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How do you find the antiderivative for the absolute value function #f(x) = |x|#?

Answer 1

Christopher Agresta

You can't do it without splitting the absolute value, so:

If #x>=0#, than #|x|=x# and #F(x)=intxdx=x^2/2+c#.

If #x<0#, than #|x|=-x# and #F(x)=int-xdx=-x^2/2+c#.

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Answer 2

Samantha Adkison

To find the antiderivative of the absolute value function ( f(x) = |x| ), you can split the function into two cases:

When ( x \geq 0 ): In this case, ( |x| = x ).
When ( x < 0 ): In this case, ( |x| = -x ).

So the antiderivative of ( |x| ) would be:

[ F(x) = \begin{cases} \frac{x^2}{2} + C & \text{if } x \geq 0 \ -\frac{x^2}{2} + C & \text{if } x < 0 \end{cases} ]

where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.