Derivatives At 0

Calculus Level 3

Let \(f(x)\) be continuous and differentiable everywhere, and assume that \(f(x)\) is analytic. Furthermore assume that for all non-negative integers \(n\), we have \(f^{(n)}(0)=n\), where \(f^{(n)}(x)\) denotes the \(n\)th derivative of \(f\) at \(x\). Find \(f(1)\).

Give your answer to 3 decimal places.

Bonus: As a fun extension, calculate \(f(1)\) if we say that for non-negative \(n\), we have \(f^{(n)}(0)=n^k\) for some defined positive integer \(k\).

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