Chapter 2: Q. 67 (page 198)

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = |1 - 2 x|$

Short Answer

Expert verified

The derivative of the function is $f' (x) = \{\begin{cases} 2, i f x > \frac{1}{2} \\ D N E, i f x = \frac{1}{2} \\ - 2 . i f x < \frac{1}{2} \end{cases}$ .

Step by step solution

Step 1. Given Information

The given function is $f (x) = |1 - 2 x|$ .

Step 2. Rewrite the function

The given function can be rewritten as, $f (x) = \{\begin{cases} - (1 - 2 x), i f x \geq \frac{1}{2} \\ 1 - 2 x, i f x > \frac{1}{2} \end{cases}$ .

Step 3. Find the derivative

It is known that, the derivative of the linear function is $(m x + b)' = m$ .
Find the derivative for $x > \frac{1}{2}$ .

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (- (1 - 2 x)) \\ = & \frac{d}{d x} (2 x - 1) \\ = & 2 \end{array}$

Find the derivative for $x > \frac{1}{2}$ .

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} (1 - 2 x) \\ = & - 2 \end{array}$

Since $2 \neq - 2$ , the derivatives at left and right pieces are not equal at $x = \frac{1}{2}$ . The derivative does not exists at $x = \frac{1}{2}$ .
So, the derivative of the function is, $f' (x) = \{\begin{cases} 2, i f x > \frac{1}{2} \\ D N E, i f x = \frac{1}{2} \\ - 2 . i f x < \frac{1}{2} \end{cases}$ .

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Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = |1 - 2 x|$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Rewrite the function

Step 3. Find the derivative

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Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72. f(x)=1-2x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Rewrite the function

Step 3. Find the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.
$f (x) = |1 - 2 x|$