concave up i put a min and everytimet here was a concave down i put a max. right?

 

[Hootenanny]

There will be an infinite number of roots. What is the interval you are required to solve for?

 

[arildno]

You are using TEXAS, right?

Are you absolutely sure you were asked to do this by aid of a calculator?

 

[frenkie]

need to do it by hand...i know that to find the root i need to set the original equation to 0 and solve, but I'm stuck because i never did it with trig functions. i also konw that to find mins and maxs you are suppose to set the second derivative = to 0, but again, stuck.

and the intervals are -infinity to infinity.;-(

 

[arildno]

Well, let's take the roots first:
Letting y=x/2, when is sin(y)=0?

 

[frenkie]

when x=0, is that the only root? or is there more?

 

[Hootenanny]

when x=0, is that the only root? or is there more?

Think about a sin function. Where does it cross the x-axis?

 

[frenkie]

from the interval of -10 to 10 sin function crosses the x-axis at 3, 6, 9 same for the negative. and sin(x/2) crosses at -6,0,6...so these are the 3 roots of the equation from -10 to 10?

 

[Hootenanny]

I assume your working in radians.Your answers are correct, however it is more usual to give them in terms of \pi, for example, \pi , 2\pi , 3\pi etc.

Now you need to think about your function f(x) = \sin\left( \frac{x}{2} \right), where will the crossing points be?

 

[frenkie]

i believe they will be at 0, negative pie and 2pie. since one cycle i pie. and there are 2 complete cycles.

 

[arildno]

i believe they will be at 0, negative pie and 2pie. since one cycle i pie. and there are 2 complete cycles.

So, can you find some GENERAL formula for the zeroes out of this?
(Hint: It has something to do with multiples of a famous number).

 

[frenkie]

plug in Pi for x in the original equation? :-(

 

[d_leet]

plug in Pi for x in the original equation? :-(

But if the original equation is y = sin(x/2) then letting x = pi you get

y = sin(pi/2) = 1 So that certainly isn't a zero.

 

[arildno]

Well:
What do you think the following expressions equals:
\sin(-3\pi), \sin(4\pi), \sin(7\pi)

What is the common feature with these expressions?

 

[frenkie]

yeah true..I don't konw what the equation is...anybody know how to find POI of the equation and min/max? i know how to find it on the graph but I don't know how to do it and show work.

 

[d_leet]

yeah true..I don't konw what the equation is...anybody know how to find POI of the equation and min/max? i know how to find it on the graph but I don't know how to do it and show work.

First and second derivative tests maybe...

 

[frenkie]

you mean set the first derivative equal to 0? and then the numbers you get you plug into the original equation? because when i set the first derivative equal to 0 i get x=0 as my only answer.

 

[d_leet]

you mean set the first derivative equal to 0? and then the numbers you get you plug into the original equation? because when i set the first derivative equal to 0 i get x=0 as my only answer.

The first derivative of that function certainly has more than 1 zero, and x=0 is definitely not one of them.

 

[frenkie]

derivative of sin(x/2) is cos(x/2)? and the second derivative is -sin(x/2)?

 

[d_leet]

derivative of sin(x/2) is cos(x/2)? and the second derivative is -sin(x/2)?

Close, but you need to remember the chain rule.

 

[frenkie]

what is the derivative of (x/2)?

 

[frenkie]

1st derivative of y=sin(x/2) i found to be y=cos(x/2)/4 and second derivative y=-1sin(x/2)/4...but I am not sure how i got it. any ideas?

 

[d_leet]

1st derivative of y=sin(x/2) i found to be y=cos(x/2)/4 and second derivative y=-1sin(x/2)/4...but I am not sure how i got it. any ideas?

How are you not sure how you got it? And oddly enough your second derivative is correct but the first derivative you found is wrong..

 

[matt grime]

Please don't take this the wrong way, but why don't you know the properties of sin (and cos etc) when you're expected to work out all these things. I don't see how you've got to be in a situation like this where you need to ask what the derivative of x/2 is.

This isn't me saying 'gosh, how can someone not know *that*' but asking 'how can someone who doesn't know that be in a class that asks them to find the points of inflexion of sin(x/2)'?

 

[frenkie]

I have this sketch pad program and it does it automatically, but I'm not sure how. first derivative is instead of the /4 it is /2

 

[quantumdude]

Why are you using a program? This is easy to do without one.

Here are two hints:

If u is a differentiable function of x then we have:

\frac{d}{dx}\sin(u)=\cos(u)u'
\frac{d}{dx}\cos(u)=-\sin(u)u'

As for the derivative of \frac{x}{2} simply recall that \frac{x}{2}=\frac{1}{2}x and use the appropriate differentiation rule.

 

[matt grime]

you're using a computer program for this? cough, splutter, ahem, various 'what's the world coming too' noises. This question is the equivalent of asking what 2+2 is, really, which is why I would really like to understand the set of circumstances that has led you to need to answer this question and not understand how to differentiate x/2

 

[frenkie]

I don't know why there is a 2 and a 4 at the end and a -1 in front...i know that sinx=cosx and that cosx=-sinx..?