I have y= |X^2 -4| How can I rewrite this as a piece wise, and find the derivative?
J jetter2 New member Joined Nov 2, 2010 Messages 17 Nov 28, 2010 #1 I have y= |X^2 -4| How can I rewrite this as a piece wise, and find the derivative?
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Nov 28, 2010 #2 Hello, jetter2! \(\displaystyle \text{I have: }\: y \:=\: |x^2 -4|\) \(\displaystyle \text{How can I rewrite this as a piece-wise function, and find the derivative?}\) Click to expand... \(\displaystyle \text{When is }x^2-4\text{ positive?}\) . . \(\displaystyle x^2- 4 \:\> \:0 \quad\Rightarrow\quad x^2 \:>\:4 \quad\Rightarrow\quad |x| \:>\:2\) \(\displaystyle \text{When is }x^2-4\text{ negative?}\) . . \(\displaystyle x^2-4\:<\:0 \quad\Rightarrow\quad x^2 \:<\:4 \quad\Rightarrow\quad |x| \:<\:2\) \(\displaystyle \text{Therefore: }\;f(x) \;=\;\begin{Bmatrix}x^2-4 && \text{if }|x|\,\ge\,2 \\ \\[-3mm] 4-x^2 && \text{if }|x|\,<\,2 \end{array}\) \(\displaystyle \text{Hence: }\;f'(x) \;=\;\begin{Bmatrix}2x && \text{if }|x|\,\ge\,2 \\ \\[-3mm] -2x && \text{if }|x|\,<\,2 \end{array}\)
Hello, jetter2! \(\displaystyle \text{I have: }\: y \:=\: |x^2 -4|\) \(\displaystyle \text{How can I rewrite this as a piece-wise function, and find the derivative?}\) Click to expand... \(\displaystyle \text{When is }x^2-4\text{ positive?}\) . . \(\displaystyle x^2- 4 \:\> \:0 \quad\Rightarrow\quad x^2 \:>\:4 \quad\Rightarrow\quad |x| \:>\:2\) \(\displaystyle \text{When is }x^2-4\text{ negative?}\) . . \(\displaystyle x^2-4\:<\:0 \quad\Rightarrow\quad x^2 \:<\:4 \quad\Rightarrow\quad |x| \:<\:2\) \(\displaystyle \text{Therefore: }\;f(x) \;=\;\begin{Bmatrix}x^2-4 && \text{if }|x|\,\ge\,2 \\ \\[-3mm] 4-x^2 && \text{if }|x|\,<\,2 \end{array}\) \(\displaystyle \text{Hence: }\;f'(x) \;=\;\begin{Bmatrix}2x && \text{if }|x|\,\ge\,2 \\ \\[-3mm] -2x && \text{if }|x|\,<\,2 \end{array}\)
