# What is the arclength of #f(t) = (t^3-t^2+5t,9t)# on #t in [1,4]#?

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To find the arc length of ( f(t) = (t^3 - t^2 + 5t, 9t) ) on ( t ) in the interval ([1,4]), we first need to compute the derivative of ( f(t) ), then integrate the square root of the sum of the squares of the derivatives over the given interval. The arc length formula is given by:

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

Substituting the derivatives of ( f(t) ) into the formula and evaluating the integral over the interval ([1,4]) will give us the arc length.

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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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