What is the arclength of #f(x)=arctan(2x)/x# on #x in [2,3]#?
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To find the arc length of ( f(x) = \frac{\arctan(2x)}{x} ) on the interval ([2, 3]), we use the formula for arc length:
[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
First, we find the derivative of ( f(x) ) with respect to ( x ):
[ f'(x) = \frac{d}{dx}\left(\frac{\arctan(2x)}{x}\right) ]
Using the quotient rule and the chain rule:
[ f'(x) = \frac{(1+x^2)(2)}{x^2(1+4x^2)} - \frac{\arctan(2x)}{x^2} ]
Now, we can plug this into the formula for arc length and integrate over the interval ([2, 3]):
[ L = \int_{2}^{3} \sqrt{1 + \left(\frac{(1+x^2)(2)}{x^2(1+4x^2)} - \frac{\arctan(2x)}{x^2}\right)^2} , dx ]
After integrating this expression, you will obtain the arc length of the function ( f(x) ) on the interval ([2, 3]).
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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.
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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