I have solved this exercise:
Seen from the south an island has the shape like the graph of
y = f(x) = 2/(x^2 +1) - 1.
There's treasure burried some place where the angle 10% in east/west direction // is 0.1 from the x-axis.
A) Show that there are such points on the graph.
B) Use Newton-Raphsons method or fixed-point notation to find a point where the treasure can be burried. The point should be given with an error of less than 0.0005 and marked with an "x"
I have added two files with my solution of this. I was wondering if I could get some feedback and/or guidance as to if I have solved this correctly because I am uncertain and I have no one else to ask. Thanks in advance.
I have written:
A) f has a kritical point in x = c if f(c) exists and if one of the two following statements are true: '
1. f'(c) = 0
2. f'(c) does not exists.
Then I found the derivative and second derivative of f, found the kritical points and stated that f is decreasing where f' is negative. Then I stated that 'if f'(x0)= 0 and f''(x0) = 0, then we can conclude that f has a critical point in x0 = 0, that can be a local max/min, or an inflection point, that means that the concavity of the graph of f is different on different sides of x0.
f is continously differentiable over all of R, which means that the graph y = f(x) is a continous line that crosses the xy-axis at at least one point, which means that such points exists on the graph.
B) I calculated the derivative and used NR-method ( x_n+1 = x_n - (f(x_n)/f'(x_n)) ) with the start value of 0.9.
The problem here is that I think I found the nullpoints and not the point they're asking for in the text and I dont know what to do differently.