I need help wraping my head around how to do this, and /why/ it works out the way it does.
The purpose of this exercise is to illustrate how a transcendental equation can be solved using limits. Let f(x) = cosx for x E R We intend to solve the equation f(x) = x. Denote the solution by \(\displaystyle \x\). Then \(\displaystyle \x\) is the unique real number such that \(\displaystyle f(\x) = \x\). To find \(\displaystyle \x\), we start with an initial quess \(\displaystyle x_0\). The choice of \(\displaystyle x_0\) isn't too important. Any number in the interval [-10..10] should work, but for simplicity you take \(\displaystyle x_0 = 0\). Now calculate
\(\displaystyle \L s_1 = f(x_0), \,\,\,x_2 = f(x_1) = f(f(x_0)),\,\,\, x_3 = f(x_2) = f(f(f(x_0))), ...\)
\(\displaystyle \L x_n = f(x_{n-1}) = f(...f(f(x_0))...)\)
You can do this by repeatedly pushing the cosine button on your calculator. Be sure to use radian mode. After 20 cosine evaluations (n = 20) you should have a number \(\displaystyle x_{20}\) whose first two or three decimal digits agree with those of the actual solution \(\displaystyle \x\) Infact: \(\displaystyle \L \x = \lim_{x\to\infty}x_n\)
Give the first 6 decimal digits of \(\displaystyle \x\)
First of all - I really don't understand /why/ this infinite function composition approaches a specific positive value: If I graph f(x), f(f(x), f(f(f(x))), .... I see the cosine function stretching out into /almost/ a horizontal line at around 0.7389
EDIT: And how would I show this without just doing cos(cos(cos(x))).. on my calculator? I know this is not the place for tutoring, but I hope someone can help explain this to me.
The purpose of this exercise is to illustrate how a transcendental equation can be solved using limits. Let f(x) = cosx for x E R We intend to solve the equation f(x) = x. Denote the solution by \(\displaystyle \x\). Then \(\displaystyle \x\) is the unique real number such that \(\displaystyle f(\x) = \x\). To find \(\displaystyle \x\), we start with an initial quess \(\displaystyle x_0\). The choice of \(\displaystyle x_0\) isn't too important. Any number in the interval [-10..10] should work, but for simplicity you take \(\displaystyle x_0 = 0\). Now calculate
\(\displaystyle \L s_1 = f(x_0), \,\,\,x_2 = f(x_1) = f(f(x_0)),\,\,\, x_3 = f(x_2) = f(f(f(x_0))), ...\)
\(\displaystyle \L x_n = f(x_{n-1}) = f(...f(f(x_0))...)\)
You can do this by repeatedly pushing the cosine button on your calculator. Be sure to use radian mode. After 20 cosine evaluations (n = 20) you should have a number \(\displaystyle x_{20}\) whose first two or three decimal digits agree with those of the actual solution \(\displaystyle \x\) Infact: \(\displaystyle \L \x = \lim_{x\to\infty}x_n\)
Give the first 6 decimal digits of \(\displaystyle \x\)
First of all - I really don't understand /why/ this infinite function composition approaches a specific positive value: If I graph f(x), f(f(x), f(f(f(x))), .... I see the cosine function stretching out into /almost/ a horizontal line at around 0.7389
EDIT: And how would I show this without just doing cos(cos(cos(x))).. on my calculator? I know this is not the place for tutoring, but I hope someone can help explain this to me.
