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Welcome to March Calculus Madness!
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What is the derivative and the indefinite integral of the absolute value?
By defintion:
| x | = x when x ≥ 0, -x when x < 0
Hence:
d/dx | x | = 1 when x ≥ 0, and -1 when x < 0
and
∫ | x | dx = x^/2 + C when x ≥ 0, abnd -x^2/2 + C when x < 0
What about |a∙x + b|?
The function |a∙x + b| hits the x-axis when:
a∙x + b = 0
a∙x = -b
x = -b/a
|a∙x + b| =
(a∙x + b) when x ≥ (-b/a),
and -(a∙x + b) when < (-b/a)
d/dx |a∙x + b| =
a when x ≥ (-b/a),
and -A when < (-b/a)
∫ |a∙x + b| dx =
A ∙ x^2/2 + C when x ≥ (-b/a),
and -A ∙ x^2/2 + C when < (-b/a)
Eddie
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