# What is the arc length of #f(x) =x -tanx # on #x in [pi/12,(pi)/8] #?

Arc length is given by:

Simplify:

Integrate directly:

Reverse the last substitution:

Insert the limits of integration:

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To find the arc length of the curve ( f(x) = x - \tan(x) ) on the interval ([ \frac{\pi}{12}, \frac{\pi}{8} ]), we can use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx ]

where ( f'(x) ) is the derivative of ( f(x) ). Let's first find ( f'(x) ) which is ( 1 - \sec^2(x) ). Now, we can substitute these into the formula and integrate over the given interval. After evaluating the integral, we get the arc length.

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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.

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.

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.

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