Differentiable function

Calculus Level pending

Let \[ f(x)=\begin{cases} 4x-3 \text{ for } 0\le x<1 \\ ax^{ 2 }+bx \text{ for } 1\le x\le 2, \end{cases} \] where \( a\) and \( b \) are constants. If \(f(x)\) is differentiable in the open interval \( (0, 2), \) what is the value of \[ \lim _{ n\to \infty }{ \frac { 2 }{ n } \sum _{ k=1 }^{ n }{ f\left( \frac { 2k }{ n } \right) } }? \]

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