(Solved): part c Q2 [30 points] Derivatives of piecewise functions, Derivative from a limit, Derivatives of tr ...

Q2 [30 points] Derivatives of piecewise functions, Derivative from a limit, Derivatives of trig and exponential functions, Elasticity in demand, Rules of differentiation. Given the piecewise function \[ f(x)=\left\{\begin{array}{ll} \frac{x^{2}+e^{-x}+\cos x-2}{x}, & -1 \leq x<0 \\ \left(x^{2}-\frac{23}{21}\right)\left(x^{30}+2\right), & 0 \leq x<1 \\ \frac{x^{2}+k}{2 x^{2}-1}, & 1 \leq x<2 \\ 2400-200 x, & 2 \leq x \leq 12 \end{array}\right. \] (a) [10 pts] Find the left-sided limit of \( f(x) \) at \( x=0 \) through the derivative of \( g(x) \) with justification. You should guess a function \( g(x) \) in the limit of difference quotient with \( g(0) \neq 0 \). (b) [10 pts] Knowing that \( f(x) \) is continuous at \( x=1 \), determine the value \( k \) such that \( f(x) \) is differentiable at \( x=1 \) with justification. (c) [10 pts] View \( 2400-200 x, 2 \leq x \leq 12 \) as a demand function \( D=2400-200 p, 2 \leq p \leq 12 \). Find the elasticity \( E(p) \) at the price \( p=4 \), and interpret the result. Show all steps or lose points.