# What is the arc length of #f(x)=((4x^5)/5) + (1/(48x^3)) - 1 # on #x in [1,2]#?

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To find the arc length of the function ( f(x) = \frac{4x^5}{5} + \frac{1}{48x^3} - 1 ) on the interval ([1,2]), you can use the formula for arc length:

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

where ( f'(x) ) is the derivative of the function. First, find ( f'(x) ):

[ f'(x) = \frac{4}{5}x^4 - \frac{1}{16x^4} ]

Then, plug ( f'(x) ) into the formula for arc length and integrate over the interval ([1,2]):

[ L = \int_{1}^{2} \sqrt{1 + \left(\frac{4}{5}x^4 - \frac{1}{16x^4}\right)^2} , dx ]

This integral represents the arc length of the function ( f(x) ) on the interval ([1,2]). You can solve it numerically using calculus techniques or numerical methods to find the exact value.

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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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