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Chapter 2: Problem 58

Use the alternative form of the derivative to find the derivative at \(x=c\) (if it exists). \(f(x)=|x-4|, \quad c=4\)

Short Answer

Expert verified
The derivative of the function \(f(x) = |x-4|\) at \(x = c = 4\) does not exist as the right-hand limit is not equal to the left-hand limit.

Step by step solution

01

Break down the absolute value function

Firstly, write the absolute value function \(f(x)\) in its piecewise form. Since \(f(x) = |x-4|\), write it as: \(f(x) = \{(x-4), x \geq 4; -(x-4), x < 4\}\).
02

Find the right-hand limit for derivative

The right limit means the derivative of \(f(x)\) when \(x\) approaches \(c = 4\) from the right. Here it means the part of \(f(x)\) that is equal to \((x-4)\) for \(x \geq 4\). Using the alternative form of derivative, the right-hand limit is: \(\lim_{{h \to 0^+}}\frac{f(4+h)-f(4)}{h} = \lim_{{h \to 0^+}}\frac{(4+h-4)-(4-4)}{h} = 1\).
03

Find the left-hand limit for derivative

The left limit implies the derivative of \(f(x)\) when \(x\) approaches \(c = 4\) from the left. Here it represents the part of \(f(x)\) that is equal to \(-(x-4)\) for \(x < 4\). Using the alternative form, the left-hand limit is: \(\lim_{{h \to 0^-}}\frac{f(4+h)-f(4)}{h} = \lim_{{h \to 0^-}}\frac{-(4+h-4)-(4-4)}{h} = -1\).
04

Compare the right and left limit

The derivative of \(f(x)\) at a particular point \(x = c\) exists only when the right-hand limit equals the left-hand limit. As found in the previous steps, the right-hand limit and left-hand limit at \(x = c = 4\) are not the same (1 鈮� -1).

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