I am given the function x ( t ) = 4 arctan ⁡ t and told that routine computations will show that x ( 0 ) = 0 and x ( 1 ) = π.I must determine a differential equation for x ( t ) of the form x ′ ( t ) = f ( t , x ).I must then use f to approximate π.I will use Euler's Method to find minimum number of iterations ( n) needed to yield the approximation π ≈ 3.1400, such that n − 1 iterations is strictly less than 3.1400.How do I find n? It seems to me that it can't be found given the instruction set that I am provided.

Question

I am given the function   x  (  t  )  =  4  arctan  ⁡  t and told that routine computations will show that   x  (  0  )  =  0 and   x  (  1  )  =  π.I must determine a differential equation for   x  (  t  ) of the form       x    ′    (  t  )  =  f  (  t  ,  x  ).I must then use   f to approximate   π.I will use Euler&#039;s Method to find minimum number of iterations (  n) needed to yield the approximation   π  ≈  3.1400, such that   n  −  1 iterations is strictly less than 3.1400.How do I find   n? It seems to me that it can&#039;t be found given the instruction set that I am provided.

Kasey Bird · Accepted Answer

I think you are expected to divide [0,1] into n intervals, use Euler&#039;s method to integrate       1          1      +              t        2             over the interval and see what you get for x=1. For n=5, you would compute   x  (      1    5    )  =  x  (  0  )  +      1    5        x    ′    (  0  )  =  0  +      1    5    ⋅  1  =  0.2,   x  (      2    5    )  =  x  (      1    5    )  +      1    5        x    ′    (      1    5    )  ≈  0.2  +      1    5    ⋅  0.961538  ≈  0.392308  ,, continue stepping until you get to x(1), and compare it with       π    4  . The problem I have is that the derivative is decreasing over the interval, so the forward Euler&#039;s method overestimates the function, while the problem assumes it underestimates the function. At n=10 I get   x  (  1  )  ≈  0.809981, an error of about 0.025. Since the global truncation error goes as       1    n  , we would expect to need 10 times more terms to get the error to be below 0.0025 so the error in   π is less than 0.01 and that is what I find in Excel. You would need to look more closely to find the exact n requiblack.

I am given the function x(t)=4arctant and told that routine computations will show that x(0)=0 and x(1)=pi.. I must determine a differential equation for x(t) of the form x′(t)=f(t,x). I must then use f to approximate pi.

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