# Differentiability with Floor Function

Calculus Level 4

$f(x) = \begin{cases} \dfrac{\sin (\lfloor x^2 \rfloor \pi)}{x^2-3x-18} +ax^3+b & \text{for } 0 \le x \le 1 \\ 2 \cos (\pi x)+ \tan^{-1} x & \text{for } 1 \le x \le 2 \end{cases}$

If $$f(x)$$ is differentiable in $$[0,2]$$, find the value of $$\left| \dfrac{\pi}{4} - b - a \right|$$.

Note:

• $$\lfloor \cdot \rfloor$$ denotes the floor function
• $$|\cdot |$$ denotes the modulus function.