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

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To find the arc length of the curve defined by ( f(t) = (t^3 - 1, t^2 - 1) ) on the interval ( t ) in ([2,3]), we use the formula for arc length:

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

For the given function ( f(t) ), its derivatives are:

[ \frac{dx}{dt} = 3t^2 ] [ \frac{dy}{dt} = 2t ]

Now, substitute these derivatives into the arc length formula and integrate over the interval ([2,3]):

[ L = \int_{2}^{3} \sqrt{(3t^2)^2 + (2t)^2} , dt ]

[ L = \int_{2}^{3} \sqrt{9t^4 + 4t^2} , dt ]

This integral is usually computed numerically since it doesn't have a simple closed-form solution.

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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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- What is the arclength of #f(t) = (sin^2t-cos2t,t/pi)# on #t in [-pi/4,pi/4]#?
- What is the slope of #f(t) = (t^2+2t,2t-3)# at #t =-1#?
- What is the derivative of #f(t) = (e^(t^2-1)+3t, -t^3+t ) #?

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