2. (a) Let \[ f(x)=\left\{\begin{array}{ll} a \sin x+b & \text { if } x \leq 0, \\ \frac{\tan ^{-1}(x)}{x} & \text { if } x>0 . \end{array}\right. \] (I) If \( f \) is continuous at \( x=0 \), find the value of \( b \). (II) Furthermore, if \( f^{\prime}(0) \) exists, find \( a \). [5 marks] [5 marks]
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We are given a piecewise function \( f(x) \) that is defined differently for \( x \leq 0 \) and \( x > 0 \). We need to find the values of \( a \) and \( b \) such that \( f \) is continuous at \( x = 0 \) and \( f^{\prime}(0) \) exists. Show more…
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