# What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) = 6-t# from #t in [0, 2]#?

The average speed is

The velocity is the integral of the acceleration

So,

We can calculate the average value of the velocity

So,

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The average speed of the object from ( t = 0 ) to ( t = 2 ) can be calculated using the formula for average speed:

[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} ]

First, we need to find the total distance traveled by the object during this time interval. This can be done by integrating the velocity function ( v(t) ) over the given time interval:

[ v(t) = \int_{0}^{t} a(\tau) d\tau = \int_{0}^{t} (6 - \tau) d\tau ]

[ v(t) = 6t - \frac{t^2}{2} ]

To find the total distance, we integrate the absolute value of the velocity function over the time interval:

[ \text{Total Distance} = \int_{0}^{2} |6t - \frac{t^2}{2}| dt ]

[ \text{Total Distance} = \int_{0}^{2} (6t - \frac{t^2}{2}) dt ]

[ \text{Total Distance} = \left[ 3t^2 - \frac{t^3}{6} \right]_{0}^{2} = 6 \text{ units} ]

The total time is simply ( 2 ) units.

[ \text{Average Speed} = \frac{6}{2} = 3 \text{ units per time unit} ]

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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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