Question

problem 9
the pitch of a singers recorded voice is represented by the piecewise - function (p(t)) and its graph consists of pieces a, b, and c.
explain how you could change the function to eliminate any breaks or jumps in the recording.
p(t)=\begin{cases}2t + 2&0leq t<2\\-t + 10&2leq t<8\\2&8leq t<12end{cases}

Explanation:

Step1: Analyze function at boundaries

Check the values of \(p(t)\) at \(t = 2\) and \(t=8\). For \(t = 2\), when \(0\leq t<2\), \(p(t)=2t + 2\), so \(p(2^{-})=2\times2 + 2=6\). When \(2\leq t<8\), \(p(t)=-t + 10\), so \(p(2^{+})=-2 + 10 = 8\). For \(t = 8\), \(p(8^{-})=-8 + 10=2\) and when \(8\leq t<12\), \(p(t)=2\), so \(p(8^{+}) = 2\).

Step2: Suggest modification

To estimate breaks or jumps, we can use the concept of one - sided limits more formally. We can define new functions that approach the values from the left and right more precisely. For example, we can create a new piece - wise function with more sub - intervals near the boundaries \(t = 2\) and \(t = 8\). We could introduce a small positive number \(\epsilon\) and define functions on intervals like \((2-\epsilon,2)\), \((2,2 + \epsilon)\), \((8-\epsilon,8)\) and \((8,8+\epsilon)\) and calculate the limits as \(t\) approaches 2 and 8 from the left and right respectively. This would give a better estimate of the jumps in the function.

Answer:

Create a new piece - wise function with sub - intervals near the boundaries \(t = 2\) and \(t = 8\) (using a small \(\epsilon\)) and calculate one - sided limits to estimate breaks or jumps.