What is the arclength of #f(x)=(1+x^2)/(x-1)# on #x in [2,3]#?

Answer 1

#approx 1.07667#

We get by the Quotient rule

#f'(x)=(2x(x-1)-(1+x^2))/(x-1)^2# #f'(x)=(x^2-2x-1)/(x-1)^2# so our integral is given by
#int_2^3sqrt(1+((x^2-2x-1)/(x-1)^2)^2)dx#
A numerical result is given by #approx 1.07667#
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Answer 2

To find the arc length of the function ( f(x) = \frac{1+x^2}{x-1} ) on the interval ( [2,3] ), follow these steps:

  1. Compute the derivative of ( f(x) ), denoted ( f'(x) ).
  2. Use the formula for arc length: [ L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx ], where ( a = 2 ) and ( b = 3 ).
  3. Integrate ( \sqrt{1 + [f'(x)]^2} ) over the interval ( [2,3] ).
  4. The result of the integration is the arc length of ( f(x) ) on the interval ( [2,3] ).
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Answer 3

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

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

where (a = 2), (b = 3), and (y = f(x)). Then, we'll find the derivative of (f(x)) with respect to (x), which will give us ( \frac{{dy}}{{dx}} ), and then integrate.

[ f(x) = \frac{{1+x^2}}{{x-1}} ]

[ \frac{{df}}{{dx}} = \frac{{d}}{{dx}}\left(\frac{{1+x^2}}{{x-1}}\right) ]

[ = \frac{{(2x)(x-1) - (1+x^2)(1)}}{{(x-1)^2}} ]

[ = \frac{{x^2 - 2x - 1 - x^2}}{{(x-1)^2}} ]

[ = \frac{{-2x - 1}}{{(x-1)^2}} ]

Now, we'll plug this into the formula for arc length and integrate:

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

This integral can be solved numerically using calculus techniques or numerical approximation methods.

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Answer from HIX Tutor

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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