I am given the function $x(t)=4 \arctan t$ 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 $\pi$.
I will use Euler's Method to find minimum number of iterations ($n$) needed to yield the approximation $\pi ≈ 3.1400$, 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.
I think you are expected to divide $[0,1]$ into $n$ intervals, use Euler's method to integrate $\frac 1{1+t^2}$ over the interval and see what you get for $x=1$. For $n=5$, you would compute $x(\frac 15)=x(0)+\frac 15 x'(0)=0+\frac 15 \cdot 1=0.2$, $x(\frac 25)=x(\frac 15)+\frac 15x'(\frac 15)\approx 0.2+\frac 15 \cdot 0.961538\approx0.392308,$ continue stepping until you get to $x(1)$, and compare it with $\frac \pi 4$. The problem I have is that the derivative is decreasing over the interval, so the forward Euler's method overestimates the function, while the problem assumes it underestimates the function. At $n=10$ I get $x(1)\approx 0.809981$, an error of about $0.025.$ Since the global truncation error goes as $\frac 1n$, we would expect to need $10$ times more terms to get the error to be below $0.0025$ so the error in $\pi$ is less than $0.01$ and that is what I find in Excel. You would need to look more closely to find the exact $n$ required.