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

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To find the arc length of (f(x) = x\cos(x - 2)) on the interval ([1,2]), we use the arc length formula for a function (y = f(x)) over ([a,b]):

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

First, compute the derivative (f'(x)):

[ f'(x) = \frac{d}{dx}[x\cos(x - 2)] = \cos(x - 2) - x\sin(x - 2) ]

Then, square (f'(x)):

[ \left( f'(x) \right)^2 = [\cos(x - 2) - x\sin(x - 2)]^2 = \cos^2(x - 2) - 2x\cos(x - 2)\sin(x - 2) + x^2\sin^2(x - 2) ]

Now, substitute into the arc length formula:

[ L = \int_1^2 \sqrt{1 + \cos^2(x - 2) - 2x\cos(x - 2)\sin(x - 2) + x^2\sin^2(x - 2)} , dx ]

This integral does not simplify to a basic form that can be easily integrated with elementary functions. It typically requires numerical integration techniques for an exact solution.

To proceed with a numerical solution, one could use techniques such as the Simpson's rule, trapezoidal rule, or numerical integration capabilities of software or calculators.

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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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