Let \( f \) be defined as \[ f(x)=\left\{\begin{array}{l}\sqrt[3]{x}+k x, \\ \ln x, \quad x \geq 1\end{array}\right. \] for some constant \( k \). For what value of \( k \) will \( f \) be differentiable over its whole domain? (A) -2 (B) -1 (C) \( \frac{2}{3} \) (D) 1 (E) None of the above

d) \( \lim _{x \rightarrow 1} \frac{\ln x}{x^{2}-1}= \)

Answer any THREE. \( \lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{1}{\ln x}\right) \)

Calculate the volume of the solid formed by rotating the area between \( y = \sqrt{x} \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \) about the line \( y = -1 \) utilizing the washer method.

\( \lim _ { x \rightarrow 9 } ^ { 9 : 03 } ( \frac { \sqrt { x - 9 } } { \sqrt { x } - 3 } ) \)

Compute the average value of \( f(t) = \sin(t) + \cos(t) \) over the interval \([0, \pi]\).