# What is the average distance between a point in the interval [-4,3] and the origin?

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To find the average distance between a point in the interval [-4, 3] and the origin, we need to calculate the average value of the distance function over that interval. The distance between any point (x, 0) and the origin (0, 0) in the coordinate plane is given by the absolute value of x. Therefore, the distance function in this case is |x|.

To find the average distance, we integrate |x| over the interval [-4, 3] and then divide by the length of the interval, which is 7 (3 - (-4)).

[ \text{Average distance} = \frac{1}{7} \int_{-4}^{3} |x| , dx ]

[ = \frac{1}{7} \left[ \int_{-4}^{0} -x , dx + \int_{0}^{3} x , dx \right] ]

[ = \frac{1}{7} \left[ \left[ -\frac{x^2}{2} \right]*{-4}^{0} + \left[ \frac{x^2}{2} \right]*{0}^{3} \right] ]

[ = \frac{1}{7} \left[ \left( 0 - \left( -\frac{(-4)^2}{2} \right) \right) + \left( \frac{3^2}{2} - 0 \right) \right] ]

[ = \frac{1}{7} \left[ \left( 0 - 8 \right) + \left( \frac{9}{2} \right) \right] ]

[ = \frac{1}{7} \left( -8 + \frac{9}{2} \right) ]

[ = \frac{1}{7} \left( -8 + \frac{9}{2} \right) ]

[ = \frac{1}{7} \left( -8 + \frac{9}{2} \right) ]

[ = \frac{1}{7} \left( -\frac{7}{2} \right) ]

[ = -\frac{1}{2} ]

So, the average distance between a point in the interval [-4, 3] and the origin is ( \frac{1}{2} ).

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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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