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

Answer 1

#1.01941#

Differentiaitng #f(x)=arctan(2x)/x# with respect to #x# we get
#f'(x)=2/(x(1+4x^2))-arctan(2x)/x^2# so we have to solve the integral #int_2^3sqrt(1+(2/(x*(1+4x^2))-arctan(2x)/x^2)^2)dxapprox 1.01941#
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Answer 2

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

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

First, we find the derivative of ( f(x) ) with respect to ( x ):

[ f'(x) = \frac{d}{dx}\left(\frac{\arctan(2x)}{x}\right) ]

Using the quotient rule and the chain rule:

[ f'(x) = \frac{(1+x^2)(2)}{x^2(1+4x^2)} - \frac{\arctan(2x)}{x^2} ]

Now, we can plug this into the formula for arc length and integrate over the interval ([2, 3]):

[ L = \int_{2}^{3} \sqrt{1 + \left(\frac{(1+x^2)(2)}{x^2(1+4x^2)} - \frac{\arctan(2x)}{x^2}\right)^2} , dx ]

After integrating this expression, you will obtain the arc length of the function ( f(x) ) on the interval ([2, 3]).

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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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