What is the arc length of #f(x)= 1/sqrt(x1) # on #x in [2,4] #?
The length is given by
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To find the arc length of ( f(x) = \frac{1}{\sqrt{x1}} ) on the interval ( x \in [2,4] ):

Use the formula for arc length: [ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

Calculate ( \frac{dy}{dx} ) which is the derivative of ( f(x) ).

Substitute ( \frac{dy}{dx} ) into the formula.

Integrate the expression over the interval ( x \in [2,4] ).
[ L = \int_{2}^{4} \sqrt{1 + \left(\frac{d}{dx} \left(\frac{1}{\sqrt{x1}}\right)\right)^2} , dx ]
[ = \int_{2}^{4} \sqrt{1 + \left(\frac{1}{2(x1)^{3/2}}\right)^2} , dx ]
[ = \int_{2}^{4} \sqrt{1 + \frac{1}{4(x1)^3}} , dx ]
This integral needs to be evaluated, which might involve some techniques like trigonometric substitution or other methods depending on the complexity of the function. After integration, you'll get the arc length ( L ) of the function ( f(x) ) over the interval ( x \in [2,4] ).
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When evaluating a onesided 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 onesided 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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