# What is the arclength of #(t-1,t/(t+5))# on #t in [-1,1]#?

The arc length is approximately

The arc length of parametric functions is

Use a calculator to evaluate this tricky integral. Thus

Hopefully this helps!

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To find the arc length of the curve ( (t-1, \frac{t}{t+5}) ) on the interval ( [-1, 1] ), you would integrate the square root of the sum of the squares of the first derivative of each component function with respect to ( t ) over the interval.

The first derivative of ( t-1 ) is ( 1 ), and the first derivative of ( \frac{t}{t+5} ) is ( \frac{5}{(t+5)^2} ).

So, the arc length integral becomes:

[ \int_{-1}^{1} \sqrt{1 + \left(\frac{5}{(t+5)^2}\right)^2} , dt ]

Solving this integral will give you the arc length of the curve on the given interval.

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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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