Piecewise Domain Analysis of the Tangent Function
Consider the function \(f(x) = \tan(x)\) defined piecewise such that its domain is restricted to intervals between its vertical asymptotes. Describe how the domain of \(f(x)\) can be written as the union of open intervals, and explicitly determine the interval that contains \(x=0\). Provide justification for the restrictions applied.
Extended Analysis of Piecewise Trigonometric Functions with Domain Constraints
Consider the piecewise function defined on the interval \( [0, \pi] \) as follows: \[f(x) = \begin{cases} \sin(x) & \text{for } 0 \leq x \leq \frac{\pi}{2}\\ 1 – \cos(x) & \text{for } \frac{\pi}{2} < x \leq \pi \end{cases}\] (a) Find the values of \(x\) within each subinterval where \(f'(x) = 0\). (b) Evaluate the function […]
Part b
Using the segments defined in part (a), evaluate the discontinuities observed in the sensor data anomalies. Classify each discontinuity as removable, jump, or infinite, and explain your reasoning based on the behavior of the sensor readings at the segment boundaries.