Provide an Appropriate Response $f ( x ) = x ^ { 2 }$

question 299

Essay

Provide an appropriate response.
-Consider the functions $f ( x ) = x ^ { 2 }$ and $g ( x ) = x ^ { 3 }$ and their linearizations at the origin. Over some interval $- \varepsilon \leq \mathrm { x } \leq \varepsilon$ , the approximation error for $\mathrm { g } ( \mathrm { x } )$ is less than the approximation error for $\mathrm { f } ( \mathrm { x } )$ for all $\mathrm { x }$ within the interval. Derive a reasonable approximation for the value of $\varepsilon$ . Show your work. (Hint, the absolute value of the second derivative of each function gives a measure of how quickly the slopes of the function and its linear approximation are deviating from one another.)

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