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What is the arc length of #f(x)=-xln(1/x)-xlnx# on #x in [3,5]#?

Answer 1

Aurora Alvarado

#2#

Use the rule #log(a^b)=blog(a)# to simplify the function:

#f(x)=-xln(x^-1)-xln(x)#

Bringing the #-1# out:

#f(x)=xln(x)-xln(x)#

#f(x)=0#

This is the straight line #y=0#. Thus its arc length on #x in [3,5]# is just the line segment with length #2#.

Using #f(x)=0# we can apply the arc length formula for #f# on #x in [a,b]# for the same result:

#L=int_a^bsqrt(1+(f'(x))^2)dx#

#L=int_2^4sqrt(1+0)dx#

#L=int_2^4dx#

#L=x]_2^4#

#L=2#

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Answer 2

Axel Alvarez

To find the arc length of ( f(x) = -x \ln\left(\frac{1}{x}\right) - x \ln(x) ) on the interval ([3, 5]), you can use the formula for arc length:

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

First, find the derivative ( f'(x) ). Then plug it into the formula along with the given interval [3, 5] to evaluate the integral. This will give you the arc length ( L ) of the function over the specified interval.

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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.