Finding Trigonometric Limits through Calculator Techniques

[lightningz]

Homework Statement

Let f(x)=cos (x). Denote f^2(x) = ff(x), f^3(x)=fff(x) and so on.
For x=100 radians, find f^n(x) for n=1 to 20
Develop a calculator technique to find f^n(x) for any given value of n. Hence, find correct to 7 decimal places, the limit of f^n(x) for x= 100 radians and n approaching infinity.

Homework Equations

The Attempt at a Solution

sorry,i have no idea how to tackle this question.can any1 help me?=)

 

[Dick]

Any vague ideas? The problem just asks for a 'calculator technique'. I would just set my calculator for radian mode, put in 100 and then just keep pushing 'cos', 'cos', 'cos'...

 

[lightningz]

oh yeah...i tot we need to develop another different formula or method to calculate the value rather than using the calculator and cos cos cos.

 

[lanedance]

hi lightningz

so use the caclulator to see what happens with repeated applications, then see if you can work out why...

conceptually, you could imgine the process as taking:
x₁ point = 100radians, y₁ = cos(x ₁)

then transfer that back to the x-axis using the line y = x, so x₂ = y₁ = cosx, now repeat the process
y₂ = cos(x ₂)= cos(cos(x ₁))

try drawing it out for a few cycles (after the first it will be near the y axis)
is there any pattern you can see? can you describe it?

 

[lightningz]

the graph become smaller and smaller(ie converging to a certain value) and when n=infinty, it actually becomes a straight line. in this case, limit of cos^n(x),(x--->infinity), for all value of x is that particular value..that's what i got...urm..is that what the question want?=)

 

[lanedance]

which graph?

i assume you are plotting:
$$f^n(100) = Cos(f^{n-1}(100))$$ against n

did you try what i mentioned? plot the line f(x) = cos(x) and g(x) = x and trace the operation of the function by moving a point between those 2 lines, this should illuminate what is going on

based on your question its probably enough to hit cos on your calculator enough times that the numbers down to the seventh decimal stop changing... probably about 40 or so times (easier in a speadhseet)

though there is a bit more going on here - have you heard of a fixed point of a function before?

 

[lightningz]

sorry.i never heard of fixed point.
well the intersection point of the two graphs f(x)= cos (x) and g(x)=x..i got it by plotting in my calculator..which is the value for limit of cos^n(x),(x--->infinity), is that point called fixed point? thanks a lot for helping all the time =)

 

[lanedance]

fixed point of a function where f(x) = x, so repeated application of the function does not cahneg the value at the fixed point

have a look here, it actually has a picture of f(x) = cos(x) as I was trying to explain
http://en.wikipedia.org/wiki/Fixed_point_(mathematics )

 

[lightningz]

thanks.i think i understood already =) such a nice guy you are..