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Chapter 1: Problem 36

Determine the slope function for the following functions. $$f(x)=|x|$$

Short Answer

Expert verified
Answer: The slope function of $$f(x) = |x|$$ is a piecewise function given by: $$ f'(x) = \left\{ \begin{array}{ll} 1 & \quad x > 0 \\ -1 & \quad x < 0 \\ \text{Undefined} & \quad x = 0 \end{array} \right. $$

Step by step solution

01

Differentiate for positive and negative x

We will find the derivative for x > 0 and x < 0, treating the function as x and -x in each case. For x > 0: $$f(x) = x$$ $$f'(x) = 1$$ For x < 0: $$f(x) = -x$$ $$f'(x) = -1$$
02

Define the slope function for x=0

The function |x| is not differentiable at x=0, as it doesn't have a well-defined tangent line at this point. However, we can use a piecewise function to represent the derivative for all x values.
03

Create the piecewise function for the slope function

By combining our results in Step 1 and Step 2, we can construct a piecewise function for the slope function (f'(x)): $$ f'(x) = \left\{ \begin{array}{ll} 1 & \quad x > 0 \\ -1 & \quad x < 0 \\ \text{Undefined} & \quad x = 0 \end{array} \right. $$ This piecewise function represents the slope function of $$f(x) = |x|$$.

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