2. Assuming that \( f(x) \) is a function \( f(0) = 1 \) and that \( f'(x) = f(x) \), we want to estimate \( f(1) \) as we did for the previous exercise (with a larger number of steps).
(a) Use the interactive SageMath app below to approximate \( f(1) \) for the number of steps \( n \) given in your lab report.
(b) What if we want a larger number of steps? When \( n = 1 \), we get \( f(1) \approx 1 + 1 = 2 \). When \( n = 2 \), we found that \( f(1) \approx \frac{9}{4} \). When \( n = 3 \), we get \( f(1) \approx \frac{81}{27} \). In general, for any number of steps \( n \), the formula for \( f(1) \) is \[ f(1) \approx \left( \frac{n+1}{n} \right)^n \] The actual answer for \( f(1) \) will then be \[ \lim_{n \rightarrow \infty} \left( \frac{n+1}{n} \right)^n \] We don't have the tools for computing this limit yet, but we can use the computer to estimate the limit for large values of \( n \). Use the interactive SageMath app below to approximate \( f(1) \) for the number of steps \( n \) given in your lab report.