k(x) = |x - 2|
Compute the derivative (that is, the derivative function) by evaluating the limit
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Note that the definition of |x| means that
k(x) =
x-2 for x-2 ≥ 0
-(x-2) for x-2 < 0
To compute the derivative of the function k(x) = |x - 2|, we need to evaluate the limit as h approaches 0 of the difference quotient:
lim(h->0) [k(x + h) - k(x)] / h
Let's start by finding the difference quotient.
k(x + h) = |(x + h) - 2| = |x + h - 2|
k(x) = |x - 2|
Substituting these expressions into the difference quotient, we get:
lim(h->0) [|x + h - 2| - |x - 2|] / h
Now, we need to simplify this expression. The absolute value function can be defined as:
|a| = a, if a >= 0
-a, if a < 0
Using this definition, we can simplify the expression further:
lim(h->0) [(x + h - 2) - (x - 2)] / h
The x terms cancel out:
lim(h->0) [h] / h
Simplifying, we get:
lim(h->0) 1
Therefore, the derivative of k(x) = |x - 2| is:
k'(x) = 1
The derivative is a constant value of 1, meaning the slope of the graph of k(x) is always 1.