# What is the arc length of #f(x)=(3x)/sqrt(x-1) # on #x in [2,6] #?

Writing

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To find the arc length of the function (f(x) = \frac{3x}{\sqrt{x - 1}}) on the interval ([2, 6]), we use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

First, we find (\frac{dy}{dx}):

[ \frac{dy}{dx} = \frac{d}{dx} \left(\frac{3x}{\sqrt{x - 1}}\right) = \frac{3\sqrt{x - 1} - \frac{3x}{2\sqrt{x - 1}}}{x - 1} = \frac{3(2 - x)}{2(x - 1)\sqrt{x - 1}} ]

Now, we plug this into the arc length formula and integrate over the interval ([2, 6]):

[ L = \int_{2}^{6} \sqrt{1 + \left(\frac{3(2 - x)}{2(x - 1)\sqrt{x - 1}}\right)^2} , dx ]

This integral is quite complex and doesn't have a simple closed-form solution. Numerical methods, such as numerical integration or software like MATLAB or Python, can be used to approximate the value of this integral and thus the arc length of the curve.

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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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