How do you find the arc length of the curve #f(x)=2(x-1)^(3/2)# over the interval [1,5]?
Use the arc length formula.
Arc length is given by:
Simplify:
Integrate directly:
Hence:
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To find the arc length of the curve ( f(x) = 2(x-1)^{\frac{3}{2}} ) over the interval ([1, 5]), you can use the arc length formula:
[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} , dx ]
First, find ( \frac{dy}{dx} ) by differentiating ( f(x) ) with respect to ( x ):
[ f'(x) = \frac{d}{dx} [2(x-1)^{\frac{3}{2}}] ]
Then apply the arc length formula with the limits of integration from 1 to 5. Calculate the integral, and that will give you the arc length of the curve over the interval ([1, 5]).
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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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