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..?

 

[d_leet]

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..?

NO!

It is absolutely not true that

sin(x) = cos(x) for all x.

or

cos(x) = -sin(x) for all x.

Do you understand what the problem is with what you wrote.

However it is true that teh first derivative of sin(x) is equal to cos(x) and that the first derivative of cos(x) is equal to -sin(x).

 

[quantumdude]

I don't know why there is a 2 and a 4 at the end and a -1 in front...

I gave you the differentiation rules. Have you tried to use them? If so then present your work and we'll show you what's wrong with it.

i know that sinx=cosx and that cosx=-sinx..?

No. Put a derivative operator in front of the left side of each of those equations, and then they will become true.

 

[frenkie]

how do you find end behavior of the function sin(x/2)

 

[d_leet]

how do you find end behavior of the function sin(x/2)

What do you know about the function y = sin(x)?

Surely the behavior of y = sin(x/2) should be similar..

 

[frenkie]

end behavior is a straight line going to negative and positive infinity?