# An object with a mass of #5 kg# is traveling at #2 m/s#. If the object is accelerated by a force of #f(x) = x^2 -x +3 # over #x in [1, 9]#, where x is in meters, what is the impulse at #x = 8#?

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To find the impulse at x = 8, you need to calculate the change in momentum of the object. Impulse is defined as the change in momentum, which can be calculated using the integral of force with respect to time or by finding the area under the force-time graph.

First, calculate the force at x = 8 using the given force function f(x) = x^2 - x + 3:

f(8) = (8)^2 - 8 + 3 f(8) = 64 - 8 + 3 f(8) = 59 N

Now, to find the impulse, integrate the force function over the given interval [1, 9]:

Impulse = ∫[1, 9] f(x) dx

Impulse = ∫[1, 9] (x^2 - x + 3) dx

Now, integrate the function:

Impulse = [((1/3)x^3 - (1/2)x^2 + 3x)] from 1 to 9

Impulse = ((1/3)(9)^3 - (1/2)(9)^2 + 3(9)) - ((1/3)(1)^3 - (1/2)(1)^2 + 3(1))

Impulse = ((1/3)(729) - (1/2)(81) + 27) - ((1/3)(1) - (1/2)(1) + 3)

Impulse = (243 - 40.5 + 27) - (1/3 - 0.5 + 3)

Impulse = 229.5 - 2.167

Impulse ≈ 227.333 Ns

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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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